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My former IMO coach, Gregory Galperin, converted a famous joke into a logic puzzle, adding a variation.
Puzzle. Ten logicians walked into a cafe. Each knew whether they wanted tea or coffee, but no one knew each other's preferences. When they sat at a table, the waiter asked loudly, "Will everyone be having coffee?" Then the waiter went around the table, writing down each person's answer.
There were three possible answers: "I don't know", "Yes", and "No". All answers were truthful and spoken loudly so that all group members heard them.
- Suppose the first nine people said, "I don't know", and the tenth person said, "Yes". How many of them wanted coffee?
- Suppose the sixth and the seventh answers were not the same. How many people said, "I don't know", how many said, "No", and how many said, "Yes". Find the smallest number of people who for sure would have ordered coffee and the smallest number who for sure would have wanted tea?
Here are some problems that I liked from the YuMSh (Youths Math School in St. Petersburg) Olympiad.
Problem for 6th grade. Twenty people from an island of knights and knaves have a party. Knights always tell the truth, and knaves always lie. Each party-goer got a card with a different number from 1 to 20. When they were asked about their numbers, each answered with a number from 1 to 20. The sum of all the answers is 156. What is the minimum possible number of liars that have to be at the party?
Problem for 7th grade. Alice and Bob bought a deck of playing cards (52 cards total) and took turns gluing the cards on the wall one at a time. Alice was first. The game is lost if, after a move, the wall has 4 cards of the same suit or 4 cards of consecutive values (for example, 8-9-10-jack). Can Alice or Bob guarantee themselves a win, regardless of their opponent's moves?
Problem for 7th grade. Buddhist monks gather in an infinite cave where a finite number of prime numbers are written on the wall. The numbers might not be distinct. Every second, one of the monks performs one of the following operations.
Problem for 8th grade. The incenter of a triangle is equidistant from the midpoints of the sides of the triangle. Prove that the triangle is equilateral.
Problem for 9th grade. Bob was given 30 distinct natural numbers. He wrote down all the 435 pairwise sums. It appears that among those sums, 230 are divisible by 3. How many of the original 30 numbers are divisible by 3?
I recently posted a cute puzzle about poisoned wine. Today, I would like to discuss this puzzle's variation with N total glasses, of which two are poisoned.
Puzzle. N glasses of wine are placed in a circle on a round table. S sages are invited to take the following challenge. In the presence of the first sage, N − 2 glasses are filled with good wine and the other two with poisoned wine, indistinguishable from the good wine. After drinking the poisoned wine, the person will die a terrible tormented death. Each sage has to drink one full glass. The first sage is not allowed to give any hints to the other sages, but they can see which glass he chooses before making their own selection. The sages can agree on their strategy beforehand. For which S can you find a strategy to keep them all alive?
What does this have to do with rulers, and what are those? I am grateful to Konstantin Knop for showing me a solution with rulers. But first, let's define them.
A sparse ruler is a ruler in which some distance marks may be missing. For example, suppose we have a ruler of length 6, with only one mark at a distance 1 from the left. We can still measure distances 1, 5, and 6. Such a ruler is often described as {0,1,6} to emphasize its marks, endpoints, and size.
A complete sparse ruler is a sparse ruler that allows one to measure any integer distance up to its full length. For example, the ruler {0,1,6} is not complete. It can't measure distances 2, 3, and 4. Thus, if we add the marks at 2, 3, and 4, such a ruler becomes complete.
A complete sparse ruler is called minimal if it uses as few marks as possible. In our previous example, the ruler {0,1,2,3,4,6} is not minimal. The distance between marks 1 and 4 is 3, so if we remove mark 3, we can still measure any distance. We can remove mark 2 too. The ruler {0,1,4,6} with marks 1 and 4 is minimal.
Oops. I forgot that we have a round table. This means we need to look at cyclic rulers: the idea is the same, but the numbers wrap around. For example, consider the cyclic ruler {0,1,4,6} of length 6, where 0 and 6 is the same point. This ruler has three marks at 0, 1, and 4.
Going back to the puzzle, suppose N = 6, aka there are six glasses around the table. The sages need to agree on a complete cyclic ruler, for example, the one described above. As this ruler contains any possible difference between the marks, the first sage can mentally place the ruler on the table so that the marks cover poisoned glasses. He signals the position of the ruler by drinking his glass. The sages can agree that the glass drunk by the first sage corresponds to position −1 on the ruler, and the other sages avoid the first, second, and fifth glass clockwise from the chosen glass.
In this case, three glasses are not covered by the ruler's marks. This means three sages can be saved.
To summarize, the sages need a complete ruler, as such a ruler can always cover two glasses at any distance from each other with its marks. The number of sages that can be saved by such a ruler is N minus the number of marks. To save more sages, we want to find a minimal ruler.
There are actually more cool rulers. A ruler is called maximal if it is the longest complete ruler with a given number of marks. For example, the non-cyclic ruler {0,1,4,6} is maximal. A ruler is optimal if it is both maximal and minimal. Thus, the ruler {0,1,4,6} is also optimal.
There are other types of rulers called Golomb rulers. They require measured distances to be distinct rather than covering all possibilities. Formally, a Golomb ruler is a ruler with a set of marks at integer positions such that no two pairs of marks are the same distance apart. If, however, a Golomb ruler can measure all the distances, it is called a perfect Golomb ruler. As we can deduce, a perfect Golomb ruler is a complete sparse ruler. I leave it to the reader to show that a perfect Golomb ruler must be a minimal complete sparse ruler.
The rulers rule!
I wrote a lot about the inventiveness of my students. Here is more proof.
Puzzle. A police officer saw a truck driver going the wrong way down a one-way street but didn't try to stop him. Why?
Many of my students came up with the expected answer:
The truck driver was walking.
They also found some legit ways for a truck driver to not be stopped.
Some more ideas, rather far-fetched.
Some funny ones.
I often receive letters from parents of math geniuses — "My twelve-year-old is reading an algebraic geometry book: accept him to PRIMES," or "My ten-year-old finished her calculus course: here is her picture to post on your blog," or "My two-year-old knows the multiplication table, can you write a research paper with him?" The last letter was a sarcastic extrapolation.
I am happy to hear that there are a lot of math geniuses out there. They are potentially our future PRIMES and PRIMES STEP students. But, it is difficult to impress me. The fact that children know things early doesn't tell me much. I've seen a student who didn't know arithmetic and managed to pass calculus. I've also met a student claiming the full knowledge of fusion categories, which later appeared to be from half-watching a five-minute YouTube video.
There are a lot of products catering to parents who want to bring up geniuses. My grandson received a calculus book for his first birthday: Introductory Calculus For Infants. Ten years later, he still is not ready for calculus.
Back to gifted children. Once a mom brought us her kid, who I can't forget. The child bragged that he solved 30 thousand math problems. What do you think my first thought was? Actually, I had two first thoughts: 1) Why on Earth would anyone count all the problems they solved? 2) And, what is the difficulty of the problems he solved 30 thousand of?
From time to time, I receive an email from a parent whose child is a true math genius. My answer to this parent is the same as to any other parent: "Let your child apply to our programs. We do a great job at working with math geniuses."
Our programs' admissions are done by entrance tests. Surprisingly, or not surprisingly, the heavily advertised kids often do poorly on these tests. It could be that the parents overestimate their children's abilities. But sometimes, the situation is more interesting and sad: I have seen children who sabotage the entrance tests so as not to be accepted into our programs. We also had students give us hints on their application forms that they were forced to apply.
In the first version of this essay, I wrote funny stories of what these students did. Then, I erased the stories. I do not want the parents to know how their children are trying to free themselves.
Dear parents, do not push your children into our programs. If they do not want to be mathematicians, you are decreasing their chances of getting into a good college. Imagine an admission officer who reads an essay from a student who wants to be a doctor but wastes ten hours a week on a prestigious math research program. Such a student doesn't qualify as a potential math genius, as their passion lies elsewhere. Nor does this student qualify as a future doctor, as they didn't do anything to pursue their claimed passion. In the end, the student is written off as a person with weak character.
On the other hand, the students who do want to be in our programs, thrive. They often start breathing mathematics and are extremely successful. Encourage your children to apply to our programs if they have BOTH: the gift for mathematics and the heart for it.
This puzzle was in last week's homework.
Puzzle. How can an egg fly three meters and not break?
The expected answer:
Some students tried to protect the egg:
Other students specified qualities of an egg making it more resistant:
Here are some more elaborate explanations:
To conclude this essay, here is a punny answer:
I am so tired of spam emails. I keep thinking about how we can fight spam, and here is an idea.
Gmail should change its system: every email you send to me would cost 1 dollar, payable to me. We can add an exception for people on my contact list. Everyone else, pay up!
I do not often contact strangers. But if I do, it is always important. So paying 1 dollar seems more than fair. On the other hand, this system will immediately discourage mass emails to strangers. Spam would go down, and I would stop receiving emails inviting me to buy a pill to increase the size of a body part I do not have.
This idea of getting paid for reading an email is not new. It was implemented by Jim Sanborn, the creator of the famous Kryptos sculpture. Kryptos is located at the CIA headquarters and has four encrypted messages. People tried to decrypt them and would send Jim their wrong solutions. Jim got tired of all the emails and administered Kryptos fees. Anyone who wants Jim to check their solution, can do so by paying him 50 dollars. I wonder if Jim would still charge the fee if someone sent him the correct solution.
Thinking about it, I would like the payable email system to be customizable, so I can charge whatever I want. After all, I do value my time.
Gmail could get a small percentage. Either Gmail, together with me, gets rich, or spam goes away. Both outcomes would make my life easier.
Puzzle. A goat was on a 10-meter leash. Yet it managed to go 300 meters away from the post. How come?
The standard answer. The leash wasn't attached to the post.
My students scrutinized the puzzle and found some other possible ambiguities. For example, there might be two posts: the goat was leashed to one and was far away from the other. In another example, the timing is not given. It is possible that the goat was on the leash at one time and unleashed and far away from the post at another time. Here is my favorite answer.
My favorite answer. The goat ate the leash.
Puzzle. Find the largest and the smallest 4-digit numbers n such that when you erase the first two digits of n, you get the sum of the digits of n.
I recently posted a gnome puzzle by Alexander Gribalko.
Puzzle. Nine gnomes repeat the following procedure three times. They arrange themselves on a 3 by 3 chessboard with one gnome per cell and greet all of their orthogonal neighbors with handshakes. Prove that not all pairs of gnomes greet each other.
As often happens with my blog puzzles, I used this puzzle as homework for my students. They calculated that the total number of handshakes needed for all nine gnomes to greet each other is 36. On the other hand, each arrangement of gnomes creates 12 handshakes. This means that the numbers are tight: no greeting can be wasted, and every pair of gnomes need to greet each other exactly once. The students then studied different cases to prove this was impossible.
In each arrangement, a gnome can have either 2, 3, or 4 handshakes. Hence, we can distribute handshakes over three placements as 2+2+4 or 2+3+3. It follows that if a gnome is ever in the center of the grid, he has to be in a corner for the other two arrangements. Therefore, the three gnomes who end up in the center for one of the arrangements never greet each other.
However, I always love solutions that involve coloring the board in a checkerboard manner. Here is my solution.
Solution. Let's color the board in a checkerboard manner. We assign each gnome a binary string, of length 3, describing the colors of the cells where the gnome was in each placement. There are 8 different possible strings. It follows that at least two gnomes are assigned the same string. But they can't greet each other if they are standing on the same colors in each arrangement!
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I hate getting into debates about Möbius strips. They're always one-sided.
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North Korea's ballistic missile test failed due to a bug in Windows. The next missile containing a bug report has been automatically sent to Microsoft.
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4 out of 3 people have trouble with fractions.
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Do you know what seems odd to me?
Numbers that aren't divisible by two.
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Why was algebra so easy for the Romans?
X was always 10.
My readers know that I love hat puzzles. This is why I decided to turn a number trick by Konstantin Knop (in Russian) into a hat trick.
Hat Trick. The audience has a bottomless supply of hats in ten different colors. They arrange ten people in a line and put one of the hats on each person. Then the magician's assistant comes in and removes a hat from one of the ten people. After that, the magician appears and, abracadabra, guesses the color of the removed hat. The magician and the assistant agreed on a strategy beforehand. What is it?
Keep in mind that this trick won't work with fewer than ten colors. As a bonus, can you explain why?
As my readers know, I am devoted to my students. When I need something I can't buy, I try to make it. That is why I crocheted a lot of mathematical objects. One day, I resolved to have in my possession a Seifert surface bounded by Borromean rings (a two-sided surface that has Borromean rings as its border).
However, my crocheting skills were not advanced enough, so I signed up for a wet and needle felting workshop. When I showed up, Linda, our teacher, revealed her lesson plan: a felted soap with a nice pink heart on top. It looked cool to have soap inside a sponge, not to mention that wool is anti-bacterial. But I had bigger plans than soap and eagerly waited for no one else to show up.
When my dream materialized, and, as I had hoped, no one else was interested in felt, I asked Linda if we could drop the hearty soap and make my dream thingy. She agreed, but my plan didn't survive for long. As soon as Linda saw a picture of what I wanted, she got scared. Seifert surfaces were not in the cards, so soap it was. I told her that there was no way I was going to needle-felt a pink heart onto my felted soap. I ended up with a blue Sierpiński gasket.
We had a great time. Linda was teaching me felting, and I was teaching her math. I am a good teacher, so even felters working on a farm enjoy my lessons.
After the workshop, I went online and found my dream surface on Shapeways. In the end, I was happy to just buy it and not have to make it.
But my felting workshop wasn't a waste of time: tomorrow I will wash myself with a gasket.
A combinatorial games section? At a math competition? I have never heard of it before. But here it is, at the Lomonosov Tournament. The first problem is from the 2012 Tournament (the link is to a Russian version). The problem is by Alexander Shapovalov. I picked it for its elegance and simplicity.
A rectangular band. You have a paper rectangle of size m by n with grid lines, where both m and n are greater than 1. Two players play the following game. The first player cuts the rectangle along any grid line into two rectangles. The next player picks one of the rectangles and cuts it again along a grid line, and so on. The winner is the player who, after their move, can arrange all the rectangles into a band of width 1. Which player is guaranteed a win regardless of the moves of their partner? Consider two cases.
- One of the numbers, m or n, is even;
- Both m and n are odd.
The next problem, by I. Raskin, is from the 2015 Tournament (in Russian). The problem is about fruits, and I know a lot of great puzzles with bananas and oranges, so I immediately got attracted to this one.
The original version had three types of fruit starting with the letters a, b, and c in Russian. They were oranges, bananas, and plums. I reused bananas and replaced oranges with apples, but I got stuck on the letter c. The players in this game eat their fruits, so using cantaloupes seemed like overkill. Plus, I am on a diet, so I decided on cherries.
Fruits. There are a apples, b bananas, and c cherries on the table. Two players play a game where one move consists of eating two different fruits. The person who can't move loses. Assuming the players use their best strategies, would the first or the second player win in the following cases?
- a = 1 (the answer might depend on the values of b and c);
- a = 6, b = 8, c = 10;
- a = 7, b = 9, c = 15;
- a = 19, b = 20, c = 21.
The last problem is from the 2012 Tournament (in Russian). It has great sentimental value for me, as it was created by John Conway. It also reminds me of the famous Frobenius (chicken McNugget) problem.
Coin mintage. Once upon a time, in a faraway kingdom, two treasurers were minting coins. They decided to make it into a game, taking turns minting coins. Each turn, the player chooses a particular integral denomination and mints an infinite supply of coins of this denomination. The rules of the game forbid choosing a new denomination that can be paid with the existing coins. The treasurer who is forced to mint a coin of denomination 1 loses.
- Prove that if the first treasurer starts with denomination 2 or 3, s/he loses.
- Is it profitable for the first treasurer to start with denomination 4?
- Is it profitable for the first treasurer to start with denomination 6?
- Suppose the first treasurer minted coins of denomination 5, and the second treasurer of denomination 6. What is the winning strategy for the first treasurer after that?
- Suppose the first treasurer minted coins of denomination 5, and the second treasurer of denomination k. Prove that the largest denomination available for minting is 4k − 5.
- Prove that the first treasurer can win by starting with denomination 5. (Hint: Suppose the second treasurer replied with denomination k, and the first treasurer minted 4k − 5 after that. If this strategy wins, the problem is solved. However, if after that, the second treasurer wins by minting denomination m, then minting denomination 4k − 5 was the wrong move. What was the right move?)
Here is another exciting puzzle posted on Facebook by Konstantin Knop.
Puzzle. Eight glasses of wine are placed in a circle on a round table. Three sages are invited to take the following challenge. In the presence of the first sage, five glasses are filled with good wine and the other three with poisoned wine, indistinguishable from the good wine. After drinking the poisoned wine, the person will die a terrible tormented death. Each sage has to drink one full glass. The first sage is not allowed to give any hints to the other sages, but they can see which glass he chooses before making their own selection. The three sages can agree on their strategy beforehand. What is the strategy to keep them all alive?
An extra question. Does a strategy exist if fewer than eight glasses are placed around the table?
Let's start with two trivial variations. If there is only one sage, s/he knows which glass to drink. Now, suppose there is only one poisoned glass and any number of sages. If the total number of good glasses is not less than the number of sages, the solution is obvious. The first sage drinks the glass clockwise from the poisoned one, and the other sages continue clockwise.
The next slightly less trivial case involves two sages and two poisoned glasses. If the total number of glasses is at least five, the sages are safe. The reason: there are at least two good glasses in a row. So the sages can agree that the first one drinks a good glass followed clockwise by another good glass. If the total number of glasses is less than 5, there is no reliable strategy, as the reader can check.
The above idea works if the number of sages is S, the number of poisoned glasses is P, and the total number of glasses is T, where T is greater than SP. Then, the strategy is the same since there is a guaranteed continuous stretch of S good glasses.
On the other hand, one can argue that for S and P more than 1, and T = S + P, it is impossible to find a strategy.
Our original problem corresponds to the case S = P = 3, and T = 8. Presumably, the strategy doesn't exist if S = P = 3, and T < 8. If the 8-glasses problem is difficult, here is a much easier version, corresponding to the case S = 2, P = 3, and T = 6.
An Easier Puzzle. Six glasses of wine are placed in a circle on a round table. Two sages are invited to take the following challenge. In the presence of the first sage, three glasses are filled with good wine and the other three with poisoned wine, indistinguishable from the good wine. After drinking the poisoned wine, the person will die a terrible tormented death. Each sage has to drink one full glass. The first sage is not allowed to give any hints to the other sages, but they can see which glass he chooses before making their own selection. The three sages can agree on their strategy beforehand. What is the strategy to keep them all alive?
I am trying to make a point that, mathematically, the Sleeping Beauty problem is resolved, and the Wikipedia article about it should stop assuming that there is an ongoing debate.
The Sleeping Beauty problem. Sleeping Beauty participates in the following experiment. On Sunday, she is put to sleep, and a fair coin is flipped. Regardless of the result of the coin flip, she is awakened on Monday and asked whether she thinks the coin was heads or tails. Regardless of her answer, if the coin was heads, the experiment ends. However, if the coin was tails, she is put back to sleep with her memory erased and awakened again on Tuesday and asked the same question. In this case, the experiment stops on Tuesday. She knows the protocol. She is awakened one morning. From her point of view, what is the probability that the coin was heads?
Here is the solution. If it is Monday, then the probability that the coin is heads is one half. So the probability of Monday/heads is the same as Monday/tails. If the coin is tails, Sleeping Beauty can't distinguish between Monday and Tuesday. So the probability of Monday/tails is the same as Tuesday/tails. Thus, the three cases Monday/heads, Monday/tails, and Tuesday/tails are equally probable. It follows that when she is awakened, the probability of heads is one third.
However, there are still people — called halfers — who think that the probability of heads is one half.
But suppose we ask a different question.
Different question. She is awakened one morning. From her point of view, what is the probability that the day is Tuesday?
As I explained before, when she is awakened, the probability of it being Tuesday is one third. Let us calculate what the halfers think. Suppose they think that the probability of the day being Tuesday is x, then the probability of Monday is 1 − x. Let's calculate the probability of the coin being heads from here. The probability of heads, if today is Tuesday, is zero. The probability of heads, if today is Monday, is 1/2. Therefore, the probability of heads equals 0 · x + (1 − x)/2 = (1 − x)/2. In halfers' view, the resulting calculation equals 1/2. In other words, (1 − x)/2 = 1/2. It follows that x is zero. Doesn't make much sense that the probability of the day being Tuesday is zero, does it?
Halfers are wrong. Wikipedia should update the article.
The Sleeping Beauty problem. Sleeping Beauty participates in the following experiment. On Sunday, she is put to sleep, and a fair coin is flipped. Regardless of the result of the coin flip, she is awakened on Monday and asked whether she thinks the coin was heads or tails. Regardless of her answer, if the coin was heads, the experiment ends. However, if the coin was tails, she is put back to sleep with her memory erased and awakened again on Tuesday and asked the same question. In this case, the experiment stops on Tuesday. She knows the protocol. She is awakened one morning. From her point of view, what is the probability that the coin was heads?
This problem is considered controversial. Some people, called halfers, think that when she is awakened, the probability that the coin was heads is one half. Other people, called thirders, think that when she is awakened, the probability that the coin was heads is one third.
I am a thirder, so let me explain my thinking. If it is Monday, then the probability that the coin is heads is one half. So the probability of Monday/heads is the same as Monday/tails. If the coin is tails, Sleeping Beauty can't distinguish between Monday and Tuesday. So the probability of Monday/tails is the same as Tuesday/tails. Thus, the three cases Monday/heads, Monday/tails, and Tuesday/tails are equally probable, and the answer follows.
This problem is similar to the Monty Hall problem in some ways. The main difference is that The Monty Hall problem stopped being controversial, while Sleeping Beauty problem continues to be. The best argument that helped intuition and led to the resolution of the Monty Hall problem was increasing the number of doors. Similarly, for the Sleeping Beauty problem, we can increase the number of days she is put back to sleep when the coin is tails. Forgive me for the cruelty of this theoretical experiment.
The Sleeping Beauty Variation. Sleeping Beauty participates in the following experiment. On Sunday, she is put to sleep, and a fair coin is flipped. Regardless of the result of the coin flip, she is awakened on Monday and asked whether she thinks the coin was heads or tails. Regardless of her answer, if the coin was heads, the experiment ends. However, if the coin was tails, the experiment continues for 99 more days. Each day, she is put back to sleep with her memory erased and awakened the next day and asked the same question. She knows the protocol. She is awakened one morning. From her point of view, what is the probability that the coin was heads?
In this variation, when she is awakened, she should be almost sure that the coin was tails. I hope it will help halfers feel that, in this case, the probability of heads can't be one half. For me, the argument is the same as before, making the probability that the coin was heads is 1 over 101.
After I wrote this essay, I discovered that Nick Bostrom made the same argument. Though, for tails, he put Sleeping Beauty to sleep one million and one times, which is about 2,740 years. He increased my one hundred days by several orders of magnitude, amplifying our point. Surely, when Sleeping Beauty awakens, she should be almost certain that the coin was tails. After Sleeping Beauty agrees to so much torture, why is the problem still controversial?
We already blogged about the Sleeping beauty problem three times: The Sleeping Beauty Problem, Sleeping Beauty Meets Monty Hall, and Sleeping Beauty and Mondays. The posts were more than ten years ago, but the mathematical community seems to continue being stuck on this problem. So we come back to it.
Problem. Sleeping Beauty participates in the following experiment. On Sunday, she is put to sleep, and a fair coin is flipped. Regardless of the result of the coin flip, she is awakened on Monday and asked whether she thinks the coin was heads or not. If the coin was tails, however, she is put back to sleep with her memory erased, awakened on Tuesday and asked the same question again. She knows the protocol. She is awakened one morning: What is her probability that the coin was heads?
Let's raise the stakes a little bit. Suppose a prize of $1K is set aside for her correct guess of a flip. What should be her strategy?
The problem is ambiguous. We didn't tell you how the prizes are awarded. We see several reasonable scenarios:
Because of selective amnesia, she can't distinguish between her awakenings, thus her strategy has to be the same at every awakening. That is, she should say heads with probability p. The exact number depends on the scenario:
We see that though she is asked to guess the outcome of the coin flip and is rewarded for guessing correctly, her strategy is not to try to maximize the probability of guessing correctly each time. The strategy depends on the reward protocol.
We see that in the first and the sixth scenario, she gets one guess per coin flip. Not surprisingly, it doesn't matter what she chooses to do. The classic Sleeping Beauty problem corresponds to the third scenario. If she wants to improve her chances per guess, she should favor tails.
Let's look at the following variation.
Problem. Suppose Sleeping Beauty's goal is to guess right once, but if she guesses right on Monday, she wins and is not put back to sleep. If she guesses wrong on Monday and the coin was tails, she is put back to sleep with he memory erased. That is, she is given a second chance. What's the right strategy in this case?
As her memory is erased, the only strategy is to do the same thing each time she is awakened. That is to guess heads with probability p. Thus, she guesses correctly at least once with probability p/2 + (1–p)/2 + p(1–p)/2. We see that whatever strategy she chooses, she guesses with probability one half on Monday. She gets an additional chance of p(1–p)/2 to get a correct guess on Tuesday. Her worst strategy is to always guess tails or heads. Her best strategy is to flip a fair coin each time. This way, she has an additional probability of 1/8 to win on Tuesday.
Here is a question to our readers, for the best strategy above, what is her probability on waking that the coin is actually heads? Here is the problem more formally.
Problem. Sleeping Beauty participates in the following experiment. On Sunday, she is put to sleep, and a fair coin is flipped. Regardless of the result of the coin flip, she is awakened on Monday and asked whether she thought the coin was heads or not. Sleeping Beauty answers by flipping a fair coin. If she guesses correctly, or if the Sunday coin was heads, the experiment ends. If the Sunday coin was tails and different from her guess, she is put back to sleep with her memory erased, awakened on Tuesday, and asked the same question again. She knows the protocol. She is awakened one morning: What is her probability that the Sunday coin was heads?
I love Toblerone, but I never paid attention to its logo.
This Swiss chocolate was created in Bern (Berne) more than a hundred years ago. A legend says that Berne's founder loved hunting and vowed to name the city after the first animal he met on his hunt. The animal happened to be a bear, and so the town was named Berne. The bear became a big deal for the town; you even can find the bear on the town's flag and coat of arms.
Theodor Tobler, the creator of Toblerone, put the famous local mountain, Matterhorn, in the logo of Toblerone and hid a bear in the image. He obviously loved his country and town.
Tobler was clearly a wordsmith and invented a fascinating name for his chocolate. As Wikipedia explains, the name Toblerone is a portmanteau combining Tobler's name with the Italian word torrone, which is a type of nougat used in his chocolate. However, Wikipedia doesn't explain that the word Toblerone also contains a secret: the word BERNE (bear).
After writing this, I couldn't resist and bought some dark Toblerone chocolate. Now it brings me pleasure in more ways than one.
There are many cute math problems that use the trivial fact announced in the title. For example, I recently posted the following problem from the 43rd Tournament of Towns.
Problem. Find the largest number n such that for any prime number p greater than 2 and less than n, the difference n − p is also a prime number.
Solution. Prime numbers 3, 5, and 7 have different remainders modulo 3. Thus, for any n, one of the numbers n − 3, n − 5, or n − 7 is divisible by 3. If n > 10, that number that is divisible by 3 is also greater than 3, thus, making it composite. Therefore, the answer to this problem is not greater than 10. Number 10 works, thus, the answer is 10.
Here is another problem using the fact that 3, 5, and 7 have different remainders when divided by 3.
Problem. Find the maximum integer n, such that for any prime p, where p < n, the number n + 2p is prime.
Here is the current picture of my coauthor, Joel Lewis. I remember him from many years ago when he was a graduate student at MIT. I am glad he kept his big smile.
Back to the subject matter. Joel Lewis made a comment on my recent post, thinking-outside-the-box ideas. He mentioned his two theorems:
The Big Point Theorem. Any three lines intersect at a point, provided that the point is big enough.
The Thick Line Theorem. Any three points lie on the same line, provided that the line is thick enough.
I grew up a purist. Mathematics was about mathematics, not about gender. I personally would never have competed in a math competition exclusively for girls. I would have felt diminished somehow. Furthermore, suppose there were separate math competitions for boys, where girls were not allowed. I would have felt outraged. By symmetry, I should have been outraged by math competitions exclusively for girls.
When I first heard about math competitions for girls, I was uncomfortable. I also noticed that my Eastern European friends shared my feelings, while my other friends did not.
There was a stage in my life when I lived in Princeton for seven years and became friends with Ingrid Daubechies, who is NOT Eastern European. She didn't share my extreme views, and we argued a lot. Every spring, there was a big conference for Women-in-Mathematics at the Institute for Advanced Study in Princeton. But I was stubborn and completely ignored it. With one exception: when Ingrid gave a lecture there, I went just to hear her talk.
In 2008, I was living in Boston, worrying about money, as I had just resigned from my industry work to return to mathematics. Out of the blue, Ingrid called to offer me a job at that very same Women-in-Mathematics conference. I admire Ingrid and got excited about working with her. More importantly, I needed the money. So I decided to put aside my prejudice and take the job.
Being part of the conference was a revelation. Although most of the two-week conference was spent on mathematics, there were some women-and-math seminars. There, women discussed bullying by advisers and colleagues, impostor syndrome, workplace bias, the two-body problem, and other issues. And guess what? I went through all of that too. I just never realized it and never talked about it.
My biggest regret was that I hadn't attended the conference earlier. Plus, the conference didn't feel unfair towards men: all lectures and math seminars were open to the public, and some courageous guys sat in.
What about my symmetry test? What happens if we swap genders? Should there be separate math conferences for men, open to the public? The idea makes me laugh. Women are in the minority in math, and many conferences already feel like conferences for men open to the public. My symmetry test doesn't quite work.
But, while I saw how helpful the support was for female mathematicians and for me, something still bugs me. Where is the fine line between minority support and unfairness? Let's look at math clubs for girls. On the surface, there are so many math clubs, so why not have the occasional math club for girls? I can imagine a girl, not me, who would prefer to attend such a club. However, girls' clubs often have sponsors and are much cheaper to attend than regular clubs. This is unfair to boys whose parents can't afford a regular club. It also becomes counterproductive for girls. What if some girls go to such a club, not because they need a special environment but because it is a cheaper club? They are missing out on the fun of learning math together with boys. (Trust me, I know!) So, I am not sure where that line is.
I am older and wiser now. I do not run away from events organized for women in math. I think lunches and dinners for women in math are great. They help female mathematicians find mentors, build networks, and stay in mathematics.
What about my original question: Should there be separate math competitions for girls? In a truly equal society, math competitions should be for everyone. And, though I do not actively oppose girls' competitions anymore, I hope the need for them will die out in my lifetime.
Recently, I stumbled upon three lovely logic puzzles, while scrolling Facebook for news from the Russia-Ukraine war. I will start with an easy puzzle.
Puzzle. A traveler got to an island, where each resident either always tells the truth or always lies. A hundred islanders stood in a circle facing the center, and each told the traveler whether their neighbor to the right was a truth-teller. Based on these statements, the traveler was able to clearly determine how many times he had been lied to. Can you do the same?
In the next puzzle, there is another island where people are either truth-tellers or normal. There are three questions that increase in difficulty.
Puzzle. You are on an island with 65 inhabitants. You know that 63 inhabitants are truth-tellers who always tell the truth, and the other 2 are normal, who can either lie or tell the truth. You are allowed one type of question, "Are all the people on this list truth-tellers?" This question requires a list, which you can create yourself. Moreover, you can have as many different lists as you want. You can pose this question to any islander any number of times. Your goal is to find the two normal people. The easy task is to do it in 30 questions. The medium task is to do it in 14 questions. The hard task is to figure out whether it is possible to do it in fewer than 14 questions.
Here is the last puzzle.
Puzzle. You got to an island with 999 inhabitants. The island's governor tells you, "Each of us is either a truth-teller who always tells the truth, or a liar who always lies. " You go around the island asking each person the same question, "How many liars are on the island?" You get the following replies.
First person, the governor: "There is at least one liar on the island."
The second person: "There are at least two liars on the island."
This continues progressively until the 999th person says: "There are at least 999 liars on the island."
What can you tell about the number of liars and truth-tellers on this island?
I recently wrote an essay, Thinking Inside and Outside the Box, which starts with a famous nine-dots puzzle that kicked off the expression: thinking outside the box. Here is another puzzle with the same nine-dots setup.
Puzzle. What is the smallest number of squares needed to ensure that each dot is in its own region?
Usually, people who try to solve this puzzle come up with the following four-squares solution.
As with the classic nine-dots puzzle, they imagine that the dots are on a grid and try to build squares with sides parallel to the grid lines. What would be the outside-the-box idea? The sides of the squares would not need to be parallel to the grid. This way, we can solve the puzzle with three squares.
One of my MathRoots students offered a different and awesome solution also using three squares.
Problem. Find the largest number n such that for any prime number p greater than 2 and less than n, the difference n − p is also a prime number.
The most famous thinking-outside-the-box puzzle is the Nine-Dots puzzle. This puzzle probably started the expression, "To think outside the box". Here is the puzzle.
Puzzle. Without lifting the pencil off the paper, connect the nine dots by drawing four straight continuous lines that pass through all the dots.
Most people attempt something similar to the picture below and fail to connect all the dots.
They try to connect the dots with line segments that fit inside the square box around the dots, mentally restricting themselves to solutions that are literally inside the box.
To get to the correct solution, the line segments should be drawn outside this imaginary box.
Do you think that four line segments is the best you can do? Jason Rosenhouse showed me a solution for this puzzle that requires only three lines. Here, the outside-the-box idea is to use the thickness of the dots.
This and many other examples of thinking outside the box are included in my paper aptly titled Thinking Inside and Outside the Box and published in the G4G12 Exchange book.
A section of this paper is devoted to my students, who sometimes give unexpected and inventive solutions to famous puzzles. Here is an example of such a puzzle and such solutions that aren't in the paper because I collected them after the paper was published.
Puzzle. Three men were in a boat. It capsized, but only two got their hair wet. Why?
The standard answer is the following: One man was bald.
Lucky for me, my creative students suggested tons of other solutions. For example,
My favorite example, however, is the following.
I still get comments that my place is in the kitchen, and women shouldn't do math. Luckily, occurrences of such events are getting rarer. So the world is progressing in the right direction. But gender bias still exists. Multiple studies show that among people of the same level of intelligence, men are perceived as smarter. In layman's terms, people think women are stupider than they are. I will give my own examples at another time. But now, I want to discuss my model for gender bias.
Let's denote the mathematical strength of a researcher by S. In real life, if the researcher is female, people perceive her strength as smaller than S. Let us assume that there exists a bias coefficient B, a number between 0 and 1 so that a female researcher of strength S is perceived on average as having strength BS. When B is 1, then there is no bias. But the world is not there yet.
In recent years, there has been a push to hire female mathematicians. Suppose a math department has a cutoff C for hiring a math professor. Being eager to hire a female professor, the department slightly reduces the cutoff by the bias reduction coefficient R, where R is a number between 0 and 1. Thus, the cutoff to hire a female professor becomes RC, which is less than C.
Now consider Alice, who has mathematical strength S, such that S ≥ C ≥ BS ≥ RC. She is perfectly qualified to be hired by the department independent of her gender. However, the department members, on average, perceive her strength as not good enough for a male hire but good enough for a female one. Alice is hired. What happens as a result?
Many male colleagues are angry. They think that because of Alice, they couldn't hire a male candidate they believe to be stronger. They also expect Alice to be grateful for the opportunity. How will Alice react? It depends on whether Alice herself is biased. I will give two potential scenarios.
Scenario 1. Alice is biased and realizes she is only hired because she is female. Alice thinks that her strength is below the cutoff C. Alice develops impostor syndrome and doesn't contribute to the department and mathematics as much as she could have, reinforcing everyone's belief that she is weaker than she actually is.
Scenario 2. Alice is not biased but realizes that she was hired for her gender and not for her mathematics. She gets angry too because she knows she is perfectly qualified. She also realizes that the department expects her to be grateful, while another male hire is thanked for taking a similar position. In this scenario, everyone is angry.
In both scenarios, the intent is to fight the bias. But as a result, no one is truly happy. A supportive environment is not created. And the idea that male and female mathematicians differ in strength is reinforced.
Recently, I talked to a male colleague about gender bias. He was adamant that he isn't biased and, as proof, described his contribution to hiring females in his department. This is not proof of being unbiased. This is proof of trying to resolve the bias, either sincerely or due to outside pressure. However, the real proof would be to change one's attitude towards female colleagues. As a female colleague myself, I would notice this change in attitude. But while nothing changes, I can't accept the fact of female hires as proof of no bias.
To truly resolve the bias, we need B to be equal to 1. To improve the situation, people should collectively work on increasing the bias coefficient B. This is more about changing the attitude.
I am for hiring more female mathematicians as professors. But sometimes, I feel like hiring more female mathematicians without addressing the attitude is treating the symptoms while making the disease worse.
I love knights-and-knaves puzzles: where knights always tell the truth, and knaves always lie. The following puzzle has a new type of person: a sophist. A sophist only makes statements that, standing in their place, neither a truth-teller nor a liar could make. For example, standing next to a liar, a sophist might say, "We are both liars." Think about it. If the sophist was a truth-teller, then the statement would have been a lie, thus creating a contradiction. If the sophist was a liar, the statement would be true, again creating a contradiction.
Here is the puzzle with sophists. And by the way, this one is intended for sixth graders.
Puzzle. You are on an island inhabited by knights, knaves, and sophists. Once upon a time, a sophist made the following statements about the island's inhabitants:
1. There are exactly 25 liars on this island.
2. There are exactly 26 truth-tellers on this island.
3. The number of sophists on this island is no less than the number of truth-tellers.
How many inhabitants were on the island once upon that time?
I love this new sophist character in logic puzzles, but I have no clue why they are called sophists. Can anyone explain this to me?
As my readers know, I collect Russian license plates. They are actually American plates, but the letters form words readable in Russian. This is possible because the shapes of English and Russian letters overlap. Here is my new favorite plate. It is actually not the best plate, but rather makes the best picture ever.
The plate says Moscow, the Russian capital. And the car is parked next to the Ukrainian flag. I am from Moscow too, and I too support Ukraine in its war against evil Putin, who wants to restore the Russian Empire. Did you know that now he wants Alaska?
After reading my post, Shapovalov's Gnomes, Grant, one of my readers, directed me to the Brick puzzle from Section 2.3 of Xinfeng Zhou's A Practical Guide to Quantitative Finance Interviews.
Puzzle. Can you pack 53 bricks with dimensions 1-by-1-by-4 into a 6-by-6-by-6 box?
The solution in Zhou's book has some flaws. So I am posting my own solution here.
Solution. We start with a sanity check. The box contains 216 unit cube cells, and 53 bricks would take up 212 cells. So there is no contradiction with the volume. We need to look at something else.
Let's divide the box into 27 smaller 2-by-2-by-2 boxes and color the smaller boxes in a checkerboard manner. We get 13 boxes of one color, say white, and 14 boxes of another color, say black. Whichever way we place a brick inside the original box, it has to cover 2 white cells and 2 black cells. But we have a total of 104 white cells, which is only enough for 52 bricks.
Alexander Gribalko designed a new coin problem, two versions of which appeared at the 43rd Tournament of the Towns. I will start with the easier version.
Problem. Alice and Bob found eleven identical-looking coins; four of them are fake. All real coins weigh the same. All fake coins weigh the same but are lighter than the real ones. Alice offered Bob four specific coins. Before agreeing to take them, Bob wants to ensure they are all real. Can he do so by only using the balance scale twice? He can use any of the eleven coins for weighing purposes.
Surprisingly, the more difficult version has fewer coins.
Problem. Alice and Bob found eight identical-looking coins; three of them are fake. All real coins weigh the same. All fake coins weigh the same but are lighter than the real ones. Alice offered Bob three specific coins. Before agreeing to take them, Bob wants to ensure they are all real. Can he do so by only using the balance scale twice? He can use any of the eight coins for weighing purposes.
The Conway subprime Fibonacci sequence is defined as follows. Start with the Fibonacci sequence 0, 1, 1, 2, 3, 5, …, but before you write down a composite term, divide it by its least prime factor. Thus, the next term after 5 is not 8, but rather 8/2 = 4. After that, the sum gives us 5 + 4 = 9, which is also composite. Thus, you write 9/3 = 3, then 4 + 3 = 7, which is okay since it is prime, then 3 + 7 = 10, but you write 10/2 = 5, and so on. Here is the sequence: 0, 1, 1, 2, 3, 5, 4, 3, 7, 5, 6, 11, 17, 14, 31, and so on.
My coauthors, Richard Guy and Julian Salazar, and I studied this sequence in our paper Conway's subprime Fibonacci sequences, in which we allowed any two integers to be the two starting terms. Our computation showed that for small starting terms, the sequence starts cycling. In our first draft, we had a probabilistic argument as to why the sequence should always cycle. Our argument was the following.
Flawed probabilistic argument. Consider the parity of the numbers in a particular subprime Fibonacci sequence. I will leave it to the reader to see that such a sequence can't have all even numbers. As soon we get to an odd number, the even numbers in the sequence become isolated. Indeed, if two numbers have different parity, the next number must be odd. For subprime Fibonacci sequences, when we add two odd numbers, there is a 50 percent chance that after dividing by 2, the result will be odd. Assuming the result is odd, there a is 25 percent chance the next number will be odd as well. And this pattern continues. Thus, as soon as we get two odd numbers in a row, on average, we expect three odd numbers in a run of odd numbers. Hence, we would expect a typical subprime Fibonacci sequence to have three times as many odd as even numbers.
This means, on average, two consecutive terms sum to an even number half of the time. Therefore, while calculating the next term, we divide by 2 with probability 1/2 and by 3 with probability 1/6. Ignoring larger primes, on average, we expect to have divided by at least 1.698, while the usual Fibonacci growth is only by a factor φ, which is approximately 1.618. We see that the subprime Fibonacci sequence is divided faster than its expected growth rate. Thus, it has to cycle.
However, this argument doesn't work. When we submitted the paper, an anonymous reviewer pointed out that the argument was flawed. Here I want to explain why. I will start with a counterexample suggested by my sons, Alexey and Sergei.
Counterexample. Start with two real numbers. Suppose the sequence is to add the two previous numbers and divide the sum by 1.8 (which is greater than the golden ratio φ). The resulting sequence grows as a geometric progression with ratio 1.07321.
Here is a variation of the above argument.
Counterexample Variation. Suppose we have a sequence where we add the two previous numbers and then divide the sum by 2. We are dividing by a noticeably larger number than the golden ratio. By our flawed argument earlier, the sequence should decrease. But if we start such a sequence with two identical numbers n and n, the sequence will be constant, disproving the argument.
Here is an additional observation. Obviously, whenever we add two numbers and divide the sum by a number bigger than 2, the sequence has to cycle. Indeed, the maximum of two consecutive terms decreases. Can we add probabilities to this argument? Below is the averaging version of this argument for the subprime Fibonacci numbers. Unfortunately, this argument is also flawed.
Another flawed probabilistic argument. We need to show that, on average, at each step, we divide our sum by a number bigger than 2. We can ignore divisions by 2 as they are fine. Let's see what happens with sums that we do not divide by 2. With probability 1/3, we divide them by 3. If not, with probability 1/5, we divide them by 5. Otherwise, with probability 1/7, we divide them by 7. Hence, if we do not divide our sum by 2, then with probability 1/3, we divide it by 3, plus with probability 2/15, we divide it by 5, plus with probability 8/105, we divide it by 7. Thus, on average, if we do not divide the sum by 2, then we divide it by at least 3^{1/3}5^{2/15}7^{8/105} = 2.07.
Counterexample to the second argument. Why is this wrong? Let us, for example, consider a sequence with the following recursions: a_{2k+1} = a_{2k} + a_{2k-1}. And a_{2k} = (a_{2k-1} + a_{2k-2})/x, where x is a very large number. Then on average, we divide the sum by a very large number. Would this sequence converge to zero? If we look at it more closely, the subsequence of odd-indexed numbers increases. So the answer is no.
We found yet another probabilistic argument that the sequences shouldn't increase indefinitely. We are sure that one works. Our anonymous reviewer was happy with it too. You can find the argument in our paper: Conway's subprime Fibonacci sequences.
But I wrote this essay to show how tricky probabilistic arguments can be.
For students applying to PRIMES, we have a question about their research interests. RSI asks a similar question from their applicants.
I am looking at all the submissions, and this essay will help our applicants to get projects that are well-suited for them.
We, at PRIMES and RSI Math, usually have research projects lined up in advance. That means, we are not creating projects to match applicants' requests. We match existing projects to students' backgrounds and interests.
If you are applying to one of these programs, here is my advice.
Don't be too specific about what you want. Suppose you want to study the symmetries of an icosahedron. This request is too narrow: there is a high probability we do not have such a project. How will we match you to a project? Our hope, in this case, is to find clues in your essay. For example, we might discover that you heard a fascinating lecture on icosahedron's symmetries, which is why you requested the topic. In this case, we assume that another fascinating lecture on a different topic might also excite you, and you will be matched with a random project. But if your description is broader, say, if you write that you like group theory or geometry, your match won't be as random.
Specify things you do not want. Given our project distribution, you might not get a project in the area that is your first or even your second choice. On the other hand, if you write to us that you hate geometry, it is very easy to find a project without a geometric component. If there is something you definitely do not want, it is advantageous for you to mention it.
Be precise about your advanced knowledge. For example, linear algebra is one of the most powerful tools in mathematics. Not surprisingly, we often have projects that require a serious background in linear algebra and specifically look for students who know it. Unfortunately, we often receive inadequate descriptions of students' backgrounds. Even if you took a linear algebra course, it might be useful to mention which book you used and how many chapters you covered. This also applies to other advanced topics. An applicant might say they are proficient in cohomologies after half-listening to one half-hour lecture on the topic. This is not proficiency; it only indicates interest. If you claim advanced knowledge, specify the scope. The best way to start is by listing the books you have attempted to read. For each book, describe which chapters you only scanned, which chapters you read and understood, and for which chapters you solved all of the exercises.Add more information if your first choice is number theory. Almost every year, we have several students requesting number theory. This might be explained by the successes of the Ross and PROMYS summer programs. The graduates from these programs love number theory and have a good number theory background. However, modern number theory is very advanced, and we seldom have these types of projects. So, if number theory is your top choice, there are two things you can do. First, mention your second choice. Second, specify what you like about number theory. For example, if you are into the more abstract parts of number theory, another abstract project might be a good fit.
Describe your priorities in broader terms. It is beneficial for every starting mathematician to figure out the area they like by asking themselves broader questions. If you know the answers to the questions below, it is helpful to write them on the application form.
If you follow my advice, you might get a project that matches your tastes better. Not only that: figuring out the answers to these questions will help you build the life you love.
I recently posted a cute Shapovalov's puzzle about gnomes. Here is another easier gnome puzzle, also by Alexander Shapovalov.
Puzzle. All gnomes are knights or knaves: knights always tell the truth, and knaves always lie. There is a gnome on every cell of a 4 by 4 chessboard. It is known that both knights and knaves are present in this group. Every gnome states, "Among my neighbors, the number of knaves is the same as the number of knights". How many knaves are there, if by neighbors they mean orthogonally adjacent gnomes?
The next gnome puzzle has a different author, Alexander Gribalko. Gnomes in this puzzle are not knights or knaves but rather friendly and polite beings.
Puzzle. Nine gnomes repeated the following procedure three times. They arranged themselves on a 3 by 3 chessboard with one gnome per cell and greeted all their orthogonal neighbors. Prove that not all pairs of gnomes greeted each other.
Here is another lovely problem from a prolific problem writer, Alexander Shapovalov.
Problem. Every cell of a 7 by 7 chessboard has a gnome standing on it. For every pair of gnomes whose cells share an edge, their beards' lengths differ by no more than 1 inch. Now, we take these gnomes and sit them around a table. Prove that we can do so in a way that any two gnomes sitting next to each other have their beards' lengths differ by no more than 1 inch.
A standard chessboard is 8 by 8. Why would this problem have a 7 by 7 board? Let's see.
For even-sized boards, the problem is easy. I will explain why, but first, let me construct a graph related to a board, in this case, any board.
Each cell is a vertex, and two vertices are connected by an edge if the corresponding cells are orthogonal neighbors (share an edge) on the board. A cycle that goes through a graph and visits every vertex exactly once is called a Hamiltonian cycle. A Hamiltonian cycle is a potential way to sit the gnomes around the table and solve the problem. When we sit the gnomes in a circle following a Hamiltonian cycle, two neighbors at the table are also neighbors on the board, and so they have their beards' lengths differ by no more than 1 inch.
The problem is easy for even-sized boards because it is easy to draw a Hamiltonian cycle on them. An odd-sized chessboard can't have a Hamiltonian cycle. To prove this, let me color the board in a checkerboard manner. Then, cells that share an edge are different colors. And you can't make a cycle through the board, where you switch colors at each step, but the total number of steps is odd.
It follows that for odd-sized boards, it is impossible to solve the problem by just connecting neighboring cells on the board. There should be another way. Can you find it?
I want to discuss a problem from a test I gave recently.
Problem. My dog, Fudge, likes books. He brought two books to his corner in the morning and three more books in the evening. How many books will he read tonight?
As I expected, many students didn't pay attention and just summed up the two numbers in the problem and gave five as the answer. Here are three answers that I especially liked.
Answer 1. Zero, because most dogs can't read.
This cautious student added most to be on the safe side.
Answer 2. You cannot tell how many books he will read. Just because he brings books to his corner doesn't mean he will read them.
The second answer demonstrates great logical thinking, were Fudge a human. But the third answer made me laugh.
Answer 3. Three. If he brought three more books to his corner in the evening, it means he finished the two from that morning, so there are three books left for him to read.
Thinking about genders, we used to have only two options: male and female. Now we have more. I have a lot of young acquaintances who are non-binary. So I started to rethink my gender identity.
I was a typical girl: at least, I thought I was. I hated playing with boring dolls. I preferred cars, or even better, construction sets and board games. In second grade, I wanted to be a ballerina but later fell in love with Sherlock Holmes. My dreams switched to becoming a detective or a spy. In fifth grade, I signed up for rifle-shooting training. That same year, my school forced me to compete in an orienteering event, and I won.
Orienteering became my favorite sport, and I did it for many years. I was really good with maps. I would go to a competition, leisurely walk my course and win. Other kids were running around like crazy, but I always knew where to go and was overall faster than them. With time, other kids learned to read maps better, but I myself never learned to run faster. So I stopped winning, but I enjoyed the sport anyway.
It goes without saying: I loved math. Solving math problems was the best entertainment ever.
Later, to my surprise, I discovered that most girls liked shopping and wasted a lot of time on make-up. Not many girls were even interested in math. I actually liked that. I started having crushes on boys since second grade and enjoyed being the only girl in math clubs, having all these nerds to myself.
I grew up in Soviet Russia. While growing up, I wasn't bombarded with gender stereotypes. My first eye-opening experience was when I was 17 years old. My long-time boyfriend knew that I liked mathematics, and this was okay. But when I told him that I planned to go to college to study mathematics, he didn't approve. I broke up with him on the spot.
My mom used to tell me that most men do not like brainy women. My reply was that there are more men who like brainy women than brainy women. I got a new boyfriend the day after my breakup.
My gender identity didn't bother me much in Russia. What bothered me was the Russian tradition for both spouses to work, but the house chores and child-rearing duties fell only on women. I read somewhere that, on average, in Russia, women worked for 4 hours a day more than men. Life was unfair to women but not to their self-esteems.
As I said, I grew up thinking I was a typical girl until I came to the US. This happened 30 years ago, and I was 30 at the time. In the US, I got bombarded with gender stereotypes: they made me feel inadequate and doubt my femininity. Just for reference: by that time, I was in my third marriage breastfeeding my second child. Still, according to those stereotypes, I was not a real woman.
For some time, I wondered what was wrong with me. Then, I was elated to find a book called Brain Sex: The Real Difference Between Men and Women by Anne Moir and David Jessel. (This was many years ago.) Among other things, this book talks about the differences between men and women with respect to the brain. According to the book, men are better on average at abstract thinking and spatial visions, aka math and maps. Women, on the other hand, are better at connecting with people and have higher social intelligence. Boys are less interested in games related to storytelling, aka dolls, preferring more concrete activities. And so on.
The book also describes situations in which girls don't fit the paradigm. The authors attribute this variation to the mother's hormones during pregnancy. I found myself described perfectly in the section titled "girls who have been exposed to male hormones in the womb." I am pretty sure that my mom didn't take any hormone supplements while pregnant with me back in Soviet Russia. On the other hand, the description in the book was spot-on.
This book classifies me as "a male brain in a female body."
I was glad to find myself after a period of self-doubt. I was glad that I wasn't alone and even fit into a special category with a name.
Several years later, I met Sue Katz, a writer who also has a blog Sue Katz: Consenting Adult. She made me realize how ridiculous the whole story was: I was pressured by gender stereotypes to feel bad about myself. Then I was grateful for a book based on those same stereotypes, only because it described women like me and gave me permission to exist. I liked the book because I accepted those stereotypes in the first place. If there were no stereotypes, there wouldn't be any problems at all.
Why can't I just be me?
Over the past few years, I have become happier than I have ever been. I do not care what society thinks about my gender. I am no longer ashamed of not feeling 100 percent female.
I like that people in the modern world embrace the idea of individuals being themselves. For example, my daughter-in-law, Robin Dahan, designed a whole line, You-Be-You, for her company, Dash of Pep.
Am I non-binary? I do not know and do not care. I am just me, proudly wearing my You-Be-You socks.
Assume that we have n men and n women, all of whom want to get married. They rank each other without ties. After that, we can match them into n pairs for marriage. The matching is called stable if there are no rogue couples. A rogue couple is defined as a man and a woman who prefer each other over their assigned future spouses.
A famous theorem says that, whatever all the rankings are, a stable matching always exists. But how good could a stable matching be? There is a way to assign a quality score to a matching, called the egalitarian cost. The egalitarian cost of any matching is the sum of the rankings that each person gave their assigned partner. The best potential outcome is when all people are matched with their first choices. This corresponds to the minimum egalitarian cost of 2n. But what is the maximum egalitarian cost of a stable matching? I couldn't find it in the literature, so I proved that it is n(n+1).
Proof. It is easy to see that the egalitarian cost of n(n+1) is achievable. For example, if all men gave an identical ranking to the women, and vice versa, the matching algorithm will end up with couples having mutual rankings (j,j) for different values of j. Another example is a Latin preference profile. Each woman ranks a man n+1−x whenever he ranks her x. In this case, every potential couple's mutual rankings sum to n+1. Thus, any matching for such rankings ends up with the egalitarian cost of n(n+1).
The next step is to prove that the egalitarian cost can't be greater. Suppose the cost of a stable matching is C and is greater than n(n+1). Then, for every person, we count the number of people who are better (ranked smaller) than their assigned partner and sum these numbers over all the people. The result must be C−2n, which is the number of pairs of people of opposite genders such that the first person prefers the second one over their assigned partner. Moreover, the result is greater than n(n−1).
The total number of possible couples is n^{2}. Thus, the number of unrealized potential couples is n(n−1). We can conclude that we counted one of these unrealized couples twice. In such a couple, two people prefer each other over their assigned partners. Thus, they form a rogue couple, contradicting our assumption that the matching is stable.
Here is an old goodie.
Puzzle. A criminal is sentenced to death. He is offered the last word. He is allowed to make one statement. If the statement is true, the criminal will be electrocuted. If the statement is false, he will be hanged. Can you find a good piece of advice for this man?
The standard answer to this puzzle is for the criminal to say, "I will be hanged." This creates a paradox. If he is hanged, the statement is true, and he should be electrocuted. If he is electrocuted, the statement is false, and he should be hanged.
Thinking about it, any paradox works. If the man says, "This statement is false," then the type of punishment he gets can't be determined.
Thinking about it some more, asking a question instead of making a statement will confuse his executioners all the same.
One of my students advised the criminal to stay silent. This was not my favorite solution, as staying silent requires some concentration. My favorite solution was the statement, "It will rain exactly a thousand years from now." This way, the criminal can relax, at least for a thousand years.
Today I discuss ideas for solving the puzzle I posted previously in my blog: A Scooter Riddle.
Riddle. Alice, Bob, and Charlie are at Alice's house. They are going to Bob's house, which is 33 miles away. They have a 2-seat scooter that goes 25 miles per hour with 1 rider, or 20 miles per hour with 2 riders. Each of the 3 friends has a walking pace of 5 miles per hour. What is the fastest time that all three of them can end up at Bob's house?
Let's start.
Now this is just algebra. Each person covers 20x miles on the scooter and 33 − 20x miles on foot. The walking takes 33/5 − 4x hours. Thus the total trip for each person takes 33/5 − 3x hours.
Now we calculate what the scooter does. The scooter covers 20x miles with Alice and the same number of miles with Bob. It covers 40x − 33 miles alone. Thus the scooter spends 2x + (40x − 33)/25 hours in transit. The two times must be the same. So we have an equation: 6.6 − 3x = 2x + 1.6x − 1.32. Thus, x = 1.2, and the answer to the puzzle is 3 hours.
If you can figure out my number without the Internet, call me.
(Just in case: this is a joke and not my actual phone number.)
This is my second crocheting project to help finance our new program, Yulia's Dream, in honor of Yulia Zdanovska, a young Ukrainian math talent killed in the war.
I made four Whitehead links in the colors of the Ukrainian flag. You can read about my other crocheting project in my previous post: Hyperbolic Surfaces for Ukraine.
Fun trivia about the Whitehead links.
As you might know, my team started a project, Yulia's Dream, in honor of Yulia Zdanovska, a young Ukrainian math talent killed in the war.
In this program, we will do what we are great at — help gifted youngsters pursue advanced math. To help the program, I started crocheting hyperbolic surfaces in the colors of the Ukrainian flag. These crochets are designed as gifts to encourage individual donors.
Fun trivia about these hyperbolic surfaces.
Overall, I crocheted 10 hyperbolic surfaces. If you are interested in donating to help Ukrainian students receive coaching from our program at MIT, the details will be announced on our website shortly.
What is base 3/2? One of the ways to define such a base is to think of it in terms of exploding dots. What the heck are exploding dots? They are explained and popularized by James Tanton in his YouTube videos.
Essentially, "exploding dots" is a machine made of a row of boxes with rules describing how the dots loaded into the machine explode. As an example, let me describe the 1←2 machine, which corresponds to base 2. We load N dots into the rightmost box. Whenever there are 2 dots in one box, they explode into 1 dot in the box to the left.
For example, to write 5 in base 2, we would first load 5 dots in the rightmost box, as in the figure above. Then each group of 2 dots in the rightmost box would explode, and for each group, 1 dot would appear in the box to the left. Finally, the 2 dots in the second box would explode into 1 dot into the next box to the left. By reading the number of dots from left to right, we get 101, which is 5 in base 2.
The interesting thing here is that there is no reason this model should be exclusive to integer bases. Suppose our rule is that 3 dots explode into 2 dots in the box to the left. Such a rule is called the 2←3 machine, and it corresponds to base 3/2. To represent 5 in this base, we load 5 dots into the rightmost box, then we use the exploding rule shown in the figure below. Using this machine, 5 is represented in base 3/2 as 22.
The figures were made by my junior PRIMES STEP group, in the 2017-2018 academic year, for our paper, Variants of Base 3 Over 2.
But, in this post, I want to discuss a different paper from the same academic year. With my senior PRIMES STEP group, we wrote a paper On Base 3/2 and its Sequences. A shorter version which includes a tribute to John Conway, appeared in The Mathematical Intelligencer.
Speaking of John Conway, he liked inventing new sequences, especially ones with unusual behaviors. One of his hobbies was tweaking the Fibonacci rule to create new sequences, which he called Fibs. For example, the sorted Fibs sequence starts the same as the Fibonacci sequence with 0 and 1. To calculate the next term, we add the two previous terms and sort the digits in non-decreasing order. In base 10, this sequence is A069638: 0, 1, 1, 2, 3, 5, 8, 13, 12, 25, 37, 26, …. It is known that this sequence is periodic with a maximum value of 667.
With my senior PRIMES STEP group, we studied analogs of this sequence in base 3/2. We begin with the sorted Fibs sequence f_{n} with the same two initial values that start the Fibonacci sequence: f_{0} = 0 and f_{1} = 1. To calculate f_{n+1}, we add f_{n-1} and f_{n} in base 3/2 and sort the digits in non-decreasing order. It follows that the numbers in the sequence are written as several ones followed by several twos. Unlike base 10, the sequence is not periodic and grows indefinitely: 0, 1, 1, 2, 2, 12, 12, 112, 112, 1112, 1112, 11112, …. In recognition of the constant growth of this Fibs sequence, we call it the Pinocchio sequence.
Obviously, you can start the sorted Fibs sequence with any two numbers. But we proved an interesting theorem which stated that any sorted Fibs sequence eventually turns into either the tail of the Pinocchio sequence or the 3-cycle 112, 1122, 1122.
However, we didn't stop there. There are two natural ways to sort the digits of a number, in increasing or decreasing order. Naturally, there is another type of sequences worth considering, in which the digits are sorted in non-increasing order. We called such sequences the reverse sorted Fibs.
We defined the reverse sorted Fibs sequence r_{n} in base 3/2 as follows. To calculate r_{n+1}, we add r_{n-1} and r_{n} in base 3/2 and sort the digits in non-increasing order, ignoring zeros. It follows that after the initial terms, the terms of such a sequence are represented with several twos followed by several ones. We call the reverse sorted Fibs that start in a similar way to the Fibonacci sequence with r_{0} = 0 and r_{1} = 1, the proper reverse sorted Fibs. Here are several terms of the proper reverse sorted Fibs: 0, 1, 1, 2, 2, 21, 21, 221, 2211, 221, 221, 2211, 221, 221, 2211, …. This sequence becomes cyclic, starting from r_{7}.
We also found one reverse sorted Fibs growing indefinitely: 2211, 2211, 22211, 22211, 222211, 222211, and so on. We proved that any reverse sorted Fibs sequence eventually turns into either this sequence or a 3-cycle sequence: 221, 221, 2211. The similarity between the sorted Fibs and the reverse sorted Fibs surprised us. Up to the initial terms, they both have exactly one sequence which grows indefinitely and one 3-cycle. To emphasize this similarity, we reversed the word Pinocchio and named this growing reverse Fibs sequence the Oihcconip sequence.
Now I need to figure out how to pronounce the name of this new sequence.
I run Number Gossip, where you can input a number and get some of its cute properties. For example, the number 63 is composite, deficient, evil, lucky, and odd. In addition, it has a unique property: 63 is the smallest number out of two (the other being 69), such that the common alphabetical value of its Roman representation is equal to itself. Indeed, the Roman representation of 63 is LXIII, where L is the 12th digit, X is the 24th, and I is the 9th. Summing them up, we get 12 + 24 + 9 + 9 + 9 = 63 — the number itself.
I have a list of about 50 properties of numbers that my program checks. Each number greater than one gets at least four properties. This is because I have four groups of properties that cover all the numbers. Every number is either odd or even. Every number is either deficient, perfect, or abundant. Every number greater than one is either prime or composite. Every number is either evil or odious.
In addition, I collect unique number properties. During the website's conception, I decided not to list all possible unique properties that I could imagine but to limit the list to interesting and unusual properties. My least favorite properties of numbers are the ones that contain many parameters. For example, 138 is the smallest number whose base 4 representation (2022) contains 1 zero and 3 twos. If you are submitting a number property to me, keep this in mind.
Some parameters are more forgiving than others. For example, 361 is the smallest number which is not a multiple of 9, whose digital sum coincides with the digital sum of its largest proper divisor. In more detail, the digital sum of 361 is 3 + 6 + 1 = 10, while 19, the largest divisor of 361, has the same digital sum of 10. In this case, the parameter 9 is special: for a multiple of 9, it is too easy to find examples that work, such as 18, 27, and so on. Sequence A345309 lists numbers whose digital sum coincides with the digital sum of their largest proper divisor. The first 15 terms of the sequence are divisible by 9, and 361 is the smallest term that is not divisible by 9.
By the way, another number that is buried deep in a sequence is 945, which is the smallest odd abundant number. There are 231 abundant numbers smaller than 945; all of them are even.
A more recent addition to my collection is related to the sequence of distended numbers (A051772). Distended numbers are positive integers n for which each divisor of n is greater than the sum of all smaller divisors. It is easy to see that for distended numbers, all sums of subsets of divisors are distinct. The opposite is not true: 175 is the smallest number, where all sums of subsets of its divisors are distinct, but the number itself is not distended.
It is difficult to find special properties for larger numbers, so I am less picky with them. For example, 3841 is the number of intersections of diagonals inside a regular icosagon. The word icosagon hides a parameter, but I still like the property.
I invite people to submit number properties to me. And I received many interesting submissions. For example, the following oxymoronic property was submitted by Alexey Radul: 1 is the only square-free square.
The numbers below 200 that still lack unique properties are 116, 147, 155, 162, 166, 182, and 194. The earliest century that doesn't have unique number properties ranges between 7000 and 7100. The next ones are 8000–8100 and 9100–9200. By the way, my site goes up to ten thousand.
I also have a lot of properties in my internal database that I haven't checked yet. I am most interested in the proof of the following property: 26 is the only number to be sandwiched between any two non-trivial powers.
One of my posts about John Conway has a picture I took of him in 2015, leafing through a book about himself, Genius at Play. I allowed Wikipedia to use this image, and they did. They also retouched it for their article on John Conway in Dutch. I like the result.
Every year, after the MIT Mystery Hunt was over, I would go through all the puzzles and pick out the ones related to mathematics. This year, I didn't feel like doing it. Besides, I think it is more important to collect quality puzzles rather than all the puzzles by topic. So my new collection is the puzzles recommended to me, which I might like.
I start with math and logic puzzles.
I continue with computer science.
I carry on with some non-math fun.
I conclude with the plot device round. All the puzzles in this round are relatively easy. But our team got stuck on them until we realized that we already had the answer, which was not a single word. Here are some of the puzzles that were specifically recommended.
Putin became the 21st century Hitler. I call him Putler.
I am an American. However, I was born in Moscow and lived my youth in the Soviet Union. I speak Russian, and I have friends in both Russia and Ukraine. The war in Ukraine is the biggest tragedy of my life.
When Putler invaded Ukraine, I didn't know what to do. I wanted to pick up a rifle and go to Ukraine to fight, but then I remembered my CPAP machine and the distilled water it needs, and I didn't go. Instead, I ended up watching the news non-stop. Then I started sending money to different organizations supporting Ukraine.However, I am a mathematician, so I tried to figure out whether I could help Ukraine by doing math. At first, I posted math problems from Ukraine Olympiads. Then I started discussing what we could do with my PRIMES colleges. The result was a new program, Yulia's Dream, in honor of Yulia Zdanovska, a 21-year-old brilliant young Ukrainian mathematician killed by a Russian-fired missile. Yulia's Dream is a free enrichment program for high-school students from Ukraine who love math.
All these activities didn't help me with the pain. So I started crocheting. I bought yarn in the colors of the Ukrainian flag and crocheted a hyperbolic surface of constant curvature. The first picture shows the thingy from above. The second one is there for you to estimate its size: this is the biggest crocheting project I have ever finished.
For a free Ukraine! Let democracies win over dictatorships!
I wrote a lot about how during entrance tests for Moscow State University, the examiners were giving Jewish and other undesirable students special (e.g. more difficult) questions during the oral exams. (See, for example, our paper Jewish Problems with Alexey Radul.) Not all examiners agreed to do this. So the administration made sure that there were different exam rooms: brutal rooms with compliant examiners torturing students with difficult questions, and normal rooms with normal examiners testing preapproved students. The administration also had other methods. One of them is the topic of this essay.
The math department of Moscow State University had four entrance exams. The first was a written math test consisting of three trivial problems, a very difficult one, and a brutally challenging one. At the end, I will show you a sample: a trivial problem and a very difficult one from 1976, my entrance year.
What was the point of such vast variation in difficulty, you may ask? There were two reasons.
But first, let me explain some entrance rules. The exam was scored according to the number of solved problems. A score of two or less was a failing score. People with such scores would be disqualified from the next exam. Any applicant with a smidge of mathematical intelligence would be able to solve all three trivial problems. Almost all applicants who qualified for the next test would have the same score of three on the first test, as they wouldn't be able to solve the last two problems. Thus, mathematical geniuses and people who barely made the cut got the same score.
There was another rule. Officially, people with a gold medal from their high school (roughly equivalent to a valedictorian) could be accepted immediately if they scored 5 on the first exam. So one of the administrative goals was to prevent anyone getting a 5, thus, blocking Jewish applicants from sneaking in after the first exam.
Another goal was to have all vaguely qualified people get the same score. The same goal applied to other exams. After the four admission exams, the passing score, say X, was announced. A few people with a score higher than X were immediately accepted. Then there were hundreds of applicants with a score of X, way more than the quota of people the department was planning to accept. An official rule allowed the math department to pick and choose whoever they wanted from everyone who scored X.
I heard a speech by the famous Russian mathematician, Vladimir Arnold, directed at decent examiners who tested "approved" students at the oral math exam, which was the second admissions exam. His suggestion was brilliant and simple. If the students are good and belong in the department, give them an excellent grade of 5. If not, give them a failing grade of 2. Arnold's plan boosted the chances of good students doing better than the cutoff passing score X and removed mediocre students from the competition. His idea was not only brilliant and simple but also courageous: he was risking his career by trying to fight the system.
I never experienced the entrance exams firsthand. By ministry order, as a member of the USSR IMO team, I was accepted without taking any exams. I already wrote an essay, A Hole for Jews, about how getting on the IMO team was the only way for Jewish students to get into the Moscow State University, and how the University tried to block them.
But I still looked at the entrance exam problems I would have had to solve to get in. The last two problems scared me. Now I found them again online (in Russian) at: the 1976 entrance test. The trivial problem below is standard and mechanical, while the other problem still looks scary.
Trivial problem. Solve for x:
Solution. We were drilled in school to solve these types of problems, so this one was trivial. First, make a substitution: y = 3^{x}. This leads to an equation: (2y - 1)(y - 3)/(y^{2} - 2)(y - 1) ≤ 0. From this we get ranges for y: (-∞, -√2], [1/2,1], [√2, 3]. The last step is to take a logarithm.
Very difficult problem. Three spheres are tangent to plane P and to each other. Two of the spheres are the same size. The apex of a circular cone is on P, and the cone's axis is perpendicular to the plane P. All three spheres are outside the cone and tangent to it. Find the cosine of the angle between the cone's generatrix and the plane P, if one of the angles of the triangle formed by the intersection points of the spheres and the cone is 150 degrees.
I stumbled upon one of Smullyan's puzzle on Facebook, in Russian. I couldn't find the original text, so I just translated it back for my students.
Puzzle. You are on an island where only truth-tellers and liars live. The truth-tellers always tell the truth, and the liars always lie. You meet an islander who sits with you for a long time, then says, "I already said this sentence." Is he a truth-teller or a liar?
I expected the following solution. If this islander is a truth-teller, then there should have been a time when he said, for the first time, "I already said this sentence." But this would create a contradiction.
However, my students used this puzzle as an opportunity to teach me some intricacies of the English language. They explained to me the ambiguities of my translation. Here is a shortened and lightly edited quote from one of them:
There are two different linguistic opinions that give different answers to this problem. The first is that the truth of a statement is decided at the moment it starts to be delivered: in this case, when the islander starts saying his statement. With this interpretation, for the statement to be true, he had to have said the sentence before, and for that to be true, he had to have said it even before that, and this continues indefinitely. Clearly, he cannot have been alive forever, so he has to be a liar.
The other opinion is that the verity of a statement is decided at the exact conclusion of its deliverance. Then, when the islander finishes saying his sentence, its truth is judged, and he has at that same instant "already" said the sentence, so he is telling the truth. By this interpretation, the islander is a truth-teller.
Another student had a different brilliant idea. Depending on the islander's intonation, it is possible that he says, "I already said 'this sentence'." In that case, there are no self-referencing sentences, and the islander could be either a truth-teller or a liar.
I consulted my best English consultant: my son, Alexey, and here is his reply. "The basic answer is that neither truth nor semantic meaning are absolute, and edge cases will be judged differently by different observers. A sentence whose truth is time-dependent on the same scale as the duration of uttering the sentence is clearly an edge case. That's why mathematicians intentionally try to eliminate ambiguity from their communication."
He suggested the following fix for the puzzle's translation.
Fixed puzzle. You meet an islander who says, "I have said this sentence before." Is he a truth-teller or a liar?
Alexey didn't stop at fixes and suggested the following bonus puzzles.
Bonus puzzle 1. You meet an islander who says, "I will have said this sentence." Is he a truth-teller or a liar?
Bonus puzzle 2. You meet an islander who says, "I will say this sentence again." Is he a truth-teller or a liar?
Ukraine is on my mind. Here is a problem for 9-graders from the 1978 Ukrainian Math Olympiad:
Problem. Prove that for every natural number n, the following number is not an integer.
Oskar van Deventer is a designer of beautiful mechanical puzzles. For the recent mini-MOVES gathering at the MoMath, he asked people in his Zoom breakout room to calculate the average age in the room without revealing their actual ages. I know the following solution to this puzzle.
People agree on a large number N that is guaranteed to be greater than the sum of all the ages. The first person, say Alice, thinks of a uniformly random integer R between 0 and N. Alice adds her age to R modulo N and passes the result to the second person, say Bob. Bob adds his age modulo N and passes the result to the third person, and so on. When the result comes back to Alice, she subtracts R modulo N and announces the sum total of all the ages.
During this process, no one gets any information about other people's ages. But two people can collude to figure out the sum of the ages of the people "sitting" between them.
I gave this problem to my grandson, and he suggested the following procedure. First, people choose two trusted handlers: Alice and Bob. Then, each person splits their age into the sum of two numbers (the splitting should be random and allow one of the numbers to be negative). They then give one number to Alice and another to Bob. Alice and Bob announce the sums of the numbers they receive. After that, the sum total of everyone's ages is the sum of the two numbers that Alice and Bob announce.
The advantage of this method is that no two people, except Alice and Bob, can collude to get more information. The disadvantage is that if Alice and Bob collude, they would know everyone's age. Which method would you prefer?
Many years ago, I wrote a blog post about an amusing fact: John Conway put Moscow, the former capital of the USSR, as a coauthor: A Math Paper by Moscow, USSR. Thus, Moscow got an Erdős number 2, thanks to Conway's Erdős number 1. At that time, my Erdős number was 4, so I wondered if I should try coauthoring a paper with Moscow, my former city of birth, to improve my Erdős number.
This weird idea didn't materialize. Meanwhile, my Erdős number became 2 after coauthoring a paper with Richard Guy, Conway's Subprime Fibonacci Sequences. I relaxed and forgot all about my Erdős status. I couldn't do better anyway, or could I?
I recently heard about a cheater who applied to grad schools. In addition to a bunch of fabricated grades, the cheater submitted an arXiv link to a phony paper. What is fascinating to me is that the cheater put real professors' names from the university the cheater supposedly attended as coauthors. The professors hadn't heard of this student and had no clue about the paper. So the cheater added fake coauthors to add legitimacy to their application and boost the perceived value of the cheater's "research". As a consequence, the cheater got a fake Erdős number.
I hope that the arXiv withdrew the paper. Cheating is the wrong way to improve one's Erdős number.
But here is another story. Robert Wayne Thomason named as coauthor his dead friend, Thomas Trobaugh. The paper in question is Higher Algebraic K-Theory of Schemes and of Derived Categories and can be found at https://www.gwern.net/docs/math/1990-thomason.pdf. This paragraph in the paper's introduction explains the situation.
The first author [Robert Wayne Thomason] must state that his coauthor and close friend, Tom Trobaugh, quite intelligent, singularly original, and inordinately generous, killed himself consequent to endogenous depression. Ninety-four days later, in my dream, Tom's simulacrum remarked, "The direct limit characterization of perfect complexes shows that they extend, just as one extends a coherent sheaf." Awaking with a start, I knew this idea had to be wrong, since some perfect complexes have a non-vanishing K_{0} obstruction to extension. I had worked on this problem for 3 years, and saw this approach to be hopeless. But Tom's simulacrum had been so insistent, I knew he wouldn't let me sleep undisturbed until I had worked out the argument and could point to the gap. This work quickly led to the key results of this paper.
This precedent gives anyone hope that they might achieve an Erdős number 1. You just need to wait for Paul Erdős to come to you in your dreams with a genius idea.
Last year, our junior PRIMES STEP group studied Latin squares. We invented a lot of different types of Latin squares and wrote a paper about it, Fun with Latin Squares. Recall that a Latin square is an n by n table containing numbers 1 through n in every cell, so that every number occurs once in each row and column. In this post, I want to talk about anti-chiece Latin squares.
First, what's a chiece? A chiece is a portmanteau word made out of two words, chess and piece, and, not surprisingly, it means a chess piece. Given a chiece, an anti-chiece Latin square is a Latin square such that any two cells, where our chiece can move from one cell to the other, according to the rules of chess, can't contain the same number. Let's see what this means.
Let's start with rooks, which move along rows and columns. An anti-rook Latin square can't have the same numbers repeating in any one row or column. Ha, anti-rook Latin squares are just Latin squares. Anti-bishop and anti-queen Latin squares can't have the same numbers repeating on any diagonal.
Now, here is a picture of an anti-knight Latin square in which no two identical numbers are a knight's move apart. This particular Latin square also forms a mini-Sudoku: not only does each row and column, but also each 2 by 2 corner region, contains all distinct numbers.
Consider all instances of some number, say 1, in an anti-chiece Latin square. If the board is n by n, we get n instances of non-attacking chieces. A famous math puzzle asks to place eight non-attacking queens on a standard chessboard. So the instances of any one particular number in an anti-queen Latin square solves the problem of placing n non-attacking queens on an n by n chessboard. Thus, building an anti-queen Latin square is more complicated than solving the non-attacking queens puzzle. The former requires filling the chessboard with n non-overlapping sets of non-attacking queens. The picture below gives an example of an anti-queen 5 by 5 Latin square.
This square has some interesting properties. It can be formed by cycling the first row. It also happens to be one of the chiece Latin squares we study in our paper. A chiece Latin square is a Latin square such that for each number in a cell, there is another cell, a chiece's move apart, containing the same number. You can check that our anti-queen Latin square is at the same time a knight Latin square.
I wonder, can anyone build an anti-queen Latin square on the standard 8 by 8 chessboard?
Here is the famous barber paradox.
Paradox. The barber shaves all those and only those who don't shave themselves. Does the barber shave himself?
If he shaves himself, then he doesn't shave himself. If he doesn't shave himself, then he shaves himself.
English is not my primary language, and I am fascinated by the variety of verb tenses in English as compared to the Russian language. For example, Russian has one present tense while English has four. I wonder what would happen if we use the other present tenses in this paradox.
Present continuous. The barber shaves all those and only those who aren't shaving themselves. Does the barber shave himself?
Does this mean that the barber starts shaving himself and then has to stop, and a moment later he has to start again?
Present perfect. The barber shaves all those and only those who haven't shaved themselves. Does the barber shave himself?
Does this mean that the barber shaves himself every other day?
Present perfect continuous. The barber shaves all those and only those who haven't been shaving themselves. Does the barber shave himself?
Does this mean the barber shaves himself once in his lifetime and then never again?
Gowers starts with several assumptions.
From these assumptions, the following theorem can be deduced.
Theorem. The only possible rule satisfying these assumptions would allow any two people to have sex with each other as long as they both reached some fixed age k.
There is a problem with this type of rule. Suppose k is 18. If two people who are slightly younger than 18 have consensual sex, they can't both be predators. These are two children with raging hormones. There is no reason to punish anyone. Now imagine that one of the partners turns 18. Society would still consider this a Romeo-and-Juliet case and would tend not to punish such a partner. Now imagine a child younger than 18 having sex with a partner over 40. The older partner has no raging hormones, knows what they are doing, and probably knows how to manipulate little children into having sex. So, it might be desirable to have a rule that differentiates between these two cases. The rule would take into account the difference in ages while forgiving younger offenders and still punishing predators.
Consider the most common type of law to resolve this issue: Anyone older than 18 can have sex, and, in addition, a person who is not older than 20 can have sex with someone between the ages of 16 and 18. This law doesn't satisfy monotonicity. It could be that one day the older partner is not yet 20, and the next day, oops, they have a birthday. So, as a birthday gift, they are not allowed to have sex with each other anymore.
Here is a simple idea to resolve the issue by having the law focus on the age gap instead of the age of the older partner. We can have an adjusted law: Anyone older than 18 can have sex, and, in addition, a person can have sex with someone between the ages of 16 and 18 as long as the age gap is not more than four years. This rule doesn't satisfy the simplicity assumption above, but it is simple enough. It is close in spirit to the previous rule and satisfies monotonicity. The problem with this rule is continuity.
According to the adjusted rule, the couple with the age gap of four years and one day might have to wait two years longer to have sex than the couple with the age gap of four years. This seems unfair.
Tim Gowers suggests dropping the simplicity rule. We can use days rather than years. For example, the rule might be that if one person in a couple is under 18, but at least 16, and has age x, then the other partner has to be not more than age y, where for example, y − x = 4 + (x − 16)/2. So when one partner turns 16, their partner has to be not older than 20. When one partner is 16 and two months, the other cannot be older than 20 and three months. With the younger partner getting older, the allowable age gap is increasing slowly. By the time the younger partner is a day from turning 18, their partner can be almost five years older.
It might be complicated for two people to calculate if they are allowed to have sex according to this formula. But Gowers' big idea was that apps and websites could do this easily: two people plug in their birthdays and know whether they are allowed to have sex.
Do you know what an isosceles tetrahedron is? I didn't until recently. An isosceles tetrahedron has all of its faces congruent. Equivalently, all pairs of opposite edges are of equal length. Such tetrahedra are also called disphenoids. Here are some cute statements about them.
Disphenoids have three nontrivial symmetries: 180-degree rotations around three lines connecting the midpoints of opposite edges.
Is it possible to have a tetrahedron with fewer symmetries than disphenoids? Yes, it is. Dan Klain just published a fascinating paper about such tetrahedra: Tetrahedra with Congruent Face Pairs. The results are so elegant and simple that I was surprised that they were new. I got curious and started to google aggressively. I found an official name for this kind of tetrahedron: phyllic disphenoid, but no theorems about them. Their name is quite unappealing: no wonder people didn't want to study them much. Obviously, Dan wasn't as aggressive at googling, so he didn't find this official name and called them reversible tetrahedra. But my favorite name for them is the name Dan thought of but didn't use in his paper: bi-isosceles tetrahedra.
Here is the picture of a bi-isosceles tetrahedron from Dan's paper. It has two faces with sides a, b, and c, and two faces with sides a, b, and d. The edges of this tetrahedron are: a, a, b, b, c, and d. It has one nontrivial symmetry: a 180-degree rotation around the line connecting the midpoints of the unequal opposite edges c and d. The picture emphasizes this symmetry. The figure in the picture is a projection of a bi-isosceles tetrahedron onto a plane, such that the line of symmetry is projected onto the point of symmetry of the projection. The two cute statements above, about disphenoids, can be generalized to the case of bi-isosceles tetrahedra.
The first property is trivial, while the second one is proven in Dan's paper. Dan also shows how to calculate the volume of a bi-isosceles tetrahedron:
Here is a problem from the 1978 Kyiv Olympiad for 7 graders.
Is it possible to place seven points on a plane so that among any three of them, two will be at distance 1 from each other?
Puzzle. Two girls were born to the same mother, at the same time, on the same day, in the same month, in the same year, and yet somehow they're not twins. Why not?
I won't tell you the expected answer, but my students are inventive. They suggested all sorts of scenarios.
Scenario 1. There are two different fathers. I had to google this and discovered that, indeed, it is possible. This phenomenon is called heteropaternal superfecundation. It happens when two of a woman's eggs are fertilized by sperm from two different men. Unfortunately for my students, such babies would still be called twins.
Scenario 2. The girls are born on the same date, but not on the same day. This could happen when transitioning from the Julian to Gregorian calendar. The difference in birth times could be up to two weeks. I had to google this and discovered that twins can be born months apart. The record holders have a condition called uterus didelphys, which means that the mother has two wombs. Unfortunately for my students, such babies would still be called twins.
Scenario 3. The second girl is a clone. This scenario can potentially happen in the future. Fortunately for that student, I suspect that such babies would be called clones, not twins.
I decided to invent my own scenario outside of the actual answer, and I did.
Scenario 4. Two girls are from the same surrogate mother, but they are not twins. I had to google this and discovered that this actually happened: Surrogate mother of 'twins' finds one is hers.
Sometimes life is more interesting than math puzzles.
In his article on Möbius strips, Martin Gardner included a cute construction.
Construction. Cut out a cross from a piece of paper. Then glue one pair of opposite ends to make a cylinder and the other pair to make a Möbius strip. Then Martin instructs, "Trisect the twisted band, then bisect the untwisted one, and open up for a big surprise!"
In my effort to reuse Möbius strips, I started making them out of zippers. So Martin's construction had its destiny zipped. The first picture shows the construction before it is being dissected. I was quite happy with my plan to use zippers as it had its advantages over paper. For the surprise to work, the twisted band shouldn't be cut completely. Meaning, the middle part of the Möbius strip shouldn't be cut. I sewed my zipper monstrosity not to cause any ambiguities: there is nothing to cut, just unzip the zippers.
Little did I know that unzipping is not the issue, but zipping back up is. I tried to zip the surprise back up several times; all of them unsuccessfully, as you can see on the two other pictures. I finally had to invite the real expert — my grandson — to do it.
I used to hate crocheting. Now it's been growing on me.
To prepare for this magic trick, take all the spades out of a deck and place them in the following order: seven, ace, queen, two, eight, three, jack, four, nine, five, king, six, and ten. Turn the assembled deck face down, so that the seven is on top. Now you are ready to do the trick.
Magic trick. Transfer the top card to the bottom of your deck and deal the new top card face-up on the table. Repeat this process until all the cards are dealt. And — abracadabra — the cards are dealt in order.
I showed this trick to my grandchildren, and they decided to reproduce it. They tried to calculate where each card goes, without too much success. Then my son showed them another trick: how to arrange the cards without calculation. He started building his arranged deck from the end of the trick with all the cards in order face-up on the table with the king on top. He took the king and put it face-down into his hand. Then he repeated the following procedure until all the cards were in his hand: He took the next card from the table and put it face-down on top of the one in his hand. Then he moved the card from the bottom of his deck to the top. And — abracadabra — the cards are arranged correctly for the trick.
Next time, I should ask my grandkids to show this trick with the whole deck.
Each time I teach my students about the Möbius strip, I bring paper, scissors, and tape to class. The students make Möbius strips, and then I ask them to cut the strips in half along the midline and predict the result.
Then we move to advanced strips. To glue the Möbius strip, you need one turn of the paper. An interesting experiment is to glue strips with two, three, or more turns. In this case, too, it is fun to cut them along the midline and predict the shape of the resulting thingy. My class ends with a big paper mess.
As you might know, I am passionate about recycling. So I have always wanted to buy Möbius strips that can be cut in half and then put back together. I didn't find them, so I made them myself from zippers. Now I can unzip them along the midline and zip them back together. I hope to have less mess in my future classes. We'll see.
I recently posted my new favorite probability puzzle from the fall 2019 issue of Emissary, submitted by Peter Winkler.
Puzzle. Alice and Bob each have a biased coin that flips heads with probability 51% and 100 dollars to play with. The buzzer rings, and Alice and Bob begin flipping their coins once per minute and betting one dollar on each flip against the house. The bet is at even odds: each round, each of them either loses or wins a dollar. Alice always bets on heads, and poor Bob, always on tails. As it happens, however, both eventually go broke. Who is more likely to have gone broke first?
Follow-up question: They play the same game as above, but this time Alice and Bob are flipping the same coin (biased 51% toward heads). Again, assuming both eventually go broke, who is more likely to have lost their money first?
One might assume that Bob obviously got a bad deal. He will lose his money very fast. So the answer to both questions must be that Bob loses his money first. If you know me, you should realize that I wouldn't have given you this puzzle if the answer was that intuitive. I love counter-intuitive puzzles.
Bob has a disadvantage, so he is guaranteed to lose eventually. Alice has an advantage, so she might lose, or she might not. If she goes broke, that means there should be more tails in the flips than if she doesn't go broke. How does this influence the second scenario, in which they use the same coin? As we are expecting a lot of tails in the flips, the situation should be better for Bob as compared to the first scenario. Given that I posed this puzzle, one might decide that in the follow-up question Bob doesn't go broke first. Is it a tie?
But, if Bob is the first to go broke in the first scenario, why would I include the first part of the puzzle at all? If you know me, you should wonder what makes the first question interesting. Or maybe, you know Peter Winkler, in which case you should also wonder the same thing.
I gave you enough meta information to guess the amazing counter-intuitive answers to these two questions: in the first scenario, they tie; and in the second scenario, Alice goes broke first with higher probability.
To prove that they tie in the first scenario, let me first define a switch on a sequence of coin flips. A switch changes each head into tails and vice versa. The switch creates a one-to-one correspondence between the sequences of flips where Bob goes broke with the sequences where Alice goes broke. Suppose p is the probability that Alice goes broke with a particular sequence a, and q is the probability that Bob goes broke with a `switched' sequence b. Then p = rq, where r = (49/51)^{200}. The reason is that sequence a has exactly 200 more tails than sequence b. The same ratio r works for any pair of switched sequences. Bob is guaranteed to go broke eventually. But to calculate the conditional probability of Alice going broke with sequence a, given that she does go broke, we need to divide the probability of the sequence a occurring by the probability that Alice goes broke. The resulting probability distribution on sequences of length n has to be the same as for Bob because it has to sum to one. If Alice goes broke, then the probability that she goes broke after n flips is the same as for Bob. That means they tie.
To answer the second question, we use the switch again. The switch creates a one-to-one correspondence between sequences of flips where Alice goes broke first and Bob second, and vice versa. The sequence for which Alice loses second has 200 more tails than the corresponding sequence where Bob loses second. Thus, in each pair of two switched sequences, the probability of the sequence where Alice loses second is equal to r times the probability of the sequence where Bob loses second. Thus, the probability of Alice winning is r times the probability of Bob winning, and r is a very small number.
Here are my newly-made Borromean rings: two identical sets of them. They are an example of three linked rings, with any two of them not linked. The top set is positioned the way the Borromean rings are usually presented. You can see that any two rings are not linked by mentally ignoring the third one. For example, the red ring is on top of the green one, the green one is on top of the blue one, and the blue one is on top of the red one. They have a non-transitive ordering.
The bottom set of rings is arranged for a lazier thinker. Obviously, the blue and green rings are not linked.
I used to be proud of my Russian math education. I am still proud of my high school one, but not so much of the one I received in college. In Soviet Russia, a student had to choose their major before applying to college. I wanted to study mathematics, and I got accepted to the best place for it in Soviet Russia: mekhmat — the math school at the Moscow State University. I used to be proud of my education there, but now I have my doubts.
I had to take, on average, four math classes per semester for five years, which totals about 40 math classes. Woo hoo! I don't think American students could even choose to take that many. This was presumably good, but most of the courses were required, and their curriculum remained unchanged for many, many years. Obviously, the system was very rigid. The faculty members feared retaliation from the communist party and forgot how to take initiative. The bureaucracy prevented the department from adding new and exciting math to the outdated curriculum.
This post is not about my grades but about the actual subjects that we were taught then. But, in case anyone is wondering, my only B was in English; everything else was straight As.
Some of the classes listed below lasted two or more semesters, that's why they do not sum up to the promised 40. Unfortunately, I do not remember which ones. These were the required math classes:
An impressive list? But guess what — I remember nothing from most of these classes. As an exception, I remember bits of Differential Equations, taught by Vladimir Arnold, a charismatic teacher. I remember Linear Algebra well, not because of my Linear Algebra class, but because I read Gelfand's book on the subject and loved it. I remember that the Differential Geometry and Topology class was taught by Fomenko with great pictures and boring material. By the time I took Fomenko's class, I already knew topology from an unofficial class taught by Dmitry Fuchs, which was so much better. In fact, in order to learn what I wanted, I had to take many classes unofficially, so my total is actually way above 40.
By junior year, we were finally allowed to choose some classes which would count towards our transcripts, and this is what I picked.
I remember these classes much more vividly. I also wrote a graduate thesis: "Models of Representations of Generalized Clifford Algebras." I loved working on that paper.
We had non-math classes too: everyone had to take them.
To graduate, everyone had to pass two state exams: Mathematics and Scientific Communism. Whatever the latter might mean.
Did I mention that I am no longer proud of my former Soviet college education? What a colossal waste of time!
Pete McCabe presented his trick, Persistimis Possessiamo, at the Gathering for Gardner in 2018.
Trick. Pete asked for two volunteers, let's call them Alice and Bob. Bob took out his favorite card, the Queen of Spades, from the deck and put it back following Pete's instructions. Then Alice dealt the deck alternatively into two piles, Bob's and hers, starting with Bob's. Alice took her pile and repeated the same process several times until only one card was left. And, abracadabra: it was Bob's chosen Queen of Spades.
Pete McCabe is interested in scripting magic. In his blog post, Scripting Magic for Zoom, he describes ways to make sure that Bob inserts his card into the 22nd place without using sleight of hand, but rather using a theatrical script which makes the process magical rather than mathematical. The magic part is related to the fact that the number of letters in the trick title, Persistimis Possessiamo, is 22. As a result, he can do the trick on Zoom without ever touching the cards.
Once a magician knows how to manipulate the volunteer to insert the card into a specific place in the deck, the trick becomes deterministic and works on any-sized deck, as long as the magician can calculate where the card goes. We will now perform this calculation.
We denote our card-inserting sequence as a(n), where n is the size of the deck, and a(n) is the place where the card is inserted. For starters, a(2n+1) = a(2n): when the size of the deck is odd, the last card during the first deal goes to Bob, and doesn't effect the other deals. Now, we obviously have a recursion. First, we observe that in order to end up in Alice's pile after the first deal, Bob's chosen card should occupy an even-numbered place. Suppose we start with 2n cards. After the first deal, Bob's chosen card is in the place number a(2n)/2 from the bottom in Alice's pile. That means, the card is in the place number n + 1 − a(2n)/2 from the top. This gives us an equation: a(n) = n + 1 − a(2n)/2, which is equivalent to a recursion: a(2n) = 2(n + 1 − a(n)).
Given that each element of the sequence a(n) is doubled, we are only interested in even-indexed values. Consider b(n) = a(2n) = a(2n+1). Then b(1) = 2, and the recursion for b is b(n) = 2(n + 1 − b⌊n/2⌋).
From here, we get the sequence, which is now sequence A350652 in the OEIS:
2, 2, 4, 6, 8, 6, 8, 6, 8, 6, 8, 14, 16, 14, 16, 22, 24, 22, 24, 30, 32, 30, 32, 22, 24, 22, 24, ... .
Here is a classical puzzle I often give to my students.
Puzzle. The sultan has three red hats and two blue ones. He wants to test his three wizards, who know his hat collection. He asks them to close their eyes and puts a hat on each of their heads. After the wizards open their eyes, they see each other's hats, but not their own. The sultan asks each of them to guess the color of their own hat, without communicating with each other. In this particular test, the sultan only puts red hats on the wizards' heads. Sometime after the wizards open their eyes, one of them guesses his hat's color. How did he guess?
Here is how my students explain the solution. If a wizard sees two blue hats, he immediately knows that his hat must be red. That means, if no one immediately announces their hat's color, at least two of them are wearing red hats. In this case, if a wizard sees one red hat, he knows that his hat must also be red. So such a wizard can guess the color of his hat. If after some more time, no one announces their hat color, all the hats worn must be red.
After the students solve the problem, I run an evil experiment on them. I show the students my two blue and three red hats and ask three volunteers to close their eyes. Then, I put two red hats and one blue hat on their heads, and the blue hat goes on the fastest thinker in the group. I did this experiment many times. Half the time, the fastest thinker overestimates how fast the other students think and guesses, mistakenly, that s/he is wearing the red hat. Gotcha!
After the experiment, we discuss what is really going on in this puzzle. This is how I start my class on common knowledge.
I plan to teach knot theory to my students. So, I made four blue knots which are actually the four simplest non-trivial knots. Then I made the red ones which are mirror images of the blue ones.
Then I fiddled with the figure-eight knots, which are the second ones from the left. Out of all these four types of knots, the figure-eight knot is the only one that is amphichiral: its mirror image is equivalent to itself. It was fun to physically transform the red one to look identical to the blue one. So, I decided to crochet another pair of figure-eight knots for my students to fiddle with.
Did I mention that I hate crocheting?
I grew up relatively poor, but I wasn't aware of it and didn't care. In 7th grade, I went to a new school for children gifted in math. Looking back, I realize that most of my classmates there were privileged. My first clue about my own financial disadvantages arrived when my math teacher, Inna Victorovna, offered me several of her old dresses. I do not remember what she said to me exactly, but I remember she was tactful, so much so that I felt comfortable taking the dresses.
In an instant, I was better dressed than I had ever been. I especially loved the brown dress which I wore for my first visit to Gelfand's seminar.
A few years passed; I went to college and married Andrey. Things got somewhat better financially, but I was still struggling. My mother-in-law, Veronika, was well-off and loved clothes. She had a habit of ordering a new dress from her tailor, every season, four times a year. In Soviet Russia, this was a lot of dresses.
One day, Veronika decided to give me some of her old dresses. Unlike my math teacher, she said something that I will never forget. She told me that she was getting rid of those dresses because they were out of fashion and made her look old. I was in my twenties at the time and didn't want to look old either. However, I didn't have much choice in clothes, so I wore the dresses. I hated them.
Here is a famous matchstick puzzle based on the given picture.
Puzzle. Can you move three matchsticks to turn the fish around?
Here is my bonus puzzle.
Bonus Puzzle. Can you turn the fish by moving two matchsticks?
I usually collect puzzles related to math after each MIT mystery hunt. I just discovered that I never reviewed the 2013 hunt, though I started writing the post. Plus, I knew the puzzles of that year better than other year's puzzles: I was on the writing team. Not to mention, the hunt had some real gems.
I start with mathematical puzzles:
Logic puzzles:
Computer science puzzles:
Crypto puzzles:
Word puzzles:
Misc puzzles:
Other puzzles I worked on:
Below, you can find a lovely problem from the 2016 All-Russian Olympiad suggested by Bogdanov and Knop. I took some liberties translating it.
Problem. King Hiero II of Syracuse has 11 identical-looking metallic ingots. The king knows that the weights of the ingots are equal to 1, 2, ..., 11 libras, in some order. He also has a bag, which would be ripped apart if someone were to put more than 11 libras worth of material into it. The king loves the bag and would kill if it was destroyed. Archimedes knows the weights of all the ingots. What is the smallest number of times he needs to use the bag to prove the weight of one of the ingots to the king?
And a bonus question from me.
Bonus. Add one more weighing to prove the weight of three more ingots.
Do you know that I participated in Linguistics Olympiads in high school? They are not well-known in the US, but the Soviet Union has been running them since 1965. The first International Linguistics Olympiad was conducted in 2003, and the US joined in 2007. They are called Computational Linguistics because you are expected to discover some phenomenon in an unfamiliar language on the fly instead of knowing a lot of languages already. The problems mostly need logic and are a good fit for a person who likes mathematics. However, a feel for languages is very helpful.
I do not remember why I started attending the Olympiads, but I remember that there were two sets of problems: more difficult for senior and less difficult for non-senior years. I used to be really good at these Olympiads. When I was in 8th grade, I finished my problems before the time ran out and started the senior problems. I got two awards: first place for non-senior years and second place for senior years. In 9th grade, I got two first-place awards. I didn't know what to do in 10th grade, which was a senior year at that time in the USSR. I couldn't get two first-place awards, as I could no longer compete in the non-senior category. I felt ashamed that my result could only be worse than in the previous years, so I just didn't go.
The prizes were terrific: they gave me tons of rare language books. In the picture, a guy from the jury is carrying my prizes for me. I immediately sold the books at used-books stores for a good price. Looking back, I should have gone to the Olympiad in 10th grade: my winter boots had big holes.
What do you give a mathematician who likes only mathematics if you want to expand her geographical horizon? I just got such a gift: A math book that made me feel that I was in Australia. The book, A Dingo Ate My Math Book: Mathematics from Down Under, written by Burkard Polster and Marty Ross, has lovely essays, nice pictures, and a strong Australian flavor.
2022 is abundant, composite, even, evil, square-free, and untouchable.
In addition, 2022 is the smallest number n such that n, n+1, n+2, and n+3 have the maximal exponents in prime factorization equal 1, 2, 3, and 4 correspondingly. Indeed, 2022 = 2·3·337, 2023 = 7·17^{2}, 2024 = 2^{3}·11·23, and 2025 = 3^{4}·5^{2}.
Problem. The numbers 2^{2021} and 5^{2021} are expanded, and their digits are written out consecutively on one page. How many total digits are on the page?
Drabble cartoon, Jun 15 1987, by Kevin Fagan.
Problem. For how many prime numbers p, the expression 2^{p} + p^{2} is a prime?
* * *
My daughter was talking at her kindergarten about what her parents do for work. She said that her mom catches bugs, invokes demons, and talks to clods.
* * *
I have neither Twitter nor Instagram. I just go for a walk to tell strangers what I ate and drank and how things are at work and at home. I have three followers: a doctor and two policemen.
* * *
Life is like Rubik's cube: fix one side, better not look at the rest.
* * *
My Roomba just devoured a piece of cheese I wanted to pick up and eat. The war between humans and robots is already here.
My friend, John Conway, had a trick to help him with tricky situations. Whenever he needed to make a non-trivial decision, he would ask himself, "What would John Conway do?" As he explained to me, he had in mind the public image he himself created. He liked the image and thought this mental trick helped him be a better, more productive, and not-to-forget, flashier person.
From time to time, I catch myself in need of a decision and ask myself, "What would John Conway do?" And he gave me the answer: I should change the question and ask myself, "What would Tanya Khovanova do?"
Here are some snowball sentences suggested by my students.
Can you invent some other snowball sentences? But first, you need to figure out what they are.
Each year I look at the MIT Mystery hunt puzzles and pick ones related to mathematics, logic, and computer science. I usually give additional comments about the puzzles, but this year's titles are quite descriptive. Let's start with mathematics.
Now Nikoli-type logic puzzles. I really enjoyed "Fun with Sudoku" during the hunt.
And computer science.
I recently wrote a post, A Splashy Math Problem, with an interesting problem from the 2021 Moscow Math Olympiad.
Problem (by Dmitry Krekov). Does there exist a number A so that for any natural number n, there exists a square of a natural number that differs from the ceiling of A^{n} by 2?
The problem is very difficult, but the solution is not long. It starts with a trick. Suppose A = t^{2}, then A^{n} + 1/A^{n} = t^{2n} + 1/t^{2n} = (t^{n} + 1/t^{n})^{2} − 2. If t < 1, then the ceiling of A^{n} differs by 2 from a square as long as t^{n} + 1/t^{n} is an integer. A trivial induction shows that it is enough for t + 1/t to be an integer. What is left to do is to pick a suitable quadratic equation with the first and the last term equal to 1, say x^{2} - 6x + 1, and declare t to be its largest root.
Today I present three problems from the 41-st Tournament of the Towns that I liked: an easy one, one that reminds me of the Collatz conjecture, and a hard one.
Problem 1 (by Aleksey Voropayev). A magician places all the cards from the standard 52-card deck face up in a row. He promises that the card left at the end will be the ace of clubs. At any moment, an audience member tells a number n that doesn't exceed the number of cards left in the row. The magician counts the nth card from the left or right and removes it. Where does the magician need to put the ace of clubs to guarantee the success of his trick?
Problem 2 (by Vladislav Novikov). Number x on the blackboard can be replaced by either 3x + 1 or ⌊x/2⌋. Prove that you can use these operations to get to any natural number when starting with 1.
Problem 3 (by A. Gribalko). There are 2n consecutive integers written on a blackboard. In one move, you can split all the numbers into pairs and replace every pair a, b with two numbers: a + b and a − b. (The numbers can be subtracted in any order, and all pairs have to be replaced simultaneously.) Prove that no 2n consecutive integers will ever appear on the board after the first move.
Here is a cute old problem that Facebook recently reminded me of.
Puzzle. By mistake, a clock-maker made the hour hand and the minute hand on a clock exactly the same. How many times a day, you can't tell the current time by looking at the clock? (It is implied that the hands move continuously, and you can pinpoint their exact location. Also, you are not allowed to watch how the hands move.)
Here is the solution by my son who was working on it together with my grandson.
The right way to think about it is to imagine a "shadow minute hand", like this: Start at noon. As the true hour hand advances, the minute hand advances 12 times faster. If the true minute hand were the hour hand, there would have to be a minute hand somewhere; call that position the shadow minute hand. The shadow minute hand advances 12 times faster than the true minute hand. The situations that are potentially ambiguous are the ones where the shadow minute hand coincides with the hour hand. Since the former makes 144 circuits while the latter makes 1, they coincide 143 times. However, of those, 11 are positions where the true minute hand is also in the same place, so you can still tell the time after all. So there are 132 times where the time is ambiguous during the 12-hour period, which leads to the answer: 268.
I love the problem and gave it to my students; but, accidentally, I used CAN instead of CAN'T:
Puzzle. By mistake, a clock-maker made the hour hand and the minute hand on a clock exactly the same. How many times a day can you tell the current time by looking at the clock?
Obviously, the answer is infinitely many times. However, almost all of the students submitted the same wrong finite answer. Can you guess what it was? And can you explain to me why?
Puzzle. What is the probability of getting five Mondays in a 31-days month?
This is easy if we assume that the day of the week for the first day of the month is chosen at random. But we should know better. What is the actual probability? Bonus question: for which day of the week the probability of having this day five times in a 31-days month is the highest?
A cute puzzle found on Facebook:
Puzzle. Four wizards A, B, C, and D, were given three cards each. They were told that the cards had numbers from 1 to 12 written without repeats. The wizards only knew their own three numbers and had the following exchange.
- A: “I have number 8 on one of my cards.”
- B: “All my numbers are prime.”
- C: “All my numbers are composite. Moreover, they all have a common prime factor.”
- D: “Then I know the cards of each of you.”
Given that every wizard told the truth, what cards does A have?
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I surveyed many people who had played Russian roulette. Seems like the probability of dying is actually 0%.
* * *
What has the probability of one in five million?
Zero: there's no 1 in 5000000. Only a five and six zeros.
* * *
Two classmates:
—What did you think of our probability exam yesterday?
—All means to an end.
* * *
My classmate didn't study for our test in probability.
"I'll take my chances", he said.
* * *
I saw my math teacher with a piece of graph paper yesterday. I think he must be plotting something.
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Not all math puns are terrible. Just sum.
* * * (submitted by Sergei Bernstein)
A programmer walks into a bar, holds up two fingers, and says, "I'll have three beers please."
* * *
What is the similarity between me and an experiment involving a biased coin with two tails?
The probability of getting a head is zero.
I've been crocheting hyperbolic surfaces of constant curvature. The process is time-consuming, so while I am crocheting, I wonder about the mathematics of crocheting.
Hilbert's theorem says that I can't embed a hyperbolic plane in 3-dimensional space. The proof is rather involved. But here, I have an explanation from the point of view of a crochet hook. My hook starts with a tiny cycle of four stitches. Then for every x stitches the hook makes y stitches in the next row, where y is greater than x. The extra stitches should be evenly distributed to guarantee that locally every small area is approximately isomorphic to other areas, meaning that the surface has a constant curvature.
The ratio of stitches in the next row to the current row is r = y/x. Thus, the number of stitches in each row increases exponentially. But each row is a fixed height h. That means after k rows, my thingy has to fit inside a ball of radius kh. But the length of the last row is 4r^{k-1}. It becomes huge very fast. As the last row is a physical curve made out of stitches, there is a limit of how much of it I can fit into a given volume, creating a contradiction.
That means, if I start crocheting, something should happen that won't allow me to continue. I decided to experiment and see what actually would happen. Being lazy, I preferred the disaster to happen sooner rather than later. So I chose the ratio of three: for each stitch on my perimeter, I added three new stitches. Shortly after I started to work, the process became more and more difficult. The ball was too tight. It was challenging to hold that thing in the place where I needed to insert the hook. And the loops were getting tighter, making it more exhausting to insert the hook into the proper hole. So each new stitch was taking more and more time to complete.
To my disappointment, the thing didn't explode, as I was secretly hoping: I just couldn't work on it anymore.
I like rewarding my students. Before covid, I used to give them star stickers for good ideas. When I started to teach remotely, I wondered what I should do instead. I could tell them that they had won a star, but it felt too weak. The next idea was to show them a star and tell them that it belonged to them. But that still felt insufficient. Then I had an epiphany. I would say to them they earned a star, show it to them, and stick it to my face. So they, and all the other students, would see it for the rest of the class. The photo shows how I looked at the end of a successful lesson.
Another picture shows what my MathRoots students posted on our Discord channel.
Now that I am back teaching in person, my students asked me to continue sticking their stars to my face. Sometimes I forget about the stars and, after my class, wander around MIT star-covered.
My son, Alexey Radul, wrote a program that finds the largest numbers to start a sequence in the Online Encyclopedia of Integer Sequences (OEIS). To my surprise, the top ten are all numbers consisting of ones only. The largest number is 11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111: a number with 128 ones. The sequence is A095646: a(n) is 128 written in base n. It starts with base 1, or more precisely, the unary expansion, which indeed requires 128 ones to express the number 128.
I decided to expand my top list to 50. Again, most of the numbers are bunches of ones: 48 out of the top 50 numbers are unary expansions of numbers 81 through 128. There are two more numbers in the top 50 that are different and belong to awesome sequences.
The first sequence is sequence A033290: Ten consecutive primes in arithmetic progression. It starts with the number 100996972469714247637786655587969840329509324689190041803603417758904341703348882159067229719, which has 93 digits. This number takes 37th place on my list.
The second sequence is sequence A291042: One powerful arithmetic progression with a nontrivial difference and maximal length. The sequence corresponds to a cool puzzle that appeared in the American Mathematical Monthly in 2000. The question was, "What is the length of the longest non-constant arithmetic progression of integers with the property that the kth term is a perfect kth power?" The answer is 5. John P. Robertson proved that such progression can't have 6 terms and provided an example of a sequence with 5 terms, which is the sequence in the OEIS.
Here is how to construct this sequence. Start with an arithmetic progression 1, 9, 17, 25, 33, and multiply each term by 3^{24}5^{30}11^{24}17^{20}: the result is also an arithmetic progression. The first term is trivially a first power. The second term is 3^{26}5^{30}11^{24}17^{20} = (3^{13}5^{15}11^{12}17^{10})^{2} and a square. The third term is 3^{24}5^{30}11^{24}17^{21} = (3^{8}5^{10}11^{8}17^{7})^{3} and a cube. The fourth term is 3^{24}5^{32}11^{24}17^{20} = (3^{6}5^{8}11^{6}17^{5})^{3} and a fourth power. The fifth term is 3^{25}5^{30}11^{25}17^{20} = (3^{5}5^{6}11^{5}17^{4})^{5} and a fifth power.
The sequence starts with the number 10529630094750052867957659797284314695762718513641400204044879414141178131103515625. It has 83 digits, and it takes 48th place on my list.
My students (Matvey Borodin, Eric Chen, Aidan Duncan, Boyan Litchev, Jiahe Liu, Veronika Moroz, Matthew Qian, Rohith Raghavan, Garima Rastogi, Michael Voigt) and I recently wrote a paper connecting the stable marriage problem and Sudoku. I just blogged about it. By the way, my students are in grades 7-9.
On the way, we invented a new type of Sudoku, which we call joint-groups Sudoku. This type is in contrast to a famous type of Sudoku, called disjoint-groups Sudoku. In a disjoint-groups Sudoku, in a particular place in a 3 by 3 box, all the digits are distinct across all the boxes. For example, the top-left corners of nine boxes have all the digits 1 thought 9. This creates nine additional disconnected regions (depending on the placements inside a 3 by 3 box) to add to columns, rows, and boxes that have to contain distinct digits.
For our new type, we wanted the digits in a particular place in each box, instead of being different, to be the same as much as possible. How much of sameness is possible? The first row contains three top-left corners. Thus, by Sudoku rules, these top-left corners have to be distinct. Thus, the top-left corners in all nine boxes have to contain at least three distinct digits. So here is the rule for the joint-groups Sudoku: the nine digits in a particular place in a 3 by 3 box contain not more than three distinct digits. It is easy to see that it means they contain exactly 3 distinct digits, each of them three times.
Here are two Sudoku puzzles from our paper. Each puzzle, when completed, forms a joint-groups Sudoku.
As you may know, I run PRIMES STEP, a local program where we do mathematical research with students in grades 6-9. Last academic year, we looked at the stable marriage problem and discovered its connection to Sudoku. Our paper The Stable Matching Problem and Sudoku (written jointly with Matvey Borodin, Eric Chen, Aidan Duncan, Boyan Litchev, Jiahe Liu, Veronika Moroz, Matthew Qian, Rohith Raghavan, Garima Rastogi, Michael Voigt) is now available at the arxiv.
Consider 3 men and 3 women who want to be married to each other in heterosexual couples. They rank each other without ties. The resulting 6 permutations of numbers 1, 2, and 3 that describe the six rankings are called the preference profile of this group of people. A matching is unstable if two people would be happier to run away together than to marry into the assigned couples. The two potential runaways are called a rogue couple. A matching is called stable if no rogue couple exists. The Gale-Shapley algorithm, invented by Gale and Shapley, finds a stable matching for any preference profile, implying that stable matching is always possible.
We discovered that preference profiles form a natural bijection with ways to place one digit into a Sudoku grid. So we wrote a paper describing the stable marriage, rogue couples, the Gale-Shapley algorithm, soulmates, and such in terms of Sudoku.
Oops, I forgot to explain who the soulmates are. We invented this term to describe two people who rank each other first. Though it is possible to have several stable matchings for the same preference profile if the soulmates exist, they must always be matched together.
We also invented a new Sudoku type, which I will explain next time.
I recently posted my puzzle designed for the MoMath meet-up.
What's in the Name?
Now it is time for the solution.
The solvers might recognize some sequences and numbers. For example, numbers 6, 28, and 496 are famous perfect numbers. Otherwise, the solvers are expected to Google the numbers and the pieces of the sequences with or without X. The best resource for finding the sequences is the Online Encyclopedia of Integer Sequence.
The first “AHA!” happens when the solvers notice that the sequences’ names are in alphabetical order. The order serves as a confirmation of the correctness of the names. It also helps in figuring out the rest of the sequences’ names. The alphabetical order in such types of puzzles hints that the real order is hidden somewhere else. It also emphasizes that the names might be important. The sequences names in order are:
The second “AHA!” moment happens when the solvers realize that the Xs all have different indices. The indices serve as the final order, which in this case is the following:
The third “AHA!” moment happens when the solvers realize that the number of terms is different in different sequences. It would have been easy to make the number of terms the same. This means that the number of terms has some significance. In fact, the number of terms in each sequence matches the length of the name of the sequence. The solvers then can pick the letter from each of the names corresponding to X. When placed in order, the answer reads: NUMBERS.
The answer is related to the puzzle in two ways:
The advantage of this puzzle for zoomed group events is that the big part of the job — figuring out the sequences — is parallelizable. Additionally, it has three “AHA!” moments, which means different people can contribute to a breakthrough. The puzzle also has some redundancy in it:
How can a square be square-free? In order for square-freeness to be interesting, it must be, and is, defined in terms of divisibility by non-trivial squares. So to create this particular mathematical oxymoron, one just needs a trivial square, namely 1.
I collect exciting properties of integers on my Number Gossip website. Did you know that forty is the only number whose letters appear in alphabetical order when written in English? Or that the largest amount of money one can have in US coins without being able to make change for a dollar is 119 cents?
Recently I wrote about a weird occasion that motivated me to search for new properties. Here is a sample of some amusing new updates.
This is the puzzle I designed for yesterday's event at the Museum of Mathematics. This puzzle is without instructions — figuring out what needs to be done is part of the fun. Solvers are allowed to use the Internet and any available tools. The answer to this puzzle is a word.
I've been too busy lately, so I stopped checking my Number Gossip website. Luckily, my website has fans. So one of them, Neil, notified me that my website was hijacked, and instead of describing properties of numbers, was selling steroids. I emailed Dreamhost, my hosting provider. They requested proof that I owned the domain. Why didn't they request proof from the people selling steroids? Or were they selling steroids themselves?
I fixed my steroid issue and since I was thinking about it anyway, I decided to update Number Gossip. I ended up spending tons of time on it — I had ten years of emails with suggestions for new properties, and I went through all of them and added the interesting ones. For example, Joshua Gray emailed me a cute property of 1331 mentioned on Wikipedia: 1331 was said to be the only cube of the form x^{2} + x − 1. I didn't see how to prove it, so I posted it as a question on mathoverflow. It turns out that 1331 is actually not the only cube of this form. There are three of them: −1 (for x = 0 or −1), 1 (for x = 1 or −2), and 1331 (for x = 36 or −37). So 1331 is the only non-trivial cube with this property. I had to fix Wikipedia. By the way, did you notice a symmetry? Plugging in x and −x − 1 into the quadratic produces the same value.
After processing all the emails related to Number Gossip, I got excited, so I continued working on it and added tons of new unique properties. Some of them I invented myself, some more were inspired by sequences in the OEIS database. I now have a collection of my new favorite unique properties, which I will post soon.
This is the sequence of numbers n such that 3 times the reversal of n plus 1 is the number itself. In other words, n = 3*reversal(n)+1. For example, 742 = 3*247+1. In fact, 742 is the smallest number with this property. How does this sequence continue, and why?
I recently published Sergei Bernstein's awesome Star Battle called Swiss Cheese. Another lovely Star Battle from him is called Hooks. You can play it online at puzz.link.
Star Battle is one of my favorite puzzle types. The rules are simple: put two stars in each row, column, and bold region (one star per cell). In addition, stars cannot be neighbors, even diagonally.
My son, Sergei Bernstein, recently designed a Star Battle with a beautiful solve path. This is my favorite Star Battle so far. I like its title too: Swiss Cheese.
You can also solve it at the puzz.link Star Battle player.
A familect is a portmanteau word formed by squashing together two words: family and dialect. It means words that a family uses that are not a part of a standard vocabulary. This story is about two words that my son Sergei invented, and twenty years later, my family still uses them.
As you might know, I was married three times, and I have two sons, from two different fathers. These fathers were also married several times and had other children. My two sons are half-brothers, and they also have half-siblings through their fathers. Thus a half-sister of a half-brother became a quarter-sister. Inventing this term was quite logical for a son of two mathematicians.
As you can imagine, our family tree is complicated. One day Sergei pointed out that our tree doesn't look like a standard tree and called it a family bush.
A problem from the 2021 Moscow Math Olympiad went viral on Russian math channels. The author is Dmitry Krekov.
Problem. Does there exist a number A so that for any natural number n, there exists a square of a natural number that differs from the ceiling of A^{n} by 2?
Yesterday I stumbled upon a picture of John Conway I completely forgot I had: it was saved in a wrong folder. The photo was taken at the MOVES conference in 2015. There is a third person in the original shot, but I do not remember her. I decided to cut her out as I didn't know how to contact her for permission to post this photo.
Konstantin Knop, the world's top authority on coin-weighing puzzles, suggested the following problem for the 2019 Russian Math Olympiad.
Puzzle. Eight out of sixteen coins are heavier than the rest and weigh 11 grams each. The other eight coins weigh 10 grams each. We do not know which coin is which, but one coin is conspicuously marked as an "Anniversary" coin. Can you figure out whether the Anniversary coin is heavier or lighter using a balance scale at most three times?
Found the following cute puzzle on Facebook.
Puzzle. Discover the rule governing the following sequence to find the next term of the sequence: 8, 3, 4, 9, 3, 9, 8, 2, 4, 3.
This is one of my favorite problems given at the 2017 Moscow Olympiad to grades 6 and 7. It was suggested by one of my favorite problem writers: Alexander Shapovalov.
Problem. We are given eight unit cubes. The third of the total number of their faces are blue, and the rest are red. We build a large cube out of these cubes so that exactly the third of the unit cube's visible faces are red. Prove that you can use these cubes to build a large cube whose faces are entirely red.
For the last year, I've been obsessed with Penney's game. In this game, Alice picks a string of coin tosses, say HHH for three heads. After that, Bob picks his string of tosses of the same lengths, say HTH. Then they toss a fair coin. The person whose string shows up first wins. For example, if the tosses are THTTHHH, then Alice wins after the seventh toss. For these particular choices, Bob wins with probability 3/5.
With my PRIMES student, Sean Li, we looked at this game and asked a different question. Suppose Alice picks a pattern of three tosses in a row that are the same. Suppose after that, Bob chooses a pattern of three alternating tosses. Then they toss a fair coin. Alice is hoping for HHH or TTT, while Bob is hoping for HTH or THT. The person whose pattern shows up first wins. For example, if the tosses are THTTHHH, then Bob wins after the third toss. For these particular choices, Bob wins with probability 1/2.
In this example, what actually happens. We assume that the group of two elements acts on the alphabet of two letters. The group's non-identity element swaps letters H and T. We assume that two strings are equivalent if they belong to the same equivalency class under the group action. We call such an equivalency class a pattern.
In the new game we invented, we have an alphabet of any size and any group acting on the alphabet. Then Alice and Bob pick their patterns. After that, they play the Penney's game on these patterns. The answers to all the relevant questions are in our paper, The Penney's Game with Group Action, posted at the math.CO arxiv 2009.06080.
I wrote a paper Confirming the Labels of Coins in One Weighing together with my PRIMES STEP students Isha Agarwal, Paul Braverman, Patrick Chen, William Du, Kaylee Ji, Akhil Kammila, Shane Lee, Alicia Li, Anish Mudide, Jeffrey Shi, Maya Smith, and Isabel Tu. The paper is available at math.HO arxiv:2006.16797. Below my students describe what happens in the paper in their own words.
Tragedy has struck in an iCOINic land known as COINnecticut. One day, while everyone was minding their own business, the vault door of the bank was found to have been forcefully opened. COINcerns spread amongst the COINmoners that someone had tampered with their n sacred COINtainers of coins! The COINtainers are labeled with the integers 1 through n, which usually describe the weight of each of the countless coins inside that corresponding COINtainer. For example, the COINtainer labeled 1 should only COINtain coins that weigh 1 gram, the COINtainer labeled 2 should only COINtain coins that weigh 2 gram, and so on, you get the COINcept.
The acCOINtants COINclude that someone may have switched around the labels on the COINtainers. To resolve this COINplication, aka to check if the labels have been tampered with, they bought a balance scale for a microsCOINpic amount of money. However, they can only use the scale to COINduct one weighing as the angry COINmoners are impatient and wish to withdraw their money ASAP.
The COINfused acCOINtants COINvinced 11 COINspicuous students from Boston's COINmunity to help them. With their COINbined efforts, they COINcluded that indeed, no matter how many COINtainers there are, their labels can be COINfirmed as correct or incorrect with just one weighing! Unfortunately, sometimes, such a weighing requires the use of many coins or coins with a large COINbined weight, which could potentially break the scale. Seeing this COINundrum, the students wished to be eCOINomical and find the least amount of coins or weight they need to place on the scale.
The acCOINtants and the 11 students COINtinued examining the nature of these weighings and discovered patterns that occur within them. They COINfined their research to special weighings they called downhill. They COINfirmed the effectiveness of such weighings to solve the problem at hand. The students estimated the weight and the number of coins, thus COINpleting their task.
My PRIMES STEP students invented several variations of Penney's game. We posted a paper about these new games at math.HO arxiv:2006.13002.
In Penney's game, Alice selects a string of coin-flip outcomes of length n. Then Bob selects another string of outcomes of the same length. For example, Alice chooses HHT, and Bob chooses THH. Then a fair coin is tossed until Alice's or Bob's string appears. The player whose string appears first wins. In our example, Bob has a greater probability of winning, namely, 3/4. If the first two flips are HH, then Alice wins; otherwise, Bob wins.
The reader can check that HHT beats HTT with 2 to 1 odds. Thus, the game contains a non-transitive cycle it is famous for: THH beats HHT beats HTT beats TTH beats THH.
I already wrote about the No-Flippancy game that my students invented. It starts with Alice and Bob choosing different strings of tosses of the same length.
However, in the No-Flippancy game, they don't flip a coin but select a flip outcome deterministically according to the following rule: Let i ≤ n be the maximal length of a suffix in the sequence of "flips" that coincides with a prefix of the current player's string. The player then selects the element of their string with index i + 1 as the next "flip." Alice goes first, and whoever's string appears first in the sequence of choices wins.
My favorite game among the invented games is the Blended game, which mixes the No-Flippancy game and Penney's game.
In the game, they sometimes flip a coin and sometimes don't. Alice and Bob choose their strings as in Penney's game and the No-Flippancy game. Before each coin flip, they decide what they want by the rule of the No-Flippancy game above. If they want the same outcome, they get it without flipping a coin. If they want different outcomes, they flip a coin. Whoever's string appears first in the sequence of `flips' wins.
For example, suppose Alice selects HHT, and Bob selects THH. Then Alice wants H and Bob wants T, so they flip a coin. If the flip is T, then they both want Hs, and Bob wins. If the first flip is H, they want different things again. I leave it to the reader to see that Bob wins with probability 3/4. For this particular choice of strings, the odds are the same as in Penney's game, but they are not always the same.
This game has a lot of interesting properties. For example, similar to Penney's game, it has a non-transitive cycle of choices. Surprisingly, the cycle is of length 6: THH beats HHT beats THT beats HTT beats TTH beast HTH beat THH.
I spend a lot of time working on my blog, and I used to think it would be nice to get some money out of it.
Years ago I got two emails from different ad agencies at the same time. They wanted to place ads at particular essays for $50 a year. I decided to give them a try.
My first correspondent wanted a link on my "Does Alcohol in Teens Lead to Adult Woes?" essay, connecting to a website offering help to alcoholics. I agreed. But when I read the actual text, I couldn't stop laughing. The text they wanted to use was, "Many studies have already claimed that teenage alcoholism could lead to more problems later in life." How ironic! This ad would follow my essay explaining that one of the studies is completely bogus. I rejected them.
The second agency wanted an ad accompanying the essay "Subtraction Problems, Russian Style." I placed it. They wrote to me (and I reproduce it with all of their errors intact):
I really appreciate your efforts on this. As I checked the text link, I have seen that the text link has been label as "Sponsor ad". Kindly omit or delete the word "Sponor ad:" or you may changed it to "Recommended site or Relevant Site" but I would love to prefer the text link be seen as natural meaning no labeled inserted on it.
They wanted me to pretend that I recommend their product. I was naive enough to think that I was selling space on my page, but what they really wanted was for me to lie that I like their product.
Before this experiment, I hoped to find some honest ads for my blog. After this experiment, I realized how much stupidity and falsehood are involved. Since then, I ignore all offers of ads that come my way. That's why my blog is ad-free.
My STEP students invented a coin-flipping game that doesn't require a coin. It is called The No-Flippancy Game.
Alice and Bob choose distinct strings of length n consisting of the letters H (for heads) and T (for tails). The two players alternate selecting the outcome of the next "flip" to add to the sequence by the rule below.
The "flip" rule: Let i < n be the maximal length of a suffix of the sequence of chosen outcomes that coincides with a prefix of the current player’s string. The player then selects the element of their string with index i + 1 as the next term in the sequence.
Alice goes first, and whoever's string appears first in the sequence of choices wins. In layman terms, the game rules mean that the players are not strategizing, but rather greedily finishing their strings.
Suppose n = 2 and Alice chose HH. If Bob chooses HT, then Bob wins. Alice has to choose H for the first flip. Then Bob chooses T and wins. On the other hand, if Bob chooses TT for his string, the game becomes infinite. On her turn, Alice always chooses H, while on his turn Bob always chooses T. The game outcome is an alternating string HTHTHT... and no one wins.
Suppose n = 4, Alice chooses HHTT, and Bob chooses THHH. The game proceeds as HTHHTHHH, at which point Bob wins.
This game is very interesting. The outcome depends on how Alice's and Bob's chosen strings overlap with each other. We wrote a paper about this game, which is available at math.CO arXiv:2006.09588.
Every year I write about latest MIT Mystery Hunt puzzles that might be appealing to mathematicians. Before diving into mathy puzzles, I would like to mention two special ones:
Unfortunately math wasn't prominent this year:
On the other hand, Nikoli-type puzzles were represented very well:
Some computer sciency puzzles:
Cryptography:
A couple of puzzles with the mathy side hidden:
The academic year is over and my junior PRIMES STEP group finished their paper about a classification of magic SET squares. A magic SET square is a 3 by 3 square of SET cards such that each row, column, and diagonal is a set. See an example below. The paper is posted at the arXiv:2006.04764.
In addition to classifying the magic SET squares, my students invented the game of SET tic-tac-toe. It is played on nine cards that form a magic SET square. Two players take turns picking a card from the square. The first player who has a set wins.
One might think that this game is the same as tic-tac-toe, as a player wins as soon at they have cards from the same row, column, or diagonal. But if you build a magic SET square, you might notices that each magic SET square contains 12 sets. In addition to rows, columns, and diagonals, there are sets that form broken diagonals. The picture below shows all the sets in a magic SET square.
There are more ways to win in this game than in a regular tic-tac-toe game. My students proved that ties are impossible in this game. They also showed, that, if played correctly, the first player always wins.
You probably heard in the news that more men are dying from coronavirus than women. But not in Massachusetts. Here the proportion of women is about 52 percent. Why is this the case? Being a woman, should I be worried that I live in Massachusetts?
We know that coronavirus strikes older people harder than younger ones. Thus, we should take age into account. In the US more boys are born than girls. By the age of 40 the ratio evens out. Starting from 40 there are more women than men. With each next age group, the disparity increases. According to a recent US population report and for ages 85 and over there are about 4.22 million women versus 2.33 men: the proportion is almost 2 to 1.
As the coronavirus targets older people, were it gender-neutral, we would have had way more female deaths than male. This is not the case. So it hits males harder than females. But why are the ratios of female to male deaths different for different countries and states?
One simple explanation is that this is related to life expectancy and the age of the population. The older the population, the bigger the percentage of females. Which in turn increases the proportion of female deaths.
It could also be that Massachusetts has good health care making the average age of dying patients older than the average age for the country. This in turn will increase the proportion of females dying from coronavirus. No, I am not worried about living in Massachusetts.
I grew up in the USSR, where I was clueless about the race issues in the US. I have now lived in the US for 30 years, and still feel that there are many things about race that I do not understand. As a result, I am afraid to speak about it. I am worried that I'll say something wrong. Recent events have encouraged me to say something. This is my first piece about race.
I came back to mathematics 10 year ago and started working at MIT. I love it. With some exceptions.
Many mathematicians are introverts or snobs or gender-biased. They are not usually friendly. I often walk down a corridor and people who are coming towards don't notice me. If I say hello, they might not even reply or raise their eyes. It could be they are thinking about their next great theorem and do not notice me. It could be that I am not faculty and therefore do not deserve their attention. It could be that as a women I am not worth of their hello.
Soon after I started working at MIT, I was reminded of one of the reasons I left academia. It was this unfriendliness. But this time was different. First, I had grown a thicker skin. Second, I was working within a group. People who were working with me were nice to me. It was enough and so I stayed.
With time I adopted the same style: passing people without saying `Hello.' Mostly I got tired of people not replying to my hello.
One day I was passing this man who, as had happened many times before, purposefully didn't look at me. I thought my usual thought: another introverted/snobbish/gender-biased mathematician. Then I suddenly stopped in my tracks. My logic was wrong. This guy was Black. The unfriendliness of mathematicians is surely way worse for him than for me. It could be that he is looking at the floor for the same reason I do it: he is afraid that people will ignore his greeting. I failed to think about race deep enough before this realization. What happened next should have happened years earlier.
I took the initiative and the next couple of times I saw him, I said hello. This was all it took—two hellos—to change the whole feeling between us. The guy has a great smile.
It was reported last week that that 37 NYPD members died of covid-19. I assume that they were way below 65. It is known that the coronovirus death rate for people below 65 is a quarter of the total death rate. That means, 37 people in NYPD correspond to at least 150 people in general. Assuming that the mortality rate of coronavirus is 1 percent, the number of infected NYPD members a month ago was 15000.
By now, it could be that more than half of NYPD was infected.
NYPD members have to communicate with people a lot due to the nature of their work. That means they are more prone to being infected. At the same time, they transmit more than people in many other professions.
I can conclude, that about half of the people that are high transmitters in NY have antibodies by now. Assuming they are immune, the covid transmission rate in NY has to be down.
Assuming the immunity stays with people for a while, the second wave in NY can't be as bad as the first one.
Suppose we want to pack a unit square with non-overlapping rectangles that have sides parallel to the axes. The catch is that the lower left corners of all the rectangles are given. By the way, such rectangles are called anchored. Now, given some points in the unit square, aka the lower left corners, we want to find anchored rectangles with the maximum total area.
When the given points are close to the right upper corner of the square, the total area is small. When a single point is in the bottom left corner of the square, we can cover the whole square. The problem becomes more interesting if we add one extra assumption: one of the given points has to be the bottom left corner of the square. In the 1960's, it was conjectured by Allen Freedman that any set of points has an anchored rectangle packing with the area of at least one half. The problem is quite resistant. In 2011, Dumitrescu and Tóth showed that every set of points has a packing of area at least 0.09, which was the first constant bound found, and is the best bound currently known.
I gave this problem to my PRIMES student Vincent Bian. He wrote a paper, Special Configurations in Anchored Rectangle Packings, that is now available at the arxiv. When you look at this problem you see that the number of ways to pack depends on the relative coordinates of the points. That means you can view the points as a permutation. Vincent showed that the conjecture is true for several different configurations of points: increasing, decreasing, mountain, split layer, cliff, and sparse decreasing permutations.
An increasing permutation is easy. There are two natural ways to pack the rectangles. One way, when rectangles are horizontal and each rectangle reaches to the right side of the square (see picture above). Another way, when rectangles are vertical. When you take the union of both cases, the square is completely covered, which means at least one of the cases covers at least half of the square. The worst case scenario, that is, the case when the maximum possible area is the smallest is when your points are placed equidistantly on the diagonal.
Other cases are more difficult. For example, Vincent showed that for a decreasing permutation with n points, the worst case scenario is when the points are arranged equidistantly on a hyperbola xy = (1-1/n)^{n}. The picture shows the configuration for 15 points. The total area is more than one half.
(I wrote this piece for La Recherche. It was translated into French by Philippe PAJOT. You can find this piece and pieces by others at John Horton Conway: a magician of maths disappears.)
Unlike many other mathematicians I know, John Conway cared a lot about the way he presented things. For example, in the puzzle he invented—known as Conway's Wizards—the wizards had to be riding on a bus. Why was the bus so important? You see, the numbers in the puzzle were related to the age of one of the wizards, the number of the bus, and the number of the wizard's children. It was important to John that the readers be able to use a convenient notation a, b and c for these numbers and remember which number is which.
When I give my lecture on integers and sequences, I show my students a list of different famous sequences. The first question from the audience is almost always: "What are the Evil Numbers?" As you can guess the name for this sequence was invented by John Conway. This name was invented together with the name of another sequence which is called Odious Numbers. These two sequences are complementary in the same sense as even and odd numbers are complementary: every natural number is either evil or odious. The names are good, not only because they attract, but also because they help remember what the sequences are. Evil numbers are numbers with an even number of ones in their binary representation. I assume that you can interpolate what the odious numbers are.
When he was lecturing, John used all sorts of tricks to emphasize important points: From time to time I saw him shouting or throwing his shoes. Once I remember him staring at his statement written on the blackboard for a really long time. My neighbor in the lecture hall got uncomfortable. He assumed that John, who was at that time way over 70, was blanking out and had forgotten what he wanted to say. I calmed my neighbor down. It was my fourth time listening to the same lecture, including the same pause. John Conway didn't forget.
The picture was taken on December 21 of 2019 at Parker Life care facility right before dinner.
Every year I review MIT mystery hunt from a mathematician's point of view. I am way behind. The year is 2020, but I still didn't post my review of 2019 hunt. Here we go.
Many puzzles in 2019 used two data sets. Here is the recipe for constructing such a puzzle. Pick two of your favorite topics: Star Trek and ice cream flavors. Remember that Deanna Troi loves chocolate sundae. Incorporate Deanna Troi into your puzzle to justify the use of two data sets.
On one hand, two data sets guarantee that the puzzle is new and fresh. On the other hand, often the connection between two topics was forced. Not to mention that puzzle solving dynamic is suboptimal. For example, you start working on a puzzle because you recognize Star Trek. But then you have to deal with ice cream which you hate. Nonetheless, you are already invested in the puzzle so you finish it, enjoying only one half of it.
Overall, it was a great hunt. But the reason I love the MIT mystery hunt is because there are a lot of advanced sciency puzzles that can only appear there. For example, there was a puzzle on Feynman diagrams, or on characters of representations. This year only one puzzle, Deeply Confused, felt like AHA, this is the MIT Mystery hunt.
Before discussing mathy puzzles I have to mention that my team laughed at Uncommon Bonds.
I will group the puzzles into categories, where the categories are obvious.
Mathy puzzles.
Here are some logic puzzles, in a sense that Sudoku is a logic puzzle.
Now we have logic puzzles or another type, where you need to draw a grid. These are puzzles of the type: Who lives in the White House?
Now we have logic puzzles or yet another type, where you need to figure out which statements are true and which are false.
Now some cryptography.
And some programming.
Miscellaneous.
Every day I check coronavirus numbers in the US. Right now the number of deaths is 288 and the number of recovered is 171. More people died than recovered. If you are scared about the mortality rate, I can calm you and myself down: our government is incompetent—the testing wasn't happening—that means the numbers do not show people who had mild symptoms and recovered. The real number of recovered people should be much higher.
Scientists estimated the mortality rate of coronavirus as being between 1 and 3.5 percent. Also, they say that it usually takes three weeks to die. That means three weeks ago the number of infected people in the US was between 8,000 and 29,000. The official number of cases three weeks ago was 68. I am panicking again—our government is incompetent—three weeks ago they detected between 0.25 and 1 percent of coronavirus cases. If this trend continues, then the official 19,383 infected people as of today means, in reality, somewhere between 2 million and 8 million infected people.
I can calm you and myself down: the testing picked up pace. This means, the ratio of detected cases should be more than 1 percent today. Probably the number of infected people today in the US is much less than 8 million. I am not calm.
My former student, Dai Yang, sent me the following cute puzzle:
Puzzle. You are playing a game with the Devil. There are n coins in a line, each showing either H (heads) or T (tails). Whenever the rightmost coin is H, you decide its new orientation and move it to the leftmost position. Whenever the rightmost coin is T, the Devil decides its new orientation and moves it to the leftmost position. This process repeats until all coins face the same way, at which point you win. What's the winning strategy?
My friend Zeb, aka Zarathustra Brady, invented a new game that uses chess pieces and a chessboard. Before the game, the players put all chess pieces on white squares of the board: white pieces are placed in odd-numbered rows and black pieces are in even-numbered rows. At the beginning all white squares are occupied and all black squares are empty. As usual white starts.
On your turn, you can move your piece from any square to any empty square as long as the number of enemy neighbors doesn't decrease. The neighbors are defined as sharing a side of a square. Before the game starts each piece has zero enemy neighbors and each empty square has at least one white and one black neighbor. That means that on the first turn the white piece you move will increase the number of neighbors from zero to something. As usual, the player who doesn't have a move loses.
As you can immediately see, that number of pairs of enemy neighbors is not decreasing through the game. I tried to play this game making a move which minimizes the increase of the pairs of neighbors. I lost, twice. I wonder if there is a simple strategy that is helpful.
It is important that this game is played with chess pieces in order to confuse your friends who pass by. You can see how much time it takes them to figure out that this game is not chess, but rather a Chessnot. Or you can enjoy yourself when they start giving you chess advice before realizing that this is not chess, but rather a Chessnot.
I heard this puzzle many years ago, and do not remember the origins of it. The version below is from Peter Winkler's paper Seven Puzzles You Think You Must Not Have Heard Correctly.
Puzzle. Jan and Maria have fallen in love (via the internet) and Jan wishes to mail her a ring. Unfortunately, they live in the country of Kleptopia where anything sent through the mail will be stolen unless it is enclosed in a padlocked box. Jan and Maria each have plenty of padlocks, but none to which the other has a key. How can Jan get the ring safely into Maria's hands?
I don't know whether this puzzle appeared before the Diffie-Hellman key exchange was invented, but I am sure that one of them inspired the other. The official solution is that Jan sends Maria a box with the ring in it and one of his padlocks on it. Upon receipt Maria affixes her own padlock to the box and mails it back with both padlocks on it. When Jan gets it, he removes his padlock and sends the box back, locked only with Maria's padlock. As Maria has her own key, she can now open it.
My students suggested many other solutions. I wonder if some of them can be translated to cryptography.
Now that we've looked at the Padlock Puzzle, let's talk about cryptography. I have an imaginary student named Charlie who doesn't know the Diffie-Hellman key exchange. Charlie decided that he can adapt the padlock puzzle to help Alice send a secret message to Bob. Here's what Charlie suggests:
Suppose the message is M. Alice converts it to binary. Then she creates a random binary key A and XORs it with M. She sends the result, M XOR A, to Bob. Then Bob creates his own random key B and XORs it with what he receives and sends the result, M XOR A XOR B, back to Alice. Alice XORs the result with her key to get M XOR A XOR B XOR A = M XOR B and sends it to Bob. Bob XORs it with his key to decipher the message.
Each sent message is equivalent to a random string. Intercepting it is not useful to an evil eavesdropper. The scheme is perfect. Or is it?
Here is a logic puzzle.
Puzzle. You are visiting an island where all people know each other. The islanders are of two types: truth-tellers who always tell the truth and liars who always lie. You meet three islanders—Alice, Bob, and Charlie—and ask each of them, "Of the two other islanders here, how many are truth-tellers?" Alice replies, "Zero." Bob replies, "One." What will Charlie's reply be?
The solution proceeds as follows. Suppose Alice is a truth-teller. Then Bob and Charlie are liars. In this situation Bob's statement is true, which is a contradiction. Hence, Alice is a liar. It follows, that there is at least one truth-teller between Bob and Charlie. Suppose Bob is a liar. Then the statement that there is one truth-teller between Alice and Charlie is wrong. It follows that Charlie is a liar. We have a contradiction again. Thus, Alice is a liar and Bob is a truth-teller. From Bob's statement, we know that Charlie must be a truth-teller. That means, Charlie says "One."
But here is another solution suggested by my students that uses meta considerations. A truth-teller has only one possibility for the answer, while a liar can choose between any numbers that are not true. Even if we assume that the answer is only one of three numbers—0, 1, or 2—then the liar still has two options for the answer. If Charlie is a liar, there can't be a unique answer to this puzzle. Thus, the puzzle question implies that Charlie is a truth-teller. It follows that Alice must be lying and Bob must be telling the truth. And the answer is the same: Charlie says, "One."
You might have noticed that my blogging slowed down significantly in the last several months. I had mono: My brain was foggy, and I was tired all the time. Now I am feeling better, and I am writing again. What better way to get back to writing than to start with some jokes?
* * *
The wife of a math teacher threw him out from point A to point B.
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At the job interview at Google.
—How did you hear about our company?
* * * (submitted by Sam Steingold)
50% of marriages end with divorce. The other 50% end with death.
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People say that I am illogical. This is not so, though this is true.
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Humanity invented the decimal system, because people have 10 fingers. And they invented 32-bit computers, because people have 32 teeth.
* * *
When a person tells me, "I was never vaccinated, and, as you can see, I am fine," I reply, "I also want to hear the opinion of those who were never vaccinated and died."
* * *
I will live forever. I have collected a lot of data over the years, and in all of the examples, it is always someone else who dies.
* * *
Just got my ticket to the Fibonacci convention! I hear this year is going to be as big as the last two years put together.
* * *
I am afraid to have children as one day I will have to help them with math.
This is my favorite puzzle in the last issue of the Emissary, proposed by Peter Winkler.
Puzzle. Alice and Bob each have 100 dollars and a biased coin that flips heads with probability 51%. At a signal, each begins flipping his or her coin once per minute, and betting 1 dollar (at even odds) on each flip. Alice bets on heads; poor Bob, on tails. As it happens, however, both eventually go broke. Who is more likely to have gone broke first?
Follow-up question: As above, but this time Alice and Bob are flipping the same coin (biased 51% toward heads). Again, assume both eventually go broke. Who is more likely to have gone broke first?
Puzzle. You got two envelopes with two distinct real numbers. You chose one of them and open it. After you see the number you are allowed to swap envelopes. You win if at the end you pick the larger number. Find a strategy that gives you a probability more than 1/2 of winning.
Another cute puzzle found on Facebook.
Puzzle. A teacher wrote four positive numbers on the board and invited his students to calculate the product of any two. The students calculated only five of six products and these are the results: 2, 3, 4, 5, 6. What is the last product? What are the original four numbers?
I found this puzzle on Facebook:
Puzzle. Solve this:
1+4 = 5,
2+5 = 12,
3+6 = 21,
5+8 = ?
97% will fail this test.
Staring at this I decided on my answer. Then I looked at the comments: they were divided between 34 and 45 and didn't contain the answer that initially came to my mind. The question to my readers is to explain the answers in the comments and suggest other ones. Can you guess what my answer was?
Puzzle. The length of the day today in Boston is the same as the length of the coming night tonight. What is the total length of both?
My friend Alice reminds me of me: she has two sons and she is never straight with her age. Or, maybe, she just isn't very good with numbers.
Once I visited her family for dinner and asked her point blank, "How old are you?" Here is the rest of the conversation:
Alice: I am two times older than my younger son was 5 years ago.
Bob: My mom is 12 times older than my older brother.
Carl: My younger brother always multiplies every number he mentions by 24.
Bob: My older brother is 30 years older than me.
Carl: My mom is 8 times older than me.
Alice: My older son always multiplies every number he mentions by 2.
How old is everyone?
Last year, when I read an application file of Wayne Zhao to PRIMES, I got very excited because he liked puzzles. And I've always wanted to have a project about puzzles. After Wayne was accepted to PRIMES we started working together. Wayne chose to focus on a variation of Sudoku called Sudo-Kurve.
We chose a particular shape of Sudo-Kurve for this project, which ended up being very rewarding. It is called Cube Sudo-Kurve. The Cube Sudo-Kurve consists of three square blocks. The gray bent lines indicate how rows and columns continue. For example, the first row of the top left block becomes the last column of the middle block and continues to the first row of the bottom right block. As usual each row, column, and square region has to have 9 distinct digits.
Wayne and I wrote a paper Mathematics of a Sudo-Kurve, which has been published at Recreational Mathematics Magazine.
A Cube Sudo-Kurve needs at least 8 clues to have a unique solution. Here we have a puzzle with 8 clues that we designed for our paper.
I've been invited to help with the Puzzle Column at the MSRI newsletter Emissary. We prepared six puzzles for the Fall 2018 issue.
I love the puzzles there. Number 2 is a mafia puzzle that I suggested. Number 6 is a fun variation on the hat puzzle I wrote a lot about. Here is puzzle Number 3.
Puzzle. Let A = {1,2,3,4,5} and let P be the set of all nonempty subsets of A. A function f from P to A is a "selector" function if f(B) is in B, and f(B union C) is either equal to f(B) or f(C). How many selector functions are there?
When I was in grade school, one of the teachers called me Cave Lioness. She hated my unruly hair, which reminded her of a lion's mane. This teacher was obviously very uninformed, for female lions do not have manes.
This name calling had the opposite to the desired effect. I became proud of my mane and didn't ever want to cut it. When I grew older, I opted for convenience and started to cut my hair short&mdahs;sometimes very short.
Last year I was too busy for barbers, and my hair grew more than I intended. As it turned into a mane, I remembered the story of this nickname.
Once I was giving a math lecture and my phone, which I've never quite understood, was on the desk in front of me. Suddenly it rang. I didn't pick it up, as I proceeded with my lecture. The ringing stopped, while I was explaining a particularly interesting mathematical point. After a minute, my phone said, "I do not understand a word you are saying."
Detective Radstein is investigating a robbery. He apprehends three suspects: Anne, Bill, and Caroline. The detective knows that no one else could have participated in the robbery. During the interrogation the suspects make the following statements:
Detective Radstein also discovered that all three suspects are members of a club called The Halfsies. Every time they speak, they make two statements, one of which is a lie and the other one is true. Who committed the robbery?
I already wrote about my first experience crocheting hyperbolic surfaces. In my first surface I added two more stitches per current stitch. It took me hours to crochet the last row: the same hours it took me to crochet the rest.
For my next project, I reduced the ratio. The light blue thingy has ratio 3/2. I continued making my life simpler. The next project, the purple surface on the left, has ratio 4/3. The last project on the right has a ratio of 5/4 and is my favorite. Mostly because I am lazy and it was the fastest to make.
How much time will it take you to answer the following question?
Can the equation 29x + 30y + 31z = 366 be solved in natural numbers?
Happy 2019, the first 4 digit number to appear 6 times in the decimal expansion of Pi.
By the way, 2019 is the product of two primes 3 and 673. The sum of these two prime factors is a square.
This is not all that is interesting about factors of 2019. Every concatenation of these two prime factors is prime. Even more unusual, 2019 is the largest known composite number such that every concatenation of its prime factors is prime. [Oops, the last statement is wrong, Jan 3,2019]
Happy Happy-go-Lucky year, as 2019 is a Happy-go-Lucky number: the number that is both Happy and Lucky.
In case you are wondering, here is the definition of Happy numbers: One can take the sum of the squares of the digits of a number. Those numbers are Happy for which iterating this operation eventually leads to 1.
In case you are wondering, to build the Lucky number sequence, start with natural numbers. Delete every second number, leaving 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, …. The second number remaining is 3, so delete every third number, leaving 1, 3, 7, 9, 13, 15, 19, 21, …. The next number remaining is 7, so delete every 7th number, leaving 1, 3, 7, 9, 13, 15, 21, …. The next number remaining is 9, so delete every ninth number, etc.
Alex Bellos sent me his new book Puzzle Ninja: Pit Your Wits Against The Japanese Puzzle Masters. What has he done to me? I opened the book and couldn't close it until I solved all the puzzles.
This is a fantastic book. There are many varieties of puzzles, including some types that I've never seen before. Also, the beautifully designed puzzles are great. Often puzzles of the same type target different solving ideas or have varied cool themes.
This book is more than a bunch of puzzles; it also contains poetic stories about puzzle histories and Japanese puzzle designers. Fantastic puzzles together with a human touch: this might be my favorite puzzle book.
I present two puzzles from the book. The puzzle type is called Wolf and Sheep Slitherlink. The Slitherlink is a famous puzzle type with the goal of connecting some of the neighboring dots into a single non-self-intersecting loop. A number inside a small square cell indicates how many sides of the square are part of the loop. Wolf and Sheep Slitherlink is a variation of Slitherlink in which all sheep should be kept inside the fence (loop) and all the wolves outside.
Ignore the numbers in the title as they just indicate the order number of Wolf and Sheep Slitherlink puzzles in the book. The number of ninja heads shows the level of difficulty. (The hardest puzzles in the book have four heads.) The difficulty is followed by the name of the puzzle master who designed the puzzle.
The first puzzle above is slightly easier than the second. I like the themes of these two puzzles. In the first one, only one cell—lonely wolf—marks the relationship to the fence. In the second one, the wolf in the center—who needs to be outside the fence—is surrounded by a circle of sheep who are in turn surrounded by a circle of wolves.
My friend Alex Ryba uses interesting math questions in the CUNY Math Challenge. For the 2016 challenge they had the following problem.
Problem. Eve owns two six-sided dice. They are not necessarily fair dice and not necessarily weighted in the same manner. Eve promises to give Alice and Bob each a fabulous prize if they each roll the same sum with her dice. Eve wishes to design her two dice to minimize the likelihood that she has to buy two fabulous prizes. Can she weight them so that the probability for Alice and Bob to get prizes is less than 1/10?
The best outcome for Eve would be if she can weight the dice so that the sum is uniform. In this case the probability that Alice and Bob get the prizes is 1/11. Unfortunately for Eve, such a distribution of weight for the dice is impossible. There are many ways to prove it.
I found a beautiful argument by Hagen von Eitzen on the stack exchange: Let a_{i} (correspondingly b_{i}) be the probabilities that die A (correspondingly B) shows i + 1. It would be very useful later that that i ranges over {0,1,2,3,4,5} for both dice. Let f(z) = ∑ a_{i}z^{i} and g(z) = ∑ b_{i}z^{i}. Then the desired result is that f(z)g(z) = ∑_{j=0}^{10} z^{j}/11. The roots of the right side are the non-real roots of unity. Therefore both f and g have no real roots. So, they must both have even degree. This implies a_{5}=b_{5}=0 and the coefficient of z^{10} in their product is also 0, contradiction.
Alex himself has a straightforward argument. The probabilities of 2 and 12 have to be equal to 1/11, therefore, a_{0}b_{0} = a_{5}b_{5} = 1/11. Then the probability of a total 7 is at least a_{0}b_{5} + a_{0}b_{5}. The geometric mean of a_{0}b_{5} and a_{0}b_{5} is 1/11 (from above), so their arithmetic mean is at least 1/11 and their sum is at least 2/11. Therefore, the uniform distribution for sums is impossible.
So 1/11 is impossible, but how close to it can you get?
I just got this picture from my friend Victor Gutenmacher, which I never saw before. My 1975 IMO team is posing at our training grounds before the Olympiad trip to Bulgaria.
Left to right: Boris Yusin, Yuri Ionin, Zoya Moiseyeva (front), Gregory Galperin (back), me, Ilya Yunus, Valentin Skvortsov, Aleksandr Kornyushkin, Sergei Finashin, Sergei Fomin (front), Alexander Reznikov (back), Yuri Shmelev (front), Yuri Neretin (back), Victor Gutenmacher.
Our coaches are in the shot as well. Surprisingly, or not surprisingly, all of them moved to the USA. Yuri Ionin, now retired, was a professor at Central Michigan University. Gregory Galperin is a professor at Eastern Illinois University. Sergei Fomin is a professor at the University of Michigan. Victor Gutenmacher worked for BBN Technologies and Siemens PLM Software, and is now retired.
There are two more adults in the picture: Valentin Anatolievich Skvortsov, our leader and Zoya Ivanovna Moiseyeva, our deputy leader. Skvortsov was working at the math department of Moscow State University at that time. The University was angry that he didn't block some students with Jewish heritage from the team thus allowing them to be accepted to Moscow State University without exams. I wrote a story of how Zoya persuaded Alexander Reznikov not to go to Moscow University to help Valentin. It ruined Alexander's live, and didn't even help Valentin. 1975 was Valentin's last trip as the leader.
An orchestra of 120 players takes 70 minutes to play Beethoven's 9th Symphony. How long would it take for 60 players to play the symphony?
I do not like making objects with my hands. But I lived in Soviet Russia. So I knew how to crochet, knit, and sew — because in Russia at that time, we didn't have a choice. I was always bad at it. The only thing I was good at was darning socks: I had to do it too often. By the way, I failed to find a video on how to darn socks the same way my mom taught me.
Then I came to the US. I suddenly found myself in a rich society, where it was cheaper to buy new stuff than to spend the time doing things with my hands. So I happily dropped my craftsmanship.
After not working with my hands for 28 years, one day I needed hyperbolic surfaces for my classes and I couldn't find any to buy. Hyperbolic surfaces are famous for providing an example when the Euclid's Fifth axiom doesn't work. These hyperbolic surfaces look flat locally, so you can continue a line in any given direction. If you draw a line on such a surface and pick a point that is not on the line, then you can draw many lines through the point that are parallel to the given line.
My students are more important than my dislike of crochet, so I decided to just do it myself. I asked my friend Debbie, who knows how to crochet, for advice, and she gave me more than advice. She gave me a hook and a piece of yarn and reminded me how to work the hook. She started me with a small circle. After that all I had to do was add two stitches for each stitch on the perimeter of the circle. The finished product is this green ballish thing that looks like a brain, as in the photo.
Outside the starting circle, each small surface segment of this "brain" looks the same, making the "brain" a surface of constant curvature.
I chose a ratio of 2 to 1, adding two new stitches for each previous stitch. With this ratio, my flattish surface started looking like a ball very fast. The length of the perimeter doubled for every row. Thus each new row I crocheted took the same total amount of time that I had already spent on the whole thing. All the hours I worked on this "brain," I kept thinking: darn, it is so unrewarding to do this. Annoying as it was, the thing that kept me going was my initial decision to continue to use up all the yarn Debbie had given me. In the end, with this ratio, half the time I worked was spent making the final row.
We can inflate one permutation with another permutation. Let me define the inflation by examples and pictures. Suppose we have a permutation 132 which we want to inflate using permutation 21. The result is the permutation 216543 that can be divided into three blocks 21|65|43. The blocks are ordered as the first permutation 132, and within each block the order is according to the second permutation. This operation is often called a tensor product of two permutations. The operation is non-commutative: the inflation of 21 with 132 is 465132.
As one might guess this post is related to k-symmetric permutations, that is, permutations that contain all possible patterns of size k with the same frequency. As I mentioned in my recent post 3-Symmetric Permutations, the smallest non-trivial examples of 3-symmetric permutations are 349852167 and 761258943 in size 9.
A permutation is called k-inflatable if its inflation with k-symmetric permutation is k-symmetric. One of my PRIMES projects was about 3-inflatable permutations. The result of this project is the paper On 3-Inflatable Permutations written with Eric Zhang and posted at the arxiv.
The smallest non-trivial examples of 3-inflatable permutations are in size 17: E534BGA9HC2D1687F and B3CE1H76F5A49D2G8, where capital letters denote numbers greater than nine. Another cool property discovered in the paper is that the tensor product of two k-inflatable permutations is k-inflatable.
I am fascinated by 3-symmetric permutations, that is, permutations that contain all possible patterns of of size three with the same frequency. As I mentioned in my recent post 3-Symmetric Permutations, the smallest non-trivial examples are in size 9.
When I presented these examples at a combinatorics pre-seminar, Sasha Postnikov suggested to draw the permutations as a graph or a matrix. Why didn't I think of that?
Below are the drawings of the only two 3-symmetric permutations of size 9: 349852167 and 761258943.
As I already mentioned in the aforementioned essay the set of 3-symmetric permutations is invariant under the reversal and subtraction of each number from the size of the permutation plus 1. In geometrical terms it means reflection along the vertical midline and central symmetry. But as you can see the pictures are invariant under 90 degree rotation. Why?
What I forgot to mention was that the set of k-symmetric permutations doesn't change after the inversion. In geometrical terms it means the reflection with respect to the main diagonal. If you combine a reflection with respect to a diagonal with a reflection with respect to a vertical line you get a 90 degree rotation. Overall, the symmetries of the k-symmetric permutations are the same as all the symmetries of a square. Which means we can only look at the shapes of the k-symmetric permutations.
There are six 2-symmetric permutations: 1432, 2341, 2413, 3142, 3214, 4123. As we can see in the picture below they have two different shapes.
Here is the list of all 22 2-symmetric permutations of size 5: 14532, 15342, 15423, 23541, 24351, 24513, 25143, 25314, 31542, 32451, 32514, 34152, 34215, 35124, 41352, 41523, 42153, 42315, 43125, 51243, 51324, 52134. The list was posted by Drake Thomas in the comments to my essay. Up to symmetries the permutations form four groups. Group 1: 14532, 15423, 23541, 32451, 34215, 43125, 51243, 52134. Group 2: 15342, 24351, 42315, 51324. Group 3: 24513, 25143, 31542, 32514, 34152, 35124, 41523, 42153. Group 4: 25314, 41352. The picture shows the first permutation in each group.
In my previous post I described 3-symmetric permutations. Now I want to define 3-symmetric graphs.
A reminder: a k-symmetric permutation is such that the densities of all permutations of size k in it are the same. In particular a 2-symmetric permutation has the same number of inversions and non-inversions. How do we translate this to graphs? I call a graph 2-symmetric if it has the same number of edges as non-edges. 2-symmetric graphs are easy to construct, but they only exist if the number of vertices, n, is such that n choose 2 is even. The simplest non-trivial examples are graphs with 4 vertices and three edges.
The above definition is difficult to generalize. So I rephrase: a graph G is 2-symmetric, if the density of any subgraph H with 2 vertices in G is the same as the expected density of H in a random graph where the probability of an edge equals 1/2. This definition is easy to generalize. A graph G is k-symmetric, if the density of any subgraph H with k vertices in G is the same as the expected density of H in a random graph where the probability of an edge equals 1/2. In particular, here are the densities of all four possible subgraphs with 3 vertices in a 3-symmetric graph:
For a graph G to be 3-symmetric, the number of vertices, n, in G needs to be such that n choose 3 is divisible by 8. The first non-trivial case is n = 8. Here are the pictures of two 3-symmetric graphs. The first one is a wheel, and the second one is its complement.
I want to study patterns inside permutations. Consider a permutation of size 4: 1432. Today I am excited by ordered subset of size 3 inside this permutation. For example, I can drop the last number and look at 143. The ordering in 143 is the same as in 132, or, as a mathematician would say, 143 is order-isomorphic to 132. In my example of 1432, I have four subsets depending on which number I remove: 432 is order-isomorphic to 321, while 132, 142 and 143 are all order-isomorphic to 132.
The density of a pattern 123 in the permutation 1432 is the ratio of subsets of size 3 that are order-isomorphic to 123 to all subsets of size 3. As I already calculated, the density of 123 in 1432 is zero, while the density of 321 is 1/4, and the density of 132 is 3/4.
A permutation is called 3-symmetric if the densities of all patterns of size 3 (123, 132, 213, 231, 312, 321) are the same. The reason I love this notion, is that 3-symmetric permutations are similar to random permutations with respect to patterns of size 3.
My example permutation, 1432, is not 3-symmetric. Thinking about it, no permutation of size 4 can be 3-symmetric. The number of subsets of size 3 is four, which is not divisible by 6.
I wanted to find 3-symmetric permutations. So the size n of the permutation needs to be such that n choose 3 is divisible by 6. The numbers with this property are easy to find. The sequence starts as 1, 2, 9, 10, 18, 20, 28, 29, 36, 37, 38, 45, 46. The sequence is periodic with period 36.
Any permutation of sizes 1 or 2 is 3-symmetric as all densities are zero. Duh!
The next interesting size is 9. My student, Eric Zhang, wrote a program and found that there are two 3-symmetric permutations of size 9: 349852167 and 761258943. These numbers are so cool!. First, they are reverses of each other. This is not very surprising: if a permutations is 3-symmetric, then its reverse must also be 3-symmetric. There is another property: the permutations are rotational symmetries of each other. That is, the sum of two digits in the same place is 10. You can see that rotating a 3-symmetric permutation produces a 3-symmetric permutation.
I decided to write a program to find 3-symmetric permutations of the next size: 10. There is none. I do not trust my programming skills completely, so adjusted my program to size 9 and got the same result as Eric. I trust Eric's programming skills, so I am pretty sure that there are no 3-symmetric permutations of size 10. Maybe there are some 3-symmetric permutations in size 18.
Let's find 2-symmetric permutations. These are permutations with the same number of ascends and descends inversions and non-inversions. For n to be the size of such permutation n choose 2 needs to be divisible by 2. That means n has to have remainder 0 or 1 modulo 4. The first nontrivial case is n = 4. There are six 2-symmetric permutations: 1432, 2341, 2413, 3142, 3214, 4123. We can also group them into reversible pairs: 1432 and 2341, 2413 and 3142, 3214 and 4123. If we look at rotational symmetry we get different pairs: 1432 and 4123, 2341 and 3214, 2413 and 3142.
You can try to find non-trivial 4-symmetric permutations. Good luck! The smallest nontrivial size is 33. Finding 5-symmetric permutations is way easier: the smallest nontrivial size is 28. The sequence of nontrivial sizes as a function of n is: 1, 4, 9, 33, 28, 165, 54, 1029, 40832, 31752, 28680, 2588680, 2162700, and so on. My computer crashed while calculating it.
I found this puzzle on the Russian QWERTY channel.
Five people sit around a table playing Mafia. Among them are two innocent people, two Mafiosos, and one detective. The Mafia people know each other; the detective knows who each of them is; and the innocent people have no information whatsoever about anyone at the table.
During this particular game, the innocents and the detective always tell the truth, while mafia people always lie. They start by going around the circle making the following statements:
- A: I know who B is.
- B: I know who the detective is.
- C: I know who B is.
- D: I know who E is.
Who is who?
Once John Conway showed me a cute way to enumerate Latin squares of size 4, up to the movements of the plane. It was a joint result with Alex Ryba, which is now written in a paper Kenning KenKen.
For starters, I want to remind you that a Latin square of size n is an n by n table filled with integers 1 through n, so that every row and column doesn't have repeated integers. KenKen is a game that John Conway likes, where you need to recover a Latin square given some information about it.
Let me start by describing a particular shape of four cells that one digit can occupy in a Latin square of size 4. There are only seven different shapes. To get to the beautiful result, we need to number these seven shapes in a particular order starting from zero. The shapes are shown below.
There are 12 different Latin squares up to movements of the square and relabeling the digits. Here is how Conway and Ryba matches shapes and squares. For each Latin square, take the shapes of all four digits, remove the duplicate shape numbers and sum the leftover shape numbers. You will get a unique number from 1 to 12 that represents a particular Latin square. For example, consider the square on the picture below.
Digit 1 is shape 4, digits 2 and 4 form shape 2, and digit 3 forms shape 6. Shape 2 is used twice, and we ignore multiplicities. So we have shapes 2, 4, and 6 used. The resulting Latin square is number 2 + 4 + 6, that is 12. It is a fun exercise to try to find all the squares. For example, square 1 can only use shapes 0 and 1. But shape 1 uses exactly one corner. So the first square should use each of the digits in shape 1.
John likes finding interesting ways to remember which shape is which. You can find his and Alex's suggestions in the paper which Alex submitted to the arxiv.
Oops! While I was writing this essay, arxiv rejected the paper.
My coauthor, Konstantin Knop, sent me a coin-weighing problem that is really good. Surprisingly, it is old: it first appeared in a Russian math journal, Kvant, in 1973.
Puzzle. At a trial, 14 coins were presented as material evidence. The expert tested the coins and discovered that seven of them were fake, the rest were real, and he knew exactly which coins were fake and which were real. The court only knows that counterfeit coins weigh the same, real coins weigh the same, and fake ones are lighter than real ones. The expert wants to use not more than three weighings on a balance scales without weights to prove to the court that all the counterfeit coins he found are really fake, and the rest are real. Could he do it?
A cartoon based on my script is posted on TEDEd: Can you solve the Leonardo da Vinci riddle?.
There is only one correct answer to this puzzle. Choices:
My mom died in April of 2017. I didn't even consider flying to Russia for her funeral. April-May is my most demanding work period. We were preparing for the annual PRIMES conference. Four of the projects that I personally mentor were presented at the conference. As a head mentor, I was also helping on all the other projects. During these months, I do not have time to breath.
I felt intensely guilty missing the funeral, but I blocked my emotions and worked. I didn't shed a tear. Come June-July, I have another busy work period running Mathroots and RSI. August is often a slow month, which I usually use to finish papers that I am writing with my students. But in August, 2017, I needed to put the papers aside and give myself time to grieve. My mood was getting darker and darker. At some point I realized that I was depressed. Surprisingly, I still didn't shed a tear.
I had been depressed before, and I do not ever want to be in that place again. I ordered myself to stop mourning, and with some positive self-talk, I was able to get myself out of the depression. In the process I didn't work much in August, leaving me with a huge backlog of papers: I had about 20 papers that needed my immediate attention.
When the academic year began in September, my work was more stressful than ever. On one hand I had a pile of unfinished papers, and on the other hand our programs were growing bigger and more taxing. I limped along and did my best until April of this year. Because I had more stress than ever before. Because April-May is my most intense work time, I had to cancel my social life, stop watching TV, and drop my exercise regime to be able to prepare for our annual PRIMES conference. I was so busy I completely missed the first anniversary of my mom's death. In the year since her death I had been mourning, but I was still unable to cry. When I realized that I had forgotten this date, I felt more severe guilt than ever. I called my sister in Moscow. She told me that she had ignored the death anniversary too. She had done it on purpose. It is better to celebrate life than death, she told me, and it made me feel better.
When the PRIMES conference was over, it was clear that my work was overtaking my life. I decided to go away for a day to rethink my priorities.
I googled Googled around for a place to go, and found the Innisfree garden. The website claimed that the garden is recognized as one of the world's ten best gardens. Sounded fitting for rethinking a life.
The Innisfree Garden is different from other gardens that I have seen. With my untrained eye, I couldn't distinguish what was man-made and what was nature. Slowly it became clear that things that look like nature are in reality a work of genius. The human touch amplified the natural beauty of the land and transformed it into something out of this world: beautiful, peaceful, and serene.
I spent hours in the garden. When I was about to leave, my floodgates were open. I started crying. Mom, I love you; please forgive me.
Puzzle. A boy fell off of a 30 meter ladder, but didn't get hurt. Why not?
I was very happy to hang out with my oldest coauthor, Richard Guy, at the Gathering for Gardner conference in Atlanta in April 2018. By the way, Richard Guy is 101 years old.
John Conway came to the 2017 MOVES conference and told me that he wanted to talk to me about subprime fibs. The subprime Fibonacci sequence was invented by John Conway, and I wrote a paper about it. The paper, Conway's Subprime Fibonacci Sequences, wasn't written with John, but rather with Richard Guy and Julian Salazar, and is published in Mathematics Magazine.
I wanted to visit my friend Julia, who lives in Princeton, and this was a good opportunity to discuss the mysteries of subprime fibs with John. On my second day in Princeton, I came to the math department around 3:00 pm carrying some apples. John never goes out for lunch, as he has trouble walking, so he is always hungry by the end of his work day. Thus, each time I go visit him, I come with food. We have very different tastes in apples: unlike me, he likes his apples unwashed.
Anyway, by the time I arrived to the department, John had already left. This was somewhat unusual, so I called him. He sounded weird and not very coherent, as if he wasn't feeling well. Considering also that he had left early, I started to worry. Unfortunately, there was a lot of background noise during our conversation and I only understood that he was at a pizza place. John walks very slowly, so he couldn't have gone too far away from campus. I found him in the second pizza place I checked. It was Tiger's Pizza. He told me that he felt very sleepy and tired. However, I was gratified to see how much having an interested listener gave him energy. He started telling me stories of his trip to Germany a long time ago. He had already eaten, but decided to have some more fries. As a perfect gentleman he offered me some, but I didn't want any.
At some point he dropped a couple of fries on the floor. He tried to reach them and I jumped to help. That was a mistake. I actually know that he likes proving to me and to himself that he can do stuff independently. He accepts my help when I am subtle about it, or when it is unavoidable. Anyway, he looked at me angrily and I backed off. He picked up his fries from the floor and ate them.
I liked his T-shirt and tried to take a picture of it. As you can see, I am no photographer. The T-shirt shows a test question: Name the triangles. Then it features three triangles: an equilateral, isosceles, and right. It also provides someone's answers to this naming test: Geoffrey, Frederick, Eugene.
John asked me if I am more scared of Donald Trump or Kim Jong Un. We agreed that Trump is scarier. At this time he seemed his usual self.
I offered John a ride home, as I do whenever I visit him. He was very glad as he felt very tired. He started to get up. This time, I remembered not to try to help. He couldn't get up, I waited. He tried to push his weight off the table top, but the table was wobbly. I leaned on the table, as if to rest. We often play this sort of game in which he welcomes my help as long as we both pretend that I'm not helping.
My car was a block away and he wanted to walk the block. But after making two steps out of the pizzeria he changed his mind and asked me to bring the car to him. This was the first time ever. This visit he was so much worse than ever before.
On the drive to his place, he gave me a puzzle:
John's puzzle. Given a Mebius strip with a hole, how do you embed it in 3-D so that the two circular borders of the surface are equivalent?
I dropped him off at his house and offered to walk him to the door. He refused. I sat in my car and watched him walking very slowly along his path. I had this sinking feeling in my gut that I was seeing John for the last time. I drove away, once he disappeared behind his door.
On my way back to Boston I visited my friend Vitaly in East Brunswick, and the next day my high school friend Olga in Edison. In Edison, my car started beeping and I panicked. I was far away from home, and didn't want to be stuck in NJ. I started to look for the source of the sound. It was John's phone. As always, my gut feeling deceived me: I had to go back to Princeton.
I drove back to John's apartment. His door was unlocked and I entered. He was resting in bed. He was greatly annoyed at being disturbed. I explained the reason, and gave him his phone. He took the phone and said, "Off you go." I had this sinking feeling in my gut that these words would be the last words that I would hear from John.
Here is the crypto word search I designed as a gift exchange for G4G13 (Gathering for Gardner). The submitted file is here: Crypto Word Search.
A B C D E F G H C I F B B C D I J K L A J C I F M A C K N O O N F B I F J O P P Q G H F A R K J B
The words are: ART IDEA MAGIC MATH NOTE PI PROBLEM PUZZLE RIDDLE TRICK.
This is a math blog, but from time to time, I write about other things. Today I have something to say about puns, which I adore.
I also like gym, but rarely go there: it doesn't work out. I stopped using stairs, because they are up to something. I wanted to learn how to juggle, but I don't have the balls to do it.
I work at MIT, the work place with the best dam mascot: Tim the Beaver. My salary is not big, and I stopped saving money after I lost interest. I'm no photographer, but I have pictured myself outside of MIT too. I am a mathematician, which is the most spiritual profession: I am very comfortable with higher powers. I praise myself on great ability to think outside the box: it is mostly due to my claustrophobia. I am also a bit of a philosopher: I can go on talking about infinity forever.
I would love to tell you a joke. I recently heard a good one about amnesia, but I forgot how it goes.
My biggest problem is with English. So what if I don't know what apocalypse means? It's not the end of the world!
I never get tired of puns and here is my list of pun puzzles from the MIT Mystery Hunt:
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Business plan:
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A cafe patron ordered a pastry, then changed his mind and replaced it with a cup of coffee. When he finished his coffee, he started leaving without paying. The waiter approached him:
—You didn't pay for coffee!
—But I had it instead of the pastry.
—You didn't pay for the pastry either!
—But I didn't have the pastry.
* * *
At a farmers market stand there is a sign: 1 melon—3 dollars, 3 melons—10 dollars. A client requests one melon and pays 3 dollars, then repeats the procedure two more times. Then he says: "I bought three melons for 9 dollars, while you are trying to sell them for 10 dollars. This is really stupid." The farmer talks to himself: This happens all the time: they buy three melons instead of one, and try to teach me how to make money.
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If the government listens in on my phone conversations, should they be paying half of my phone bill?
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To get to free downloads, please, enter your credit card number.
* * *
The biggest lie of the century, "I have read and agree to the terms of ..."
* * * (submitted by Sam Steingold)
Ignorance: If your poker opponent got lucky cards four times in a row, he must get lousy cards now.
Knowledge: Nope, the deals are independent; prior observations have no bearing on the next deal.
Wisdom: The opponent is cheating; get away from the table now!
I recently posted two geometry problems. Now is the time for solutions:
Problem 1. Is it possible to put positive numbers at the vertices of a triangle so that the sum of two numbers at the ends of each side is equal to the length of the side?
One might guess that the following numbers work: (a+b-c)/2, (b+c-a)/2 and (c+a-b)/2, where a, b, and c are the side lengths. But there exists a geometric solution: Construct the incircle. The tangent points divide each side into two segment, so that the lengths of the segments ending at the same vertex are the same. Assigning this length to the vertex solves the problem. Surprisingly, or not surprisingly, this solution gives the same answer as above.
Problem 2. Prove that it is possible to assign a number to every edge of a tetrahedron so that the sum of the three numbers on the edges of every face is equal to the area of the face.
The problem is under-constrained: there are six sides and four faces. There should be many solutions. But the solution for the first problem suggests a similar idea for the second problem: Construct the inscribed sphere. Connect a tangent point on each face to the three vertices on the same face. This way each face is divided into three triangles. Moreover, the lengths of the segments connecting the tangent points to a vertex are the same. Therefore, two triangles sharing the same edge are congruent and thus have the same area. Assigning this area to each edge solves the problem.
There are many solutions to the second problem. I wonder if for each solution we can find a point on each face, so that the segments connecting these points to vertices divide the faces into three triangles in such a way that triangles sharing an edge are congruent. What would be a geometric meaning of these four points?
I was on the writing team for the 2018 MIT Mystery Hunt. I am pleased that the hunt got very positive reviews from the participants. I spent tons of hours working on the hunt and it is good that folks liked it. I edited and tested a lot of puzzles. Here is my review of these year's puzzles that are math-related.
I already posted an essay about the puzzles I wrote myself. Four of my five puzzles are math-related, so I am including them below for completeness. I will mention the topic of each puzzle unless it is a spoiler.
I start with Nikoli-type puzzles. Four elegant Nikoli-type puzzles were written or cowritten by Denis Auroux. In all of them the rules of the logic are stated at the beginning. That means the logic part doesn't contain a mystery and can be solved directly.
There were several puzzles that were very mathematical.
There were also some math-related or computer-sciency puzzles.
There were also several decryption puzzles:
Here is a famous math problem I never before wrote about:
Puzzle. Five pirates discovered a treasure of 100 gold coins. They decide to split the coins using the following scheme. The most senior pirate proposes how to share the coins, and all the pirates vote for or against it. If 50% or more of the pirates vote for it, then the coins will be shared that way. Otherwise, the pirate proposing the scheme will be thrown overboard, and the process is repeated with the next most senior pirate making a proposal.
As pirates tend to be a bloodthirsty bunch, if a pirate would get the same number of coins whether he votes for or against a proposal, he will vote against so that the pirate who proposed the plan will be thrown overboard. Assuming that all five pirates are intelligent, rational, greedy, and do not wish to die, how will the coins be distributed?
You can find the solution in many places including Wikipedia's Pirate game. The answer is surprising: the most senior pirate gets 98 coins, and the third and the fifth pirates by seniority get one coin each. I always hated this puzzle, but never bothered to think through and figure out why. Now I know.
This puzzle emphasizes the flaws of majority voting. The procedure is purely democratic, but it results in extreme inequality.
That means a democracy needs to have a mechanism to prohibit the president from blatantly benefiting himself. With our current president these mechanisms stopped working. Given that Trump does everything to enrich himself, the pirates puzzle tells us what to expect in the near future.
We, Americans, will lose everything: money, clean air and water, national parks, future climate, health, social security, and so on, while Trump will make money.
I was on the writing team of this year's hunt, which was based on the movie "Inside Out." One of our goals was to create an easy first round to allow small teams to have a full hunt experience. Our first round consisted of 34 puzzles related to five basic emotions: joy, sadness, disgust, fear, and anger. Each emotion had its own meta puzzle. And the round had a meta-meta puzzle and a runaround. As I tend to write easy puzzles, I contributed three puzzles to this emotions round. The puzzles had references to corresponding emotions that were not needed for the solve path. They were inserted there for flavor.
I also wrote another easy puzzle called A Tribute: 2010-2017 (jointly with Justin Melvin, Wesley Graybill, and Robin Diets ). Though the puzzle is easy, it is useful in solving it to be familiar with the MIT mystery hunt. This is why the puzzle didn't fit the first emotions round.
I also wrote a very difficult puzzle called Murder at the Asylum. This is a monstrosity about liars and truth-tellers.
In mathematics one of the most important questions is why. Let us consider a problem:
Problem. A number has three hundred ones and three hundred zeroes. Can it be a square?
The solution goes like this. Consider divisibility of this number by 9. The sum of the digits is 300. That means the number is divisible by 3, but not by 9. Therefore, it can't be a square.
Why do we consider divisibility by 9? The divisibility by 9 is a very powerful tool, but why was it the first thing that came to my mind? The divisibility by 9 doesn't depend on the order of the digits. Whenever I see a problem where the question talks about digits that can be in any order, the first tool to use is the divisibility by 9.
The why question, is very important in mathematics. But it is also very important in life. It took me many years to start asking why people did this or that. I remember my mom was visiting me in the US. Every time I came back from work, she complained that she was tired. Why? Because she did the laundry in the bath tub. She wouldn't use my washing machine, because she didn't have such a thing in Russia. I promised her that I'd do the laundry myself when there was a sufficient pile. However, she insisted that the dirty clothes annoyed her. I would point that my water bill went up. And so on.
We argued like this every day. We were both frustrated. Then I asked myself why. Why does she do the laundry? The answer was there. She wanted to be helpful. I calmed down and stopped arguing with her. I sucked it up and paid the water bills. Her time with me turned into the most harmonious visit we ever had. Unfortunately, it was the last.
Puzzle. Alice, Bob, and Charlie are at Alice's house. They are going to Bob's house which is 33 miles away. They have a 2-seat scooter which rides at 25 miles per hour with 1 rider on it; or, at 20 miles per hour with 2 riders. Each of the 3 friends walks at 5 miles per hour. How fast can all three of them make it to Bob's house?
* * * (submitted by Sam Steingold)
I can count to 1023 on my 10 fingers. The rudest number is 132.
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I kept forgetting my password, so I changed it to "incorrect". Now, when I make a mistake during login, my computer reminds me: "Your password is incorrect."
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—You promised me 8% interest, and in reality it is 2%.
—2 is 8—to some degree.
* * * (submitted by Sam Steingold)
Quantum entanglement is simple: when you have a pair of socks and you put one of them on your left foot, the other one becomes the "right sock," no matter where it is located in the universe.
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Teacher:
—I keep telling my students that one half can't be larger or smaller than the other. Still the larger half of my class doesn't get it.
This famous trick puzzle is very old:
Puzzle. The professor is watching across a field how the son of the professor's father is fighting with the father of the professor's son. How is this possible?
This puzzle is tricky only because of gender-bias. Most people assume that the professor is male and miss the obvious intended solution, in which a female professor is watching her brother fighting with her husband.
I just gave this problem on a test. Here are other answers that I received.
Years ago people couldn't figure out this puzzle at all. So there has been progress. I was glad that my students suggested so many ideas that work. Nonetheless, many of them revealed their gender-bias by initially assuming that the professor is a man.
I can't wait until this puzzle stops being tricky.
Puzzle. There are five houses of different colors next to each other equally spaced on the same road. In each house lives a man of a different profession.
Who lives in the white house?
Correction Nov 11, 2017. Replaced "the same distance from" with "halfway between" to eliminate the possibility of the plumber living in the yellow house. Thank you to my readers for catching this mistake and to Smylers for suggesting a correction.
Problem. Invent a connected shape made out of squares on the square grid that cannot be cut into dominoes (rectangles with sides 1 and 2), but if you add a domino to the shape then you can cut the new bigger shape.
This problem reminds me of another famous and beautiful domino-covering problem.
Problem. Two opposite corner squares are cut out from the 8 by 8 square board. Can you cover the remaining shape with dominoes?
The solution to the second problem is to color the shape as a chess board and check that the number of black and white squares is not the same.
What is interesting about the first problem is that it passes the color test. It made me wonder: Is there a way to characterize the shapes on a square grid that pass the color test, but still can't be covered in dominoes?
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Don't anthropomorphize computers: They don't like it.
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I do not have dreams any more. What did I do wrong to make them delete my account?
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How to restore justice: Create a folder named Justice. Delete it. Go to the trash bin and click restore.
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An asocial network: When you sign up, you are friends with everyone. Then you send un-friend requests.
I already wrote about two puzzles that Derek Kisman made for the 2013 MIT Mystery Hunt. The first puzzle is now called the Fractal Word Search. It is available on the Hunt website under its name In the Details. I posted one essay about the puzzle and another one describing its solution. The second puzzle, 50/50, is considered one of the most difficult hunt puzzles ever. Unfortunately, the puzzle is not available, but my description of it is.
Today let's look at the third puzzle Derek made for the 2013 Hunt, building on an idea from Tom Yue. This is a non-mathematical crossword puzzle. Derek tends to write multi-layered puzzles: You think you've got the answer, but the answer you've got is actually a hint for the next step.
Often multi-layered puzzles get solvers frustrated, but the previous paragraph is a hint in itself. If you expect the difficulty, you might appreciate the fantastic beauty of this puzzle.
Welcome to Ex Post Facto.
Every time I visit Princeton, or otherwise am in the same city as my friend John Conway, I invite him for lunch or dinner. I have this rule for myself: I invite, I pay. If we are in the same place for several meals we alternate paying. Once John Conway complained that our tradition is not fair to me. From time to time we have an odd number of meals per visit and I end up paying more. I do not trust my memory, so I prefer simplicity. I resisted any change to our tradition. We broke the tradition only once, but that is a story for another day.
Let's discuss the mathematical way of paying for meals. Many people suggest using the Thue-Morse sequence instead of the alternating sequence of taking turns. When you alternate, you use the sequence ABABAB…. If this is the order of paying for things, the sequence gives advantage to the second person. So the suggestion is to take turns taking turns: ABBAABBAABBA…. If you are a nerd like me, you wouldn't stop here. This new rule can also give a potential advantage to one person, so we should take turns taking turns taking turns. Continuing this to infinity we get the Thue-Morse sequence: ABBABAABBAABABBA… The next 2^{n} letters are generated from the first 2^{n} by swapping A and B. Some even call this sequence a fair-share sequence.
Should I go ahead and implement this sequence each time I cross paths with John Conway? Actually, the fairness of this sequence is overrated. I probably have 2 or 3 meals with John per trip. If I pay first every time, this sequence will give me an advantage. It only makes sense to use it if there is a very long stretch of meals. This could happen, for example, if we end up living in the same city. But in this case, the alternating sequence is not so bad either, and is much simpler.
Many people suggest another use for this sequence. Suppose you are divorcing and dividing a huge pile of your possessions. A wrong way to do it is to take turns. First Alice choses a piece she wants, then Bob, then Alice, and so on. Alice has the advantage as the first person to choose. An alternative suggestion I hear in different places, for example from standupmaths, is to use the Thue-Morse sequence. I don't like this suggestion either. If Alice and Bob value their stuff differently, there is a better algorithm, called the Knaster inheritance procedure, that allows each of them to think they are getting more than a half. If both of them have the same value for each piece, then the Thue-Morse sequence might not be good either. Suppose one of the pieces they are dividing is worth more than everything else put together. Then the only reasonable way to take turns is ABBBB….
The beauty of the Thue-Morse sequence is that it works very well if there are a lot of items and their consecutive prices form a power function of a small degree k, such as a square or a cube function. After 2^{k+1} turns made according to this sequence, Alice and Bob will have a tie. You might think that if the sequence of prices doesn't grow very fast, then using the Thue-Morse sequence is okay.
Not so fast. Here is the sequence of prices that I specifically constructed for this purpose: 5,4,4,4,3,3,3,2,2,2,2,1,1,0,0,0. The rule is: every time a turn in the Thue-Morse sequence switches from A to B, the value goes down by 1. Alice gets an extra 1 every time she is in the odd position. This is exactly half of her turns. That is every four turns, she gets an extra 1.
If the prices grow faster than a power, then the sequence doesn't work either. Suppose your pieces have values that form a Fibonacci sequence. Take a look at what happens after seven turns. Alice will have pieces priced F_{n} + F_{n-3} + F_{n-5} + F_{n-6}. Bob will have F_{n-1} + F_{n-2} + F_{n-4}. We see that Alice gets more by F_{n-3}. This value is bigger than the value of all the leftovers together.
I suggest a different way to divide the Fibonacci-priced possessions. If Alice takes the first piece, then Bob should take two next pieces to tie with Alice. So the sequence might be ABBABBABB…. I can combine this idea with flipping turns. So we start with a triple ABB, then switch to BAA. After that we can continue and flip the whole thing: ABBBAABAAABB. Then we flip the whole thing again. And again and again. At the end we get a sequence that I decided to call the Fibonacci fair-share sequence.
I leave you with an exercise. Describe the Tribonacci fair-share sequence.
A while ago I posted my second favorite problem from the 2015 All-Russian Math Olympiad:
Problem. A coin collector has 100 coins that look identical. He knows that 30 of the coins are genuine and 70 fake. He also knows that all the genuine coins weigh the same and all the fake coins have different weights, and every fake coin is heavier than a genuine coin. He doesn't know the exact weights though. He has a balance scale without weights that he can use to compare the weights of two groups with the same number of coins. What is the smallest number of weighings the collector needs to guarantee finding at least one genuine coin?
Now it's solution time. First we show that we can do this in 70 weighings. The strategy is to compare one coin against one coin. If the scale balances, we are lucky and can stop, because that means we have found two real coins. If the scale is unbalanced, the heavier coin is definitely fake and we can remove it from consideration. In the worst case, we will do 70 unbalanced weighings that allow us to remove all the fake coins, and we will find all the real coins.
The more difficult part is to show that 69 weighings do not guarantee finding the real coin. We do it by contradiction. Suppose the weights are such that the real coin weighs 1 gram and the i-th fake coin weighs 100^{i} grams. That means whatever coins we put on the scale, the heaviest pan is the pan that has the fake coin with the largest index among the fake coins on the scale.
Suppose there is a strategy to find a real coin in 69 weighings. Given this strategy, we produce an example designed for this strategy, so that the weighings are consistent, but the collector cannot find a real coin.
For the first weighing we assign the heaviest weight, 100^{70} to one of the coins on the scale and claim that the pan with this coin is heavier. We continue recursively. If a weighing has the coins with assigned weights, we pick the heaviest coin on the pans and claim that the corresponding pan is heavier. If there are no coins with assigned weights on pans, we pick any coin on the pans, assigned the largest available weight to it and claim that the corresponding pan is heavier.
After 69 weighings, not more than 69 coins have assigned weights, while all the weighings are consistent. The rest of the coins can have any of the leftover weights. For example, any of the rest of the coins can weigh 100 grams. That means that there is no coin that is guaranteed to be real.
I stumbled upon a couple of problems that I like while scanning the Russian website of Math Festival in Moscow 2014. The problems are for 7 graders.
Problem. Inside a 5-by-8 rectangle, Bart draws closed paths that follow diagonals of 1-by-2 rectangles. Find the longest possible path.
This problem is really very difficult. The competition organizers offered an extra point for every diagonal on top of 16. The official solution has 24 diagonals, but no proof that it's the longest. I'm not sure anyone knows if it is the longest.
Here is another problem:
Problem. Alice and Bob are playing a game. They start with two numbers: 2014 and 2015. In one move a player can do one of two things:The winner is the person who is the first to get a one-digit number. Assuming that Alice starts, who wins?
- subtract one of the numbers by one of the non-zero digits in any of the two numbers or,
- divide one number by two if the number is even.
The year is 1994. The man on the left is my first husband, Alexander Goncharov. Although we were out of touch for a decade, when I married my third husband, Joseph Bernstein (on the right), Goncharov started visiting us. It wasn't me he was interested in: he wanted to talk mathematics with my husband. I found this situation hilarious, so I took this photo.
But that's not all. My second husband, Andrey Radul, is not in the picture. But all four of us were students of Israel Gelfand. In short, my three ex-husbands and I are mathematical siblings — that is, we are all one big happy mathematical family.
The Best Writing on Mathematics 2016 is out. I am happy that my paper The Pioneering Role of the Sierpinski Gasket is included. The paper is written jointly with my high-school students Eric Nie and Alok Puranik as our PRIMES-2014 project.
At the end of the book there is a short list of notable writings that were considered but didn't make it. The "short" list is actually a dozen pages long. And it includes two more papers of mine:
To continue bragging, I want to mention that my paper A Line of Sages was on the short list for 2015 volume. And my paper Conway's Wizards was included in the 2014 volume.
I like Odd-One-Out puzzles that are ambiguous. That is why I bought the book Which One Doesn't Belong? Look at the cover: which is the odd one out? The book doesn't include answers, but it has nine more examples in each of which there are several possible odd-one-outs.
I married an American citizen and moved to the US in 1990. At the time I was a very patriotic Russian. It took me a year of pain to realize that some of my ideas had been influenced by Soviet propaganda. After I washed away the brainwashing, I fell in love with the US. For 25 years I thought that America was great. Not anymore.
For the last several months I've been worried as never before in my life. I feel paralyzed and sick. To help myself I decided to put my feelings in words.
World War. My mom was 15 when World War II started. The war affected her entire life, as well as the lives of everyone in the USSR. Every now and then my mom would tell me, "You are lucky that you are already 20 and you haven't witnessed a world war." I moved to the US while my mom stayed back in Russia. From time to time I tell myself something like, "I am lucky that halfway through my expected lifetime, I haven't had to live through a world war." It's been more than 70 years since WWII ended. To maintain the peace is a difficult job. Everything needs to be in balance. Trump is disrupting this balance. I am worried sick that my children or grandchildren will have to witness a major war.
The Red Button. I've noticed that, as a true showman, Trump likes misdirecting attention from things that worry him to fantastic plot twists that he invents. What's the best way to make people forget about his tax returns? It's the nuclear button. Dropping a nuclear bomb some place will divert people from thinking about his tax returns. As his plot twists are escalating, is he crazy enough to push the button?
Climate. The year 2014 was the warmest on record. The year 2015 was even warmer. And last year, 2016, was even warmer than that. I remember Vladimir Arnold's class on differential equations. He talked about a painting that had been hanging on a wall for 20 years. Then it unexpectedly fell off. Mathematics can explain how such catastrophic events can happen. I keep thinking about our Earth: melting ice, dead reefs, fish eating plastic, and so much more. My grandchildren might not be able to enjoy beaches and forests the way I did. What if, like the fallen painting, the Earth can spiral out of control and completely deteriorate? But Trump is ignoring the climate issues. Does he care about our grandchildren? I am horrified that Trump's policies will push climate catastrophe beyond the point of no return.
The Truth. Wiretapping is not wiretapping. Phony jobs numbers stopped being phony as soon as Trump decided that he deserved the credit. The news is fake when Trump doesn't like it. Trump is a pathological liar; he assaults the truth. Being a scientist I am in search of truth, and Trump diminishes it. I do not understand why people ignore his lies. Two plus two is four whether you are a democrat, or a republican, or whomever. Facts are facts, alternative facts are lies. I am scared that lies have become acceptable and no one cares about the truth any more.
Russia. I lived in Russia for the first unhappy half of my life, and in the US for the second happy half. I do not want to go back. There is something fishy between Putin and Trump. Whether it is blackmail or money, or both, I do not know the details yet, But Trump is under Putin's influence. Trump didn't win the elections: Putin won. This horrifies me. I do not want to go back to being under Russian rule.
Gender Issues. I grew up in a country where the idea of a good husband was a man who wasn't a drunkard. That wasn't enough for me. I dreamed of a relationship in which there would be an equal division of work, both outside and inside the home. I could not achieve that because in Russian culture both people work full-time and the wife is solely responsible for all the house chores. Moreover, Russia was much poorer than the US: most homes didn't have washing machines; we never heard of disposable diapers; and there were very long lines for milk and other necessities.
The life in USSR was really unfair to women. Most women had a full-time job and several hours of home chores every day. When I moved to the US, I thought I was in paradise. Not only did I have diapers and a washing machine, I was spending a fraction of the time shopping, not to mention that my husband was open to helping me, and didn't mind us paying for the occasional babysitter or cleaner.
For some time I was blind to gender issues in the US because it was so much better. Then I slowly opened my eyes and became aware of the bias. For some years it has felt like gender equity was improving. Now, with a misogynistic president, I feel that the situation might revert to the dark ages. When women are not happy, their children are not happy, and they grow up to be not happy. If the pursuit of happiness is the goal, the life has to be fair to all groups. But Trump insults not only women but also immigrants, Muslims, members of the LGBTQ community, as well as the poor and the sick. The list is so long, that almost everyone is marginalized. This is not a path towards a happy society.
Democracy. Trump attacks the press and attempts to exclude them. Trump has insulted the intelligence community and the courts. He seems to be trying to take more power to the presidency at the expense of the other branches of government. He ignores his conflicts of interest. Trump disregards every rule of democracy and gets away with it. I am horrified that our democracy is dying.
Tax Returns. Trump's tax returns could either exonerate him or prove that he is Putin's puppet. The fact that he is hiding the returns makes me believe that the latter is more probable. Why the Republicans refuse to demand to see his returns is beyond my understanding.
Corruption. Trump does so many unethical things. Most of his decisions as president seem to be governed by Trump trying to get richer. Let us consider his hotel in Azerbaijan—a highly corrupt country. Having lived in a highly corrupt country myself, I know how it works. For example, an Azerbaijani government official who has access to their country's money can make a deal that involves a personal kickback. This means that their government is paying more than necessary for a service or product in order to cover that kickback. This is how national money makes its way into individual pockets. Since all the deals in Azerbaijan are reputed to be like that, I imagine that when Trump built his hotel there, the Trump organization was overpaid in order to cover the bribe to local officials. Will our country become as corrupt as Azerbaijan?
Americans. The biggest shock of the election was that so many people were so gullible and actually voted for Trump. They didn't see that his agenda is focused on his own profit, and that he lies and makes promises he doesn't plan to deliver. It really terrifies me that there are some people who are not gullible but still voted for Trump.
Is there hope?
I recently wrote about my way of playing Nim against a player who doesn't know how to play. If my move starts in an N-position, then I obviously win. If my move starts in a P-position, I would remove one token hoping that more tokens for my opponent means more opportunity for them to make a mistake. But which token to remove? Does it make a difference from which pile I choose?
Consider the position (2,4,6). If I take one token, my opponent has 11 different moves. If I choose one token from the first or the last pile, my opponent needs to get to (1,4,5) not to lose. If I choose one token from the middle pile, my opponent needs to get to (1,3,2) not to lose. But the first possibility is better, because there are more tokens left, which gives me a better chance to have a longer game in case my opponent guesses correctly.
That is the strategy I actually use: I take one token so that the only way for the opponent to win is to take one token too.
This is a good heuristic idea, but to make such a strategy precise we need to know the probability distribution of the moves of my opponent. So let us assume that s/he picks a move uniformly at random. If there are n tokens in a N-position, then there are n − 1 possible moves. At least one of them goes to a P-position. That means my best chance to get on the winning track after the first move is not more than n/(n−1).
If there are 2 or 3 heaps, then the best strategy is to go for the longest game. With this strategy my opponent always has exactly one move to get to a P-position, I win after the first turn with probability n/(n−1). I lose the game with probability 1/(n−1)!!.
Something interesting happens if there are more than three heaps. In this case it is possible to have more than one winning move from a N-position. It is not obvious that I should play the longest game. Consider position (1,3,5,7). If I remove one token, then my opponent has three winning moves to a position with 14 tokens. On the other hand, if I remove 2 tokens from the second or the fourth pile, then my opponent has one good move, though to a position with only 12 tokens. What should I do?
I leave it to my readers to calculate the optimal strategy against a random player starting from position (1,3,5,7).
It is rare when a word equation coincides with a number equation.
Problem. A store sells letter magnets. The same letters cost the same and different letters might not cost the same. The word ONE costs 1 dollar, the word TWO costs 2 dollars, and the word ELEVEN costs 11 dollars. What is the cost of TWELVE?
Alex Bellos wrote a puzzle book Can You Solve My Problems? Ingenious, Perplexing, and Totally Satisfying Math and Logic Puzzles The book contains a mixture of famous puzzles and their solutions. Some of the puzzles are not mathematical in the strictest sense, but still have an appeal for mathematicians. For example, which integer comes up first when you alphabetize all the integers up to a quadrillion?
Recognize the puzzle on that book cover? You're right! That's my Odd One Out puzzle. Doesn't it look great in lights on that billboard in London?
Mine isn't the only terrific puzzle in the book. In fact, one of the puzzles got my special attention as it is related to our current PRIMES polymath project. Here it is:
A Sticky Problem. Dick has a stick. He saws it in two. If the cut is made [uniformly] at random anywhere along the stick, what is the length, on average, of the smaller part?
The beautiful Pascal triangle has been around for many years. Can you say something new about it?
Of course you can. Mathematicians always find new way to look at things. In 2012 RSI student, Kevin Garbe, did some new and cool research related to the triangle. Consider Pascal's triangle modulo 2, see picture which was copied from a stackexchange discussion.
A consecutive block of m digits in one row of the triangle modulo 2 is called an m-block. If you search the triangle you will find that all possible binary strings of length 2 are m-blocks. Will this trend continue? Yes, you can find any possible string of length 3, but it stops there. The blocks you can find are called accessible blocks. So, which blocks of length 4 are not accessible?
There are only two strings that are not accessible: 1101 and 1011. It is not surprising that they are reflections of each other. Pascal's triangle respects mirror symmetry and the answer should be symmetric with respect to reflection.
You can't find these blocks on the picture, but how do we prove that they are not accessible, that is, that you can't ever find them? The following amazing property of the triangle can help. We call a row odd/even, if it corresponds to binomial coefficients of n choose something, where n is an odd/even number. Every odd row has every digit doubled. Moreover, if we take odd rows and replace every double digit with its single self we get back Pascal's triangle. Obviously the two strings 1101 and 1011 can't be parts of odd rows.
What about even rows? The even rows have a similar property: every even-indexed digit is a zero. If you remove these zeros you get back Pascal's triangle. The two strings 1101 and 1011 can't be part of even rows. Therefore, they are not accessible.
The next question is to count the number of inaccessible blocks of a given length: a(n). This and much more was done by Kevin Garbe for his RSI 2012 project. (I was the head mentor of the math projects.) His paper is published on the arxiv. The answer to the question can be found by constructing recurrence relations for odd/even rows. It can be shown that a(2r) = 3a(r) + a(r+1) − 6 and a(2r+1) = 3a(r) + 2a(r+1) − 6. As a result the number of inaccessible blocks of length n is n^{2} − n + 2. I wonder if there exists a direct proof of this formula without considering odd and even rows separately.
This RSI result was so pretty that it became a question at our entrance PRIMES test for the year 2013. In the test we changed the word accessible to admissible, so that it would be more difficult for applicants to find the research. Besides, Garbe's paper wasn't arxived yet.
The pretty picture above is from the stackexchange, where one of our PRIMES applicants tried to solicit help in solving the test question. What a shame.
I picked four problems that I liked from the Moscow Math Olympiad 2016:
Problem 1. Ten people are sitting around a round table. Some of them are knights who always tell the truth, and some of them are knaves who always lie. Two people said, "Both neighbors of mine are knaves." The other eight people said, "Both neighbors of mine are knights." How many knights might be sitting around the round table?
Problem 2. Today at least three members of the English club came to the club. Following the tradition, each member brought their favorite juice in the amount they plan to drink tonight. By the rules of the club, at any moment any three members of the club can sit at a table and drink from their juice bottles on the condition that they drink the same amount of juice. Prove that all the members can finish their juice bottles tonight if and only if no one brings more than the third of the total juice brought to the club.
Problem 3. Three piles of nuts together contain an even number of nuts. One move consists of moving half of the nuts from a pile with an even number of nuts to one of the other two piles. Prove that no matter what the initial position of nuts, it is possible to collect exactly half of all the nuts in one pile.
Problem 4. N people crossed the river starting from the left bank and using one boat. Each time two people rowed a boat to the right bank and one person returned the boat back to the left bank. Before the crossing each person knew one joke that was different from all the other persons' jokes. While there were two people in the boat, each told the other person all the jokes they knew at the time. For any integer k find the smallest N such that it is possible that after the crossing each person knows at least k more jokes in addition to the one they knew at the start.
Spoiler for Problem 2. I want to mention a beautiful solution to problem 2. Let's divide a circle into n arcs proportionate to the amount of juice members have. Let us inscribe an equilateral triangle into the circle. In a general position the vertices of the triangle point to three distinct people. These are the people who should start drinking juices with the same speed. We rotate the triangle to match the drinking speed, and as soon as the triangle switches the arcs, we switch drinking people correspondingly. After 120 degree rotation all the juices will be finished.
I already posted a funny true story that Smullyan told me when I last visited him. Raymond Smullyan died recently at the age of 97 and my mind keeps coming back to this last visit.
The year was 2012 and I was about to drive back to Boston after my talk at Penn State. Smullyan's place in the Catskills was on the way—sort of. I wanted to call him, but I was apprehensive. Raymond Smullyan had a webpage on which his email was invisible. You could find his email address by looking at the source file or by highlighting empty space at the bottom of the page. Making your contact information invisible sends a mixed message.
While this was a little eccentric, it meant that only people who were smart enough to find it, could access his email address. I already knew his email because he had given it to me along with his witty reply to my blog post about our meeting at the Gathering for Gardner in 2010.
In our personal interactions, he always seemed to like me, so I called Raymond and arranged a visit for the next day around lunch time. When I knocked on his door, no one answered, but the door was open, and since Smullyan was expecting me, I walked right in. "Hello? Anyone there? Hello? Hello?" As I wandered around the house, I saw an open bedroom door and inside Smullyan was sleeping. So I sat down in his library and picked up a book.
When he woke up, he was happy to see me, and he was hungry. He told me that he didn't eat at home, so we should go out together for lunch. I was hungry too, so I happily agreed. Then he said that he wanted to drive. I do not have a poker face, so he saw the fear in me. My only other trip with a nonagenarian driver flashed in front of my eyes. The driver had been Roman Totenberg and it had been the scariest drive I have ever experienced.
I said that I wanted to drive myself. Annoyed, Raymond asked me if I was afraid of him taking the wheel. I told him that I have severe motion sickness and always prefer to drive myself. Raymond could see that I was telling the truth. I got the impression that he was actually relieved when he agreed to go in my car.
We went to Selena's Diner. He took out playing cards with which he showed me magic tricks. I showed him some tricks too. This was probably a bad move as he abandoned me to go to the neighboring table to show his magic tricks to a couple of young girls. They were horrified at first"his unruly hair, his over-the-top energy, his ebullient behavior"but between me and the waitress, we quickly reassured them. The girls enjoyed the tricks, and I enjoyed my visit.
Like many people, I was appalled by Trump's immigration ban. On the Internet I found many essays that explained that he did not include in the ban those majority-Muslim countries in which he has business interests. See for example, an article at Forbes with a nice map, and an article at NPR.
Now the countries that are excluded are motivated to continue to support Trump's businesses, and to offer him bribes and good deals in exchange for staying out of the ban. The countries on the list are also motivated to approach Trump and offer him a sweet business deal.
So even if the courts stopped the ban, he has already succeeded in showing every country in the world that to be on his good side requires that they pay up. And China got the hint and granted Trump a trademark he's been seeking for a decade.
Looks like Trump's vision of a great America is a very rich Mr Trump.
I recently posted the following puzzle:
Puzzle. We have 3^{2n} identical-looking coins. One of the coins is fake and lighter than the other coins, which all are real. We also have three scales: two normal and one random. Find the fake coin in the smallest total number of weighings.
Here is my son Sergei's solution. Divide the coins into nine groups of equal size and number the groups in ternary: 00, 01, 02, 10, 11, 12, 20, 21, and 22. On each scale we put three groups versus three groups. On the first scale we compare the three groups that start with 1 with the three groups that start with 2. For the second scale we do the same using the last digit instead of the first one, and for the third scale we use the sum of two digits modulo 3. Any pair of scales, if they are assumed to be normal, would point to one out of nine groups as the group containing the fake coin.
If all three pairs of scales agree on one group, then this is the group containing the fake coin. Thus in three weighings, we reduce the number of groups of coins by a factor of nine. If the pairs of scales do not agree, then the random scale produced a wrong weighing and thus can be found out. How do we do that? We have three out of nine groups of coins each of which might contain the fake coin. We compare two of the groups on all three scales. This way we know exactly which group contains the fake coin and, consequently, which scale generated a wrong weighing. If we know the random scale, we can speed up the rest of the process of finding the fake coin. Thus in the worst case we require 3n+3 weighings.
The big idea here is that as soon as the random scale shows a wrong weighing result it can be found out. So in the worst case, the random scale behaves as a normal scale and messes things up at the very end. Sergei's solution can be improved to 3n+1 weighings. Can you do that?
The improved solution is written in a paper Взвешивания на «хитрых весах» (in Russian) by Konstantin Knop, that is published in Математика в школе 2009-2. The paper contains an even stronger solution that provides a better asymptotics.
The first time I visited the US was in 1990 at the invitation of an old friend, Joseph Bernstein. After my arrival Joseph proposed and I accepted, but my essay is not about that.
Joseph reintroduced me to his daughter, Mira, who was then in her late teens. I was struck by Mira's charm. I had never before met teenagers like her. Of course, Joseph got points for that as I was hoping to have a child with him. When I moved to the US I met some other kids who were also incredibly charming. It was too late to take points away from Joseph, but it made me realize what a huge difference there was between Soviet and American teenagers. American teenagers were happier, more relaxed, better mannered, and less cynical than Soviet ones.
My oldest son, Alexey, was born in the USSR and moved to the US when he was eight. One unremarkable day when he was in middle school (Baker public school in Brookline), the principle invited me for a chat. I came to the school very worried. The principal explained to me that there was a kid who was bugging Alexey and Alexey pushed him back with a pencil. While the principal proceeded to explain the dangers of a pencil, I tuned out. I needed all my energy to conceal my happy smile. This was one of the happiest moments of my life in the US. What a great country I live in where the biggest worry of a principal in a middle school is the waving of a pencil! I remembered Alexey's prestigious school in Moscow. They had fights every day that resulted in bloody noses and lost teeth. When I complained to his Russian teacher, she told me that it was not her job to supervise children during big breaks. Plus the children needed to learn to be tough. No wonder American children are happier.
I was wondering if there were any advantages to a Soviet upbringing. For one thing, Soviet kids grow up earlier and are less naive. They are more prepared for harsh realities than those American kids who are privileged.
Naive children grow up into naive adults. Naive adults become naive presidents. I watched with pain as naive Bush ("I looked the man in the eye. I found him to be very straightforward and trustworthy.") and naive Obama (Russian reset) misunderstood and underestimated Putin.
Putin is (and, according to Forbes Magazine, has been for the last four years) the most powerful person in the world. Even though the US kept its distance from Russia, he was able to manipulate us from afar. Now that Trump wants to be close to Putin, the manipulation will be even easier. Putin is better at this game. He will win and we will lose.
May 5 of 1955 can be written as 5/5/55. How many times during the 20th century the date in the format month day and the last two digits of the year can be written with the same digit?
I had a distant relative Alla, who was brought up by a single mother, who died in a car crash when the girl was in her early teens. Alla was becoming a sweet and pleasant teenager; she was taken in by her aunt after the accident. Very soon the aunt started complaining that Alla was turning into a cheater and a thief. The aunt found a therapist for Alla, who explained that Alla was stealing for a reason. Because the world had unfairly stolen her mother, Alla felt entitled to compensation in the form of jewelry, money, and other luxuries.
I was reminded of Alla's story when I was reading The (Honest) Truth About Dishonesty: How We Lie to Everyone—Especially Ourselves by Dan Ariely. Ariely discusses a wide range of reasons why honest people cheat. But to me he neglects to look at the most prominent reason. Often honest people cheat when they feel justified and entitled to do so.
One of Ariely’s experiments went like this. One group was asked to write a text avoiding letters x and z. The other group was asked to write a text avoiding letters a and n. The second task is way more difficult and requires more energy. After the tasks were completed the participants were given a test in which they had a chance to cheat. For this experiment, the participants were compensated financially according to the number of questions they solved. Not surprisingly, the second group cheated more. The book concludes that when people are tired, their guard goes down and they cheat more. I do not argue with this conclusion, but I think another reason also contributes to cheating. Have you ever tried to write a text without using the letters a and n? I did:
I should try it here. But this is so difficult. I give up.
My son, Alexey, was way better than me:
First, God brought forth the sky with the world. The world existed without form. Gloom covered the deep. The Spirit of God hovered over the fluids. Quoth God: let there be light. Thus light existed.
Fun as it is, this is cruel and unusual punishment. The request is more difficult than most people expect at an experiment. It could be that participants cheated not only because their capacity for honesty was depleted, but because they felt entitled to more money because the challenge was so difficult.
In another experiment, the participants received a high-fashion brand of sunglasses before the test. Some of them were told that the sunglasses were a cheap imitation of the luxury brand (when they really were not). This group cheated more than the group who thought that they got a real thing. The book concludes that wearing fake sunglasses makes people feel that they themselves are fake and so they care less about their honor. Unfortunately, the book doesn't explain in detail what was actually promised. It looks like the participants were promised high-fashion sunglasses. In this case, the fake group would have felt deceived and might have felt more justified to cheat.
Dear Dan Ariely: May I suggest the following experiment. Invite people and promise them some money for a 15-minute task. Pay them the promised minimum and give them a test through which they can earn more. Construct it so that they can earn a lot more if they cheat. Then make the non-control group wait for half an hour. If I were in this group, I would have felt that I am owed for the total of 45 minutes—three times more than what I was promised. I do not know if I would cheat or just leave, but I wouldn't be surprised that in this group people would cheat more than in the control group.
If like me, you fancy Raymond Smullyan and his books, then you've heard about knights and knaves. Knights always tell the truth and knaves always lie. In addition to knights and knaves, there are normal people who sometimes tell the truth and sometimes lie. Here is a puzzle.
Puzzle. How, in one sentence, can a normal person prove that they are normal?
We can draw a parallel between people and coins. We can say that knights correspond to real coins, and knaves to fake coins that are lighter than real ones. Inspired by normal people, my coauthor Konstantin Knop invented chameleon coins. Chameleon coins can change their weight and behave like real or fake coins. I just wrote a post about chameleon coins.
Normal people are too unpredictable: they can consistently pretend to be knights or knaves. So logicians invented a simpler type of person, one who switches from telling the truth in one sentence to a lie in the next and then back to the truth. Such people are called alternators. Here is another puzzle:
Puzzle. You meet a person who is one of the three types: a knight, a knave, or an alternator. In two questions, find out which type they are.
Continuing a parallel between people and coins we can define alternator coins: the coins that switch their weight each time they are on the scale from weighing as much as real ones to weighing as much as fake ones. For the purposes of this essay, we assume that the fake coins are lighter than real ones. Unlike the chameleon coin, which might never reveal itself by always pretending to be real, the alternators can always be found. How do you find a single alternator among many real coins? There is a simple strategy: repeat every weighing twice. This strategy allows us to find an alternator among 9 coins in four weighings. Can we do better?
I used the alternator coins as a research project for my PRIMES STEP program where we do math research with students in seventh and eighth grade. The students started the alternator project and immediately discovered the strategy above. The next step is to describe a better strategy. For example, what is the maximum number of coins containing one alternator such that the alternator can always be found in four weighings?
But first we count possible outcomes. Suppose there is a strategy that finds an alternator. In this strategy we can't have two unbalanced weighings in a row. To prove that, let us suppose there was an unbalanced weighing. Then the alternator switches its weight to a real coin and whether or not the alternator is on the scale, the next weighing must balance. The beauty of it is that given a strategy each outcome has to point to a particular coin as an alternator. That means the number of outcomes bounds the total number of coins that can be processed.
Counting the number of possible outcomes that do not have two unbalances in row is a matter of solving a recurrence, which I leave to the readers to find. The result is Jacobshtal numbers: the most beautiful sequence you might never have heard of. For example, the total number of possible outcomes of four weighings is 11. Since each outcome of a strategy needs to point to a coin, the total number of coins that can be processed in four weighings is not more than 11. But 11 is better than 9 in our previous strategy. Can we process 11 coins in four weighings? Yes, we can. I will describe the first part of the strategy.
So we have 11 coins, one of which is an alternator. In the first weighing we compare 5 coins against 5 coins. If the weighing unbalances, the alternator is on a lighter pan. Our problem is reduced to finding the alternator among five coins when we know that it is in the real state. If the weighing balances, then we know that if the alternator is among the coins on the scale it must now be in the light state. For the second weighting, we pick two sets of three coins out of this ten coins and compare them against each other. Notice that 3 is a Jacobsthal number, and 5, the number of coins outside the scale, is also a Jacobsthal number. If the second weighing balances, the alternator must be among 5 coins outside the scale. All but one of these coins are in the light state, and I leave it to the readers to finish the strategy. If the weighing unbalances, we need to find the alternator among 3 coins that are in the real state now. This can be done in two weighings, and again the readers are to the rescue.
It appears that Jacobsthal numbers provide the exact lower bound of the number of coins that can be processed. This is what my middle-schoolers discovered and proved. We wrote a paper on the subject. The strategy in the paper is adaptive. That means it changes depending on the results of the previous weighings. Can we find an oblivious strategy? I will tell you in later posts.
Suppose we have 3^{n} identical-looking coins. One of the coins is fake and lighter than the other coins which all are real. We also have a random scale. That is a scale that at each weighing behaves randomly. Find the fake coin in the smallest number of weighings. Oops! That won't work! It is impossible to find the fake coin. The scale can consistently misbehave in such a way as to blame a specific real coin for being fake.
Let's try something else. Suppose we have two scales: one normal and one random. Find the fake coin.
What am I thinking? The normal scale can point to one coin and the random scale can point to another coin and we are in a "she said, he said" situation which we can't resolve.
Now, in my final try, I'll make it right. We actually have three scales, one of which is random. So here we go, with thanks to my son Sergei for giving me this puzzle:
Puzzle. We have 3^{n} identical-looking coins. One of the coins is fake and lighter than the other coins, which all are real. We also have three scales: two normal and one random. Find the fake coin in the smallest total number of weighings.
Let's start with this strategy: repeat every weighing on all three scales and have a majority vote. At least two of the scales will agree, thus pointing to the true result. This way we can use a divide-into-three-equal-groups strategy for one scale to find the fake coin. It will require 3n weighings.
Can we do better? Of course, we can. We can repeat every weighing on two scales. If they agree we do not need the third scale. If they do not agree, one of the scales is random and lying, and we can repeat the weighing on the third scale to "out" the random scale. After we identify one normal scale, the process goes faster. In the worst case we will need 2n + 1 weighings.
Can we do even better? Yes, we can. I will leave it to the readers to find a beautiful solution that is asymptotically better than the previous one.
Update on Dec 24, 2016. The total number of coins should be 3^{2n}, not 3^{n}. We are looking at the worst case scenario, when the random scale is adversarial.
We all have played with problems in which we had real coins and fake (counterfeit) coins. For this post I assume that the fake coins are always lighter than the real coins. My coauthor Konstantin Knop invented a new type of a coin: a chameleon coin. This coin can mimic a fake or a real coin. It can also choose independently which coin to mimic for each weighing on a balance scale.
You cannot find the chameleon coin in a mix with real coins if it does not want to be found, because it can consistently behave as a real coin. Let's add classic fake coins to the mix, the ones that are lighter. Still the task of identifying the chameleons using a balance scale cannot be achieved: the chameleons can pretend to be fake coins. We can't identify the fake coins either, as the chameleons can mess things us up by consistently pretending to be fake.
What we can do is to find a small number of coins some of which are guaranteed to be fake. Consider the simplest setup, when we have one fake coin and one chameleon in our mix of N coins. That is we have N − 2 real coins. Our task now is to find TWO coins, one of which has to be fake. As usual we want to do it in the smallest number of weighings that guarantees that we'll find the two coins. Let me give you a fun problem to solve:
Puzzle. The total number of coins is four. And as above we have one chameleon and one classic fake. In two weighings find two coins so that one of them is guaranteed to be fake.
If you want to learn more, we just wrote a paper titled Chameleon Coins.
* * *
—Honey, we are like two parallel lines.
—Why do you say that?
—The intersection of our life paths was a mistake.
* * *
—Q: Why did the obtuse angle go to the beach?
—A: Because it was over 90 degrees.
* * *
Ancient Roman in a clothing store: How come XL is larger than L?
* * *
—Which is the odd one out: one, three, six, seven?
—Well, three of them are odd ones.
* * *
When I am with you, I solve integrals in my head, so that blood can come back to my brain.
* * *
There are two kinds of people in this world: Those that can extrapolate from incomplete data.
* * *
Seven has the word even in it, which is odd.
I have always wanted to be an honest person and have followed my honor code. Soviet Russia had two honor codes. One code was for dealing with people and the other for dealing with businesses and the government. I remind you that in Soviet Russia, all businesses were owned by the government. The money paid for work didn't have any relation to the work done. The government paid standard salaries and the businesses did whatever. Generally, that meant that they were doing nothing. Meanwhile, the government got its income from selling oil.
All people were being screwed by the government, so they had no motivation to play fair. Just as American workers do, Soviet Russians might use the work copier to make personal copies. The difference was that we didn't feel guilty at ripping off the government. We wouldn't just make a few necessary copies; we would make copies for our friends, our family, for strangers—as many copies as possible.
I moved to the United States in 1990. Several of my friends took time to explain to me the difference between Soviet Russia and the US. One of the friends, let's call her Sarah, was working as staff at Harvard University. She told me the story of a recent visit by a famous Russian professor. After he left, the department received a bill. It appears that the professor, in a short visit, used several months-worth of the department's budget allocation for copying and phone calls. I was impressed, in a good way, by the professor who I assume spent a good deal of his time making a lot of copies of papers unavailable in Russia, presumably not only for himself but also for students and colleagues. On the other hand, it was clearly wrong.
My new friend Sarah told me that in the US money does not come from nowhere, and I should include Harvard University in my honor code for people. Actually, not only Harvard, but also any place of work and the government too. Sarah also told me that since that budget problem, she was asked to talk to every incoming Russian visitor and explain to them how capitalism works. Most Russian visitors were ready to accept the rules. I too was delighted with that. It is much easier to follow one honor code than two.
I was also very happy for my son, Alexey. He was eight when we moved to Boston. Before our move I had a dilemma. Should I tell him that Lenin and Stalin were bad guys and killed millions of people? If I gave him a truthful explanation, I would also have to teach him to lie. Otherwise, if he mentioned this at school or on the street, we would be at risk of going to prison. If I didn't teach him the truth, he might become brainwashed and grow up believing in communism, which would be very, very bad.
I was so lucky that I moved to the US: I didn't have to teach my son to lie.
Centaurs, manticores, and minotaurs roam their planet. Their society is very democratic: any two animals can become sex partners. When two different species mate, their orgasm is so potent that they merge into one creature of the third species. For example, once one centaur and one manticore make love, they are reborn as one minotaur. At the beginning of the year 2016 there were 2016 centaurs, 2017 manticores, and 2018 minotaurs. They mated non-stop and at the end of the year only one creature was left on the planet. Which one?
This is one of those puzzles that I love-hate. I hate it because it is easy to answer this puzzle by inventing a specific mating pattern that ends with one animal. It is possible to get the correct answer without seeing the beauty. I love it because there is beauty in the explanation of why, if the mating ends with one animal, it has to be a specific animal.
The solution is charming, but being a mathematician this problem makes me wonder if ii is always possible to end with one animal. So I add another puzzle.
Describe the sets of parameters for which it is impossible to end up with one creature.
Today I have two new coin puzzles that were inspired by my son, Alexey Radul:
You have N > 2 identical-looking coins. All but one of them are real and weigh the same. One coin is fake and is lighter than the real ones. You also have a balance scale which might or might not be faulty. A faulty scale differs from a normal one in reversing the sense of unbalanced weighings—it shows a lighter pan as heavier and vice versa (but still shows equal-weight pans as weighing the same). What is the smallest number of weighings you need in order to figure out whether the scale is faulty?
If you think about it, this problem is isomorphic to a known problem I wrote about before:
You have N ≥ 2 identical-looking coins. All but one of them are real and weigh the same. One coin is fake and is either lighter than the real ones or heavier than the real ones. You also have a normal balance scale. What is the smallest number of weighings you need in order to figure out whether the fake coin is lighter or heavier?
To make things more interesting let's mix the problems up.
You have N > 2 identical-looking coins. All but one of them are real and weigh the same. One coin is fake and is lighter than the real ones. You also have M > 1 identical-looking balance scales. All but one of them are normal and one is faulty. The faulty scale differs from a normal one in reversing the sense of unbalanced weighings—it shows a lighter pan as heavier and vice versa (but still shows equal-weight pans as weighing the same). What is the smallest number of total weighings needed to figure out which scale is faulty?
The following problem was at a 2016 entrance test for the MIT PRIMES STEP program.
I drew several triangles on a piece of paper. First I showed the paper to Lev and asked him how many triangles there were. Lev said 5 and he was right. Then I showed the paper to Sasha and asked him how many triangles there were. Sasha said 3 and he was right. How many triangles are there on the paper? Explain.
The intended answer was 8: there were 5 triangles on one side of the paper and 3 on the other.
Most of the students didn't think that the paper might be two-sided, but they came up with other inventive ideas. Below are some of their pictures, and I leave it to you to explain why they work. All the students who submitted these pictures got a full credit for this problem on the test.
If you are a high-school student who wants to conduct research in mathematics, you should check out the MIT PRIMES program. If you enjoy solving the problems in our entrance test, that's the first indication that you might want to apply. But to determine if the program is right for you, and you are right for the program, please read the following questions and answers which have been prepared for you by Tanya Khovanova, the PRIMES Head Mentor. (This only addresses applications to PRIMES Math, and only to the research track)
Question: I do not like math competitions. Should I apply?
Answer: Math competitions are completely separate from research in mathematics. If you enjoy thinking about mathematics for long periods of time and are fascinated by our test questions, you should apply.
Question: I am good at math, but I really want to be a doctor. Should I apply?
Answer: No. PRIMES requires a huge time commitment, so math should really be your most significant interest.
Question: I want to get into Harvard, and PRIMES looks good on a resume. Should I apply?
Answer: PRIMES does look good on a resume. But if you are more passionate about, say, climate change than math, what would Harvard's admission committee see? Our experience in the program is that if math isn't your top interest, your math student may not be sufficiently impressive to be accepted at Harvard as a math researcher. At the same time, you will not be accepted as the top climate change student as you didn't invest your time in that. Math research is a hard way to earn points for college. See also, the essay, Thoughts on research by Simon Rubinstein-Salzedo.
Question: My parents want me to apply. Should I apply?
Answer: Your parents will not be accepted to the program. Do not apply if you do not really, really want to.
Question: Your website suggests that I should spend ten hours a week on the PRIMES project. I can only spend five. But I am a genius and faster than other people.
Answer: We already assume that you are a genius and faster than anyone else you know. Five hours a week are not enough for a successful project.
Question: I looked at the past PRIMES projects and nothing excites me as much as my current interest in Pascal's triangle. I doubt I should apply.
Answer: When you start working on a project, you will learn a lot about it. You will understand why, for example, Cherednik algebras are cool. The excitement comes with knowledge and invested time. Not yet being excited about Cherednik algebras is not a good reason not to apply. Besides a lot of exciting mathematics is done between several different fields.
Question: I really want to do nothing else than study Pascal's triangle.
Answer: We try to match our projects to students' interests as much as we can. But we almost never can fulfill a specific request as above. You might get a project related to Young diagrams, which are connected to quantum Pascal's triangle. If this connection doesn't excite you, you shouldn't apply.
Question: I think I will be better positioned for research if I spend five more years studying.
Answer: There is nothing wrong with this approach. For many years the standard was to start research in graduate school. Our program is innovative. At PRIMES we are trying a different model. It may sound scary, but you will learn everything you need to know in order to do your project. If the project is in representation theory, for example, you will only learn what you need—not the whole theory. Our hope is that eventually you will take a course in representation theory and expand your grasp of it and see the bigger picture behind your project. We have a reading track for people like you who reside in Boston area.
Question: I love math, but I am not sure that I want to be a mathematician. Should I apply?
Answer: Many people start loving math early in life and then discover that there are many other things that require a similar kind of brain: computer science, cryptography, finance, and so on. We do not require from our students a commitment to become mathematicians. If you want to try research in math, you should apply. If students decide that they do not want to do research in math after finishing our program, we do not consider that a negative result. One way or another, the experience of PRIMES will help you understand better what you want to do with your life.
Question: I want to get to the International Math Olympiad. I am afraid that the time the research project takes prevents me from preparing for competitions. Should I apply?
Answer: People who are good at Olympiads often have fantastic brain power that helps in research. On the other hand, research requires a different mind set and the transition might be painful. It is possible, but not trivial to succeed in both. It is up to you to decide how you want to spend your time.
Question: I like number theory, but I do not see past PRIMES projects in number theory.
Answer: Doable number theory projects are hard to come by and we have fewer number theory projects than students who want to do number theory. There are many high-school programs that teach number theory including PROMYS and Ross programs. Our applicants like number theory because they were exposed to it. During PRIMES you will be exposed to something else and might like it as much.
Question: I found a local professor to work with on a research project. Should I apply to PRIMES?
Answer: PRIMES requires that you devote 10 hours a week to research for a year. It is unrealistic to do two research projects in parallel. Choose one. Working with someone in person may be better than by Skype at PRIMES. Also, usually our mentors are not professors, but rather graduate students. On the other hand, they are MIT grad students and projects are often suggested by professors. Our program is well structured. We guarantee weekly meetings in the Spring, we give extra help with your paper, and we have a conference. It is up to you to decide.
Alexander Shapovalov is a prolific puzzle writer. He has a special webpage of his river-crossing puzzles (in Russian). Here is one of these puzzles.
Three swindlers have two suitcases each. They approach a river they wish to cross. There is one boat that can carry three objects, where a person or a suitcase counts as one object. No swindler can trust his suitcase to his swindler friends when he is away, but each swindler doesn't mind his suitcases left alone at the river shore. Can they cross the river?
In my paper with Joshua Xiong, Nim Fractals, we produced a bijection between P-positions in the three-pile Nim and a three-branch Ulam-Warburton automaton. We also defined a parent-child relationship on games that is induced by this bijection. Namely, two consecutive P-positions in a longest optimal game of Nim are the ones that correspond to a parent-child pair in the automaton. A cell in the Ulam-Warburton automaton has exactly one parent. That means, if (a,b,c) is a Nim P-position, then exactly one of (a − 1,b − 1,c), (a − 1,b,c − 1), and (a,b − 1,c − 1) must be a P-position and a parent of (a,b,c). (See our paper for more details.)
Now I want to explicitly write out the rules of an automaton which will generate the Nim P-positions in 3D.
Let me restrict the evolution of the automaton to the non-negative octant. That is, we consider points (a,b,c) in 3D, where each coordinate is a non-negative integer. We define the neighbors of the point (a,b,c) to be the points that differ from (a,b,c) in two coordinates exactly by 1. So each point strictly inside the octant has 12 neighbors. (There are three ways to choose two coordinates, and after that four ways to choose plus or minus 1 in each of them.
There is a geometric interpretation to this notion of neighborhood. Let us correspond a unit cube to a point with integer coordinates. The center of the cube is located at the given point and the sides are parallel to the axes. Then two points are neighbors if and only if the corresponding cubes share one edge. Now it becomes more visual that a cube has 12 neighbors, as it has 12 edges.
Here is the rule of the automaton. Points never die. We start with the patriarch, (0,0,0), one point being alive. The non-alive point is born inside the non-negative octant if it has exactly 1 alive neighbor that is closer to the patriarch. In other words the point (a,b,c) is born if and only if exactly one out of three points (a − 1,b − 1,c), (a − 1,b,c − 1), and (a,b − 1,c − 1) is alive. It follows that the points that are born at the n-th step has a coordinate sum 2n.
Consider for example the starting growth. At the first step the points (0,1,1), (1,0,1) and (1,1,0) are born. At the next step the points (0,2,2) and (2,0,2) and (2,2,0) are born. while the (1,1,2) will never be born as starting from the second step it has at least two live neighbors: (0,1,1) and (1,0,1) that are closer to the patriarch.
Theorem. In the resulting automaton, the points that are born at step n are exactly the P-positions of Nim with the total of 2n tokens.
Proof. Only the points with an even total can be born. Now we proceed by induction on the total number of tokens. The base case is obvious. Suppose we proved that at step n exactly P-positions with 2n tokens are born. Consider a P-position of Nim: (a,b,c) such that a + b + c = 2n + 2. Remember, that bitwise XOR of a, b, and c is zero. Consider the 2-adic values of a, b, and c (aka the smallest powers of 2 dividing a, b, and c). There should be exactly two out of these three integers that have the smallest 2-adic value. Suppose these are a and b. Then (a − 1,b − 1,c) is a P-position, while (a − 1,b,c − 1) and (a,b − 1,a − 1) are not. That means by the inductive hypothesis (a,b,c) has exactly one alive neighbor. So the position (a,b,c) is born at time n + 1.
Now we need to proof that nothing else is born. For the sake of contradiction suppose that (a,b,c) is the earliest N-position to be born. That means it has a live neighbor that is a P-position closer to the patriarch. WLOG we can assume that this neighbor is (a − 1,b − 1,c).
If a − 1 and b − 1 are both even, then (a,b,c) is a P-position, which is a contradiction. Suppose a − 1 and b − 1 are both odd. Then their binary representations can't have the same number of ones at the end. Otherwise, (a,b,c) is a P-position. That is a and b have different 2-adic values. Suppose a has a smaller 2-adic value, Then, for (a − 1,b − 1,c) to be a P-position a and c has to have the same 2-adic value. That means (a,b − 1,c − 1) is a P-position too. Now suppose a − 1 and b − 1 are of different parities. Without loss of generality suppose a − 1 is odd and b − 1 is even, then c is odd. Then (a − 1,b,c − 1) is a P-position too. Thus we can always find a second neighbor with the same number of tokens. That is, both neighbors are alive at the same time; and the N-position (a,b,c) is never born. □
One might wonder what happens if we relax the automaton rule by removing the constraint on the distance to the patriarch. Suppose a new point is born if it has exactly one neighbor alive. This will be a different automaton. Let us look at the starting growth, up to a permutation of coordinates. At step one, positions (0,1,1) are born. At the next step positions (0,2,2) are born. At the next step positions (0,1,3), (1,2,3) and (0,3,3) are born. We see that (0,1,3) is not a P-positions. What will happen later? Will this N-position mess up the future positions that are born? Actually, this automaton will still contain all the P-positions of Nim.
Theorem. In the new automaton, the points that are born at step n and have total of 2n tokens are exactly the P-positions of Nim with the total of 2n tokens.
Proof. Only the points with an even total can be born. Now we proceed by induction on the total number of tokens. The base case is obvious. The birth of the points that have total of 2n tokens and are born at step n depend only on the points with the total of 2n − 2 tokens that are born at step n − 1. By the inductive hypothesis, those are the P-positions with 2n − 2 tokens. So the points have total of 2n tokens and are born at step n match exactly the first automaton described above. To reiterate, N-positions with 2n tokens are born after P-positions with 2n tokens, so they do not influence the birth of P-positions with 2n + 2 tokens. □
In the game of Nim you have several piles with tokens. Players take turns taking several tokens from one pile. The person who takes the last token wins.
The strategy of this game is well-known. You win if after your move the bitwise XOR of all the tokens in all the piles is 0. Such positions that you want to finish your move with are called P-positions.
I play this game with my students where the initial position has four piles with 1, 3, 5, and 7 tokens each. I invite my students to start the game, and I always win as this is a P-position. Very soon my students start complaining that I go second and want to switch with me. What should I do? My idea is to make the game last long (to have many turns before ending) to increase the chances of my students making a mistake. So what is the longest game of Nim given that it starts in a P-position?
Clearly you can't play slower then taking one token at a time. The beauty of Nim is that such an optimal game starting from a P-position is always possible. I made this claim in several papers of mine, but I can't find where this is proven. One of my papers (with Joshua Xiong) contains an indirect proof by building a bijection to the Ulam-Warburton automaton. But this claim is simple enough, so I want to present a direct proof here. Actually, I will prove a stronger statement.
Theorem. In an optimal game of Nim that starts at a P-position the first player can take one token at each turn so that the second player is forced to take one token too.
Proof. Consider a P-position in a game of Nim. Then find a pile with the lowest 2-adic value. That is the pile such that the power of two in its factorization is the smallest. Suppose this power is k. Notice that there should be at least two piles with this 2-adic value.
The first player should take a token from one of those piles. Then the bitwise heap-sum after the move is 2^{k+1}−1. Then the Nim strategy requires the second player to take tokens from a pile such that its value decreases after bitwise XORing with 2^{k+1}−1. All piles with the 2-adic value more than k will increase after xoring with 2^{k+1}−1. That means the second player has to take tokens from another pile with 2-adic value k. Moreover, the second player is forced to take exactly one token to match the heap-sum. □
In the position (1,3,5,7) all numbers are odd, so I can take one token from any pile for my first move, then the correct move is to take one token from any other pile. My students do not know that; and I usually win even as the first player. Plus, there are four different ways I can start as the first player. This way my students do not get to try several different options with the same move I make. After I win several times as the first player, I convince my students that I win anyway and persuade them to go back to me being the second player. After that I relax and never lose. I am evil.
Replace question marks with letters in the following sequence:
Y, Y, ?, ?, Y, ?, Y, ?, R, R, R, R.
Problem 1. It is true that it is possible to put positive numbers at the vertices of a triangle so that the sum of two numbers at the ends of each side is equal to the length of the side?
This looks like a simple linear algebra question with three variables and three equations. But it has a pretty geometrical solution. What is it?
Problem 2. Prove that it is possible to assign a number to every edge of a tetrahedron so that the sum of the three numbers on the edges of every face is equal to the area of the face.
Again we have six sides and four faces. There should be many solutions. Can you find a geometric one?
Do you know how to cut a cake? I mean, mathematically. There is a whole area of mathematics that studies cake cutting.
Mathematicians usually assume that each person has their own idea of what is the best part of the cake. Suppose three sisters are celebrating the New Year by having a cake. Anna likes only icing, Bella likes only chocolate chips, and Carol likes only pieces of walnuts mixed into the cake. Mathematicians want to cut cakes fairly. But what is fair? Fair is fair, right? Wrong. There are several different notions of fair cake division.
There is a proportionate division. In such a division every sister gets at least one-third of her value of the cake. This seems fair. Let's see an example. Anna gets one third of the icing, Bella gets one third of the chocolate chips, and Carol gets everything else. This is a fair proportionate division. Each of the sisters believes that she got at least one-third of the cake, in their own value. But it doesn't seem quite fair.
There is a stronger notion of fairness. It is called envy-free. In this division each sister gets at least one-third of the cake and, in addition, none of the three sisters would improve their value by swapping pieces. That means, if Anna wants only icing, not only does she get at least one-third of the icing, but also no one else gets more icing than Anna. The previous example of the proportionate division is not envy-free. Carol got two-thirds of the icing, so Anna would want to switch with her.
Let's try a different division. Anna gets one third of the icing, Bella gets the chocolate chips and another third of the icing, and Carol gets all the walnuts and another third of the icing. Formally, this is envy-free cake cutting. But poor Anna. What do you think Anna feels when she sees the smiles of contentment on the faces of her sisters? Whoever invented the name doesn't understand envy. Anna got one-third of the cake by her value, but the other sisters got the whole cake!
Luckily mathematicians understand this conundrum. So they invented another name for a cake division. They call a division equitable if everyone values all the pieces the same. So the division above is envy-free but not equitable. Let's try again. Let's give each sister one-third of all the components of the cake. This division is very good mathematically: it is proportionate, envy free, and equitable. By the way, envy-free division is always proportionate. This division seems fair. But is it a good division?
There is another term here: Pareto-efficient division means that it is impossible to make one person feel better, without making another person feel worse. All divisions above are not Pareto-efficient. Moving some icing from Carol to Anna, doesn't decrease the value for Carol, but increases the value for Anna.
There is an even better way to divide the cake. We can give the icing to Anna, the walnuts to Bella, and the chocolate chips to Carol. This division is proportionate, envy-free, and Pareto-efficient. It is perfect. Mathematicians even have a word for it. They call it a perfect division.
Mathematically this division is perfect. Unfortunately, sisters are not. I know an Anna who would still envy Bella.
Why are manhole covers round? The manhole covers are round because the manholes are round. Duh! But the cute mathematical answer is that the round shapes are better than many other shapes because a round cover can't fall into a round hole. If we assume that the hole is the same shape as the cover but slightly smaller, then it is true that circular covers can't fall into their holes. But there are many other shapes with this property. They are called the shapes of constant width.
Given the width, the shape with the largest area is not surprisingly a circle. The shape with the smallest area and a given constant width is a Reuleaux Triangle. Here is how to draw a Reuleaux triangle. Draw three points that are equidistant from each other at distance d. Then draw three circles of radius d with the centers at given points. The Reuleaux triangle is the intersection of these three circles.
Can we generalize this to 3d? What would be an analogue of a Reuleaux Triangle in 3d? Of course, it is a Reuleaux Tetrahedron: Take four points at the vertices of a regular tetrahedron; take a sphere at each vertex with the radius equal to the edge of the tetrahedron; intersect the four spheres.
Is this a shape of the constant width? Many people mistakenly think that this is the case. Indeed, if you squeeze the Reuleaux tetrahedron between two planes, one of which touches a vertex and another touches the opposite face of the curvy tetrahedron, then the distance between them is equal to d: the radius of the circle. This might give you the impression that this distance is always d. Not so. If you squeeze the Reuleaux tetrahedron between two planes that touch the opposite curvy edges, the distance between these planes will be slightly more than d. To create a shape of constant width you need to shave off the edges a bit.
Theoretically you can shave the same amount off every edge to get to a surface of constant width. But this is not the cool way to do it. The cool way is to shave a bit more but only from one edge of the pair of opposite edges. You can get two different figures this way: one that has three shaved edges forming a triangle, and the other, where three shaved edges share a vertex. These two bodies are called Meissner bodies and they are conjectured to be shapes of the constant width with the smallest volume.
On the picture I have two copies of a pair of Meissner bodies. The two left ones have three edges that share a vertex shaved off. The very left shape gives a top view of this vertex and the solid next to it has its bottom with holes looking forward. The two shapes on the right show the second Meissner body in two different positions.
I recently discovered a TED-Ed video about manhole covers. It falsely claims that the Reuleaux tetrahedron has constant width. I wrote to TED-Ed, to the author, and posted a comment on the discussion page. There was no reaction. They either should remove the video or have an errata page for it. Knowingly keeping a video with an error that is being viewed by thousands of people is irresponsible.
I am running a PRIMES-STEP program for middle school students, where we try to do research in mathematics. In the fall of 2015 we decided to study the following topic in logic.
Suppose there is an island where the following four types of people live:
See if you can solve this simple logic puzzle about people on this island.
It is known that exactly one person stole an expensive painting from an apartment. It is also known that only Alice or Bob could have done it. Here are their statements:
Alice: I am guilty. Bob is a truth-teller.
Bob: I am guilty. Alice stole it. Alice is the same type as me.
Who stole the painting and what types are Alice and Bob?
My students and I discovered a lot of interesting things about these four types of people and wrote a paper: Who Is Guilty?. This paper contains 11 cute logic puzzles designed by each of my 11 students.
I envied my students and decided to create two puzzles of my own. You have already solved the one above, so here is another, more difficult, puzzle:
A bank was robbed and a witness said that there was exactly one person who committed the robbery. Three suspects were apprehended. No one else could have participated in the robbery.
Alice: I am innocent. Bob committed the crime. Bob is a truth-teller.
Bob: I am innocent. Alice is guilty. Carol is a different type than me.
Carol: I am innocent. Alice is guilty.
Who robbed the bank and what types are the suspects?
The following cute puzzle of unknown origins was sent to me by Martina Balagovic and Vincent van der Noort:
A car travels from A to B and back again. When going uphill it goes 56 km an hour, when going downhill it goes 72 km an hour and while driving on flat surface it goes 63 km an hour. Getting from A to B takes 4 hours and getting back (over the same road) takes 4 hours and 40 minutes.
What is the distance between A and B?
I received a book Really Big Numbers by Richard Schwartz for review. I was supposed to write the review a long time ago, but I've been procrastinating. Usually, if I like a book, I write a review very fast. If I hate a book, I do not write a review at all. With this book I developed a love-hate relationship.
Let me start with love. I enjoyed reading the first 80 pages. The pictures are great, and some explanations are very well thought out. Plus, I haven't thought much about really big numbers, so the book helped me understand them. I was impressed with how this book treats very difficult ideas with simple explanations and illuminating images. I was captivated by it.
Now to hate. I had two problems with this book: one pedagogical and the other mathematical.
The pedagogical issue. The beginning of the book is suitable for small children. Most of the book is suitable for advanced middle-schoolers who like mathematics. The last part is very advanced. Is it a good idea to show children a book that looks like a children's book, but which soon becomes totally out of their reach? Richard Schwartz understands it and says many times that pieces of this book might be read several years apart. Several years? What child is ready to wait several years to finish a book? How would children feel about the book and about numbers when no matter how hard they try, they cannot understand the end of the book?
As a reviewer, I can't recommend the full book for kids who are not ready to grasp the notion of the Ackermann function or arrow notation. Even if the child is capable of understanding these ideas, there are mathematical issues that would prevent me from recommending it.
The mathematical issue. Let me start by explaining the notion of plex. We call an n-plex a number that is equal to 10^{n}. For example, 2-plex means 10^{2} which is 100, and 10-plex means 10^{10}. The fun part starts when we plex plexes. The number n-plexplex means 10 to the power n-plex which is 10^{(1010)}. We can continue plexing: n-plexplexplex means 10 to the power n-plexplex. When you are hunting for really big numbers, it is easier to write the number of plexes rather than writing plexes after plexes. Richard Schwartz introduces the following notation to help visualize the whole thing. He puts numbers in a square. Number n in a square means 1-plexed n times. For example, 2 inside a square means 10^{10}. Ten inside a square is 1-plexplexplexplexplexplexplexplexplexplex.
We can start nesting squares. For example, 2 inside a square means 1-plex-plex or 10^{10}. Let's add a square around it: 2 inside two squares means 10^{10} inside a square, which equals 1 plexed 10^{10} times. To denote 10 nested in n squares Richard Schwartz uses the next symbol: n inside a pentagon. For example, 1 inside a pentagon is 10 inside a square. I wrote this number in the previous paragraph: it is 1-plexplexplexplexplexplexplexplexplexplex. Similarly, n inside a hexagon means 10 inside n nested pentagons. We can continue this forever: n inside a k-gon is 10 inside k nested (k-1)-gons.
What bothers me is why a square? Why not a triangle? If we adopt this scheme, what is the meaning of a number in a triangle? Let's try to unravel this. Following Richard Schwartz's notation we get that n inside a square is the same as 10 inside n nested triangles. What do we do n times with 10 to get to 1 plexed n times? 1-plexed n times is 10 plexed n-1 times. There is a disconnect in notation here. For example, 10 in two nested triangles should mean 2 inside a square that is 10^{10}. 10 inside one triangle should mean 1 inside a square which is 10. This doesn't make any sense.
I started googling around and discovered the Steinhaus–Moser notation. In this notation a number n in a triangle means n^{n}. A number n in a square means the number n inside n nested triangles. A number n in a pentagon means the number n inside n nested squares. And so on. This makes total sense to me. If we move down the number of sides, we can say that the number n inside a 2-gon means n times n and the number n inside a 1-gon means n. This is perfect.
Schwartz changed the existing notation in two places. First he made everything about 10. This might not be such a bad idea except his 10 inside a square doesn't equal 10 inside a square in Steinhaus–Moser notation. In Steinhaus–Moser notation 10 inside a square means 10 plexed 10 times. The author removed one of the plexes. He made 10 inside the square to mean 1 plexed 10 times, and as a result it stopped working.
Even though the first 83 pages are delightful and the pictures are terrific, the notation doesn't work. What a pity.
I have solved too many puzzles in my life. When I see a new one, my solution is always the intended one. But my students invent other ideas from time to time and teach me to think creatively. For example, I gave them this puzzle:
Two boys wish to cross a river, but there is a single boat that can take only one boy at a time. The boat cannot return on its own; there are no ropes or similar tricks; yet both boys manage to cross the river. How?
Here is what my inventive students came up with:
And here is my standard solution: They started on different sides of the river.
I gave a talk about thinking inside and outside the box at the Gathering for Gardner conference. I mentioned this puzzle and the inventiveness of my students. After the conference a guy approached me with another answer which is now my favorite:
Consider a triangle with vertices A, B, and C. Let O be its orthocenter. Let's connect O to the vertices. We get six lines: three sides of the triangle and three altitudes. These six lines are pair-wise orthogonal: AO ⊥ BC, BO ⊥ AC, and CO ⊥ AB.
It is easy to see that A is the orthocenter of the triangle OBC, and so on: each vertex is the orthocenter of the triangle formed by the other three. We say that these four points form an orthocentric system.
I heard a talk about this structure at the MOVES 2015 conference by Richard Guy. What I loved in his talk was his call to equality and against discrimination. The point O plays the same role as the other three points. It should be counted. Richard Guy suggested calling this system an orthogonal quadrangle. I am all for equality. This is not a triangle, this is a quadrangle!
I want the world to be a better place. My contribution is teaching people to think. People who think make better decisions, whether they want to buy a house or vote for a president.
When I started my blog, I posted a lot of puzzles. I was passionate about not posting solutions. I do want people to think, not just consume puzzles. Regrettably, I feel a great push to post solutions. My readers ask about the answers, because they are accustomed to the other websites providing them.
I remember I once bought a metal brainteaser that needed untangling. The solution wasn't included. Instead, there was a postcard that I needed to sign and send to get a solution. The text that needed my signature was, "I am an idiot. I can't solve this puzzle." I struggled with the puzzle for a while, but there was no way I would have signed such a postcard, so I solved it. In a long run I am glad that the brainteaser didn't provide a solution.
That was a long time ago. Now I can just go on YouTube, where people post solutions to all possible puzzles.
I am not the only one who tries to encourage people to solve puzzles for themselves. Many Internet puzzle pages do not have solutions on the same page as the puzzle. They have a link to a solution. Although it is easy to access the solution, this separation between the puzzle and its solution is an encouragement to think first.
But the times are changing. My biggest newest disappointment is TED-Ed. They have videos with all my favorite puzzles, where you do not need to click to get to the solution. You need to click to STOP the solution from being fed to you. Their video Can you solve the prisoner hat riddle? uses my favorite hat puzzle. (To my knowledge, this puzzle first appeared at the 23-rd All-Russian Mathematical Olympiad in 1997.) Here is the standard version that I like:
A sultan decides to give 100 of his sages a test. He has the sages stand in line, one behind the other, so that the last person in line can see everyone else. The sultan puts either a black or a white hat on each sage. The sages can only see the colors of the hats on all the people in front of them. Then, in any order they want, each sage guesses the color of the hat on his own head. Each hears all previously made guesses, but other than that, the sages cannot speak. Each person who guesses the color wrong will have his head chopped off. The ones who guess correctly go free. The rules of the test are given to them one day before the test, at which point they have a chance to agree on a strategy that will minimize the number of people who die during this test. What should that strategy be?
The video is beautifully done, but sadly the puzzle is dumbed down in two ways. First, they explicitly say that it is possible for all but one person to guess the color and second, that people should start talking from the back of the line. I remember in the past I would give this puzzle to my students and they would initially estimate that half of the people would die. Their eyes would light up when they realized that it's possible to save way more than half the people. They have another aha! moment when they discover that the sages should start talking from the back to the front. This way each person sees or hears everyone else before announcing their own color. Finally, my students would think about parity, and voilà, they would solve the puzzle.
In the simplified adapted video, there are no longer any discoveries. There is no joy. People consume the solution, without realizing why this puzzle is beautiful and counterintuitive.
Nowadays, I come to class and give a puzzle, but everyone has already heard the puzzle with its solution. How can I train my students to think?
I stumbled upon a TED-Ed video with a frog puzzle:
You're stranded in a rainforest, and you've eaten a poisonous mushroom. To save your life, you need an antidote excreted by a certain species of frog. Unfortunately, only the female of the species produces the antidote. The male and female frogs occur in equal numbers and look identical. There is no way to distinguish between them except that the male has a distinctive croak. To your left you spot a frog on a tree stump. You hear a croak from a clearing in the opposite direction, where you see two frogs. You can't tell which one made the sound. You feel yourself starting to lose consciousness, and you realize that you only have time to run in one direction. Which way should you go: to the clearing and lick both frogs or to the tree stump and lick the stump frog?
My first thought was that male frogs croak to attract female frogs. That means the second frog in the clearing is probably an already-attracted female. The fact that the stump frog is not moving means it is male. I was wrong. This puzzle didn't assume any knowledge of biology. The puzzle assumes that each frog's gender is independent from other frogs. Thus this puzzle is similar to two-children puzzles that I wrote so much about. I not only blogged about this, but also wrote a paper: Martin Gardner's Mistake.
As in two-children puzzles, the solution depends on why the frog croaked. It is easy to make a reasonable model here. Suppose the male frog croaks with probability p. Now the puzzle can be solved.
Consider the stump frog before the croaking:
Consider the two frogs in the clearing before the croaking:
The probabilities corresponding to our outcome—a non-croaking frog on the stump and one croaking frog in the clearing—are in bold. Given that the stump frog is silent, the probability that it's a female is 1/(1-p). Given that one clearing frog croaked, the probability that one of them is a female is p/2 divided by p(1-p)/2. The ratio is the same 1/(1-p). It doesn't matter where you go for the antidote.
The TED-Ed's puzzle makes the same mistake that is common in the two-children puzzles. I don't want to repeat their incorrect solution. The TED-Ed's frog puzzle is wrong.
When I receive a bank statement, I review all the transactions. The problem is my failing memory. I do not remember when and where I last took money from an ATM, and how much. So I decided to create a pattern. I used to take cash in multiples of 100. Probably most people do that. A better idea is to always take the amount that has a fixed remainder modulo 100, but not 0. For example, let's say I take one of the following amounts: 40, 140, 240, or 340. Sometimes I need more money, sometimes less. This set of numbers covers all of my potential situations, but my pattern is that all the numbers end in 40. This way if someone else gets access to my account, they will almost surely take a multiple of 100. I will be able to discover a fraud without remembering the details of my last withdrawal.
In addition, when I first started doing this, I was hoping I wouldn't need to wait until I review my own statement to discover problems. My hope was that if a thief tried to take cash from my account, my bank would notice a change in the pattern and notify me immediately. Now I realize that this was wishful thinking. I doubt that banks are as smart as I am.
One day I should try an experiment. I should go to an ATM I never use and withdraw 200 dollars. I wonder if my bank would notice.
One of my jobs is giving linear algebra recitations at MIT. The most unpleasant aspect of it is dealing with late homework. Students attempt to submit their homework late for lots of different reasons: a sick parent needing help, stress, a performance at Carnegie Hall, a broken printer, flu, and so on. How do I decide which excuse is sufficient, and which is not? I do not want to be a judge! Moreover, my assumption is that people tell the truth. In the case of linear algebra homework, this assumption is unwise. As soon as students discover that I trust everyone, there's a sharp increase in the number of sick parents and broken printers. So I have the choice of being either a naive idiot or a suspicious cynic. I do not like either role.
Our linear algebra course often adopts a brilliant approach: We announce that late homework is not allowed for any reason. To compensate for emergencies, we drop everyone's lowest score. That is, we allow the students to skip one homework out of ten. They are free to use this option for oversleeping the 4 pm deadline. If all their printers work properly and they do not get so sick that they have to skip their homework, they can forgo the last homework. Happily, this relieves me from being a judge.
Or does it? Unfortunately, this policy doesn't completely resolve the problem. Some students continue trying to push their late homework on me, despite the rules. In order to be fair to other students and to follow the rules, I reject all late homework. The students who badger me nonetheless waste my time and drain my emotions. This is very unpleasant.
From the point of view of those students, such behavior makes sense. They have nothing to lose and they might get some points. There is no way to punish a person who tries to hand in homework late and from time to time they stumble onto a soft instructor who accepts the homework against the official rules. Because such behavior is occasionally rewarded, they continue doing it. I believe that wrong behavior shouldn't be rewarded. As a responsible adult, I think it is my duty to counteract the rewards of this behavior. But I do not know how.
What should I do? Maybe I should…
Maybe I should just do number 4 right now and explain what I feel. A student who persists in handing in homework late feels to me that s/he is entitled and is better than everyone else, and shows that s/he doesn't care about the rules and honor. Again, this wastes my time, puts me in a disagreeable position, and reduces my respect for that student.
Earlier I suggested that students don't have anything to lose by such behavior, but in fact, they do pay a price, even though they may not understand that.
I found the following puzzle on a facebook page of Konstantin Knop (in Russian):
Seven astronauts landed on a small spherical asteroid. They wanted to explore it and walked in different directions starting from the same location. All used the same walking algorithm: walk x kilometers forward, turn 90 degrees left and walk another x kilometers, turn 90 degree left again and walk the last x kilometers. The value of x was different for different astronauts and was one of 30, 40, 50, 60, 70, 80, and 90.
All but one astronaut finished in the same location. What was the value of x for the astronaut who finished alone? What is the size of the asteroid?
I've been collecting math jokes for many years. I thought I've seen them all. No. Inventive people continue to create them. I was recently sent a link to a math joke website that features many jokes that are new to me. Here are my favorites:
* * *
With massive loss of generality, let $n=5$.
* * *
How do you prove a cotheorem? Using rollaries.
* * *
$0\to A\to B \to C \to 0$. Exactly.
* * *
Let $\varepsilon\to 0$. There goes the neighborhood!
* * *
Take a positive integer $N$. No, wait, $N$ is too big; take a positive integer $k$.
* * *
Calculus has its limits.
* * *
There is a fine line between a numerator and a denominator.
* * *
There's a marked difference between a ruler and a straightedge.
* * *
Suppose there is no empty set. Then consider the set of all empty sets.
* * *
Q: Why is it an insult to call someone "abelian"?
A: It means they only have a 1-dimensional character, and are self-centered.
* * *
Q: What's a polar bear?
A: A rectangular bear after a coordinate transform.
* * *
A logician rides an elevator. The door opens and someone asks:
—"Are you going up or down?"
—"Yes."
Russians cherish the issues of Kvant, a famous Soviet monthly magazine for high-school students devoted to math and physics. At its peak its circulation was about 300,000, which is unparalleled for a children's science journal. I still have my old childhood issues somewhere in my basement. But one issue is very special: it has a prominent place in my office. I didn't receive it by subscription; I received it as a gift from my brother Misha.
Strictly speaking Misha is not my brother, but rather a half-brother. His full name is Mikhail Khovanov and he is a math professor at Columbia. The signed issue he gave me contains his math problem published in the journal. He invented this problem when he was a 10th grader. Here it is:
In a convex n-gon (n > 4), no three diagonals are concurrent (intersect at the same point). What is the maximum number of the diagonals that can be drawn inside this polygon so that all the parts they divide into are triangles?
He designed other problems while he was in high school. All of them are geometrical in nature. The journal is available online, and a separate document with all the math problems is also available (in Russian). His problems are M1038, M1103, M1108 (above), M1119, M1153, M1204. I like his other problems too. M1153 (below) is the shortest problem on his list: as usual I am guided by my laziness.
What's the greatest number of turns that a rook's Hamiltonian cycle through every cell on an 8 by 8 chessboard can contain?
I wanted to accompany this post with a picture of my brother at the age he was when he invented these problems—about 16. Unfortunately, I don't have a quality picture from that period. I do have a picture that is slightly off: by about ten years.
What are the Ford circles? A picture is worth a thousand words, so here is a picture.
We draw a circle for any rational number p/q between 0 and 1 inclusive. We assume that p/q is the representation of the number in the lowest terms. Then the center of the circle is located at (p/q,1/q) and the radius is 1/q. The number inside a circle is q.
Here's the game. We start with any circle on the picture, except for the two largest circles corresponding to integers 0 and 1. In one move we can switch to a larger circle that touches our circle. The person who ends up at the two largest circles corresponding to integers 0 or 1 loses. Equivalently, the person who ends in the central circle marked "2" wins.
There are other ways to describe moves in this game in terms of rational numbers corresponding to circles, that is, the x-coordinates of their centers. Two circles corresponding to numbers p/q and r/s touch each other iff one of the following equivalent statements is true:
Let me explain the last bullet. Given two rational numbers in their lowest terms a/b and c/d, we generate their mediant as: (a+c)/(b+d). We call the two numbers a/b and c/d the parents of the mediant (a+c)/(b+d). The Stern-Brocot tree starts with two parents 0/1 and 1/1. Then their mediant is inserted between them to create a row: 0/1, 1/2, and 1/1. Then all possible mediants of two consecutive numbers are inserted in a given row to get a new row. The process repeats ad infinitum. The famous theorem states that any rational number between 0 and 1 will appear in the process.
What I like about this game is a simple and beautiful description of P-positions (These are the positions you want to end your move at in order to win.) P-positions are numbers with even denominators in their lowest terms.
In the picture above P-positions are blue, while other positions are red. All circles touched by blue are red. And if we look at the larger neighbors of every red circle, one of them is blue and one is red.
Let's prove that the numbers with even denominators satisfy the conditions for P-positions. First, two circles corresponding to numbers with even denominators can't touch each other. Indeed, the cross-determinant of two such fractions is divisible by 2. Second, each red circle has to touch one blue and one red circle with larger radii. Indeed, the circles with larger radii touching a given circle are exactly the parents of the circle. If the mediant has an odd denominator, then one of the parents must have an even denominator and the other an odd denominator.
Let us start with regular (non-weighted) mediants. Suppose we have two fractions a/b and c/d, the mediant of these two numbers is the wrong way a child might sum these two fractions: (a+c)/(b+d). There is nothing wrong with this childish way of summing, it is just not a sum of two numbers, but rather an operation the result of which is called a mediant. One must be careful: the result doesn't depend on the initial numbers chosen, but on their representations. The way to deal with this is to assume that a/b, c/d, and (a+c)/(b+d) are in their lowest terms.
The numerical value of the mediant is always in between given numbers. This is probably why it is called a mediant.
Suppose you start with two rational numbers 0/1 and 1/1, and insert a mediant between them. If you continue doing it recursively, you can reach any rational number between 0 and 1. This is a well-known property of the mediants. For example, after three rounds of inserting mediants into the initial sequence (0/1, 1/1), we get the sequence: 0/1, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 1/1.
The mediants are easy to generalize if we assign weights to initial fractions. Suppose the first fraction has weight m and the second n, then their weighted mediant is (am+cn)/(bm+dn). The simplest generalization happens when the total weight, m+n is 3. In this case, given two rational numbers a/b and c/d, the left mediant is (2a+c)/(2b+d) and the right one is (a+2c)/(b+2d). Obviously the inequality property still holds. If a/b ≤ c/d, then a/b ≤ (2a+c)/(2b+d) ≤ (a+2c)/(b+2d) ≤ c/d.
James Propp suggested the following question for our PRIMES project. Suppose the starting numbers are 0/1 and 1/1. If we recursively insert two weighted mediants in order between two numbers we will get a lot of numbers. For example, after two rounds of inserting weighted mediants into the initial sequence (0/1, 1/1), we get the sequence: 0/1, 1/5, 2/7, 1/3, 4/9, 5/9, 2/3, 5/7, 4/5, 1/1. It is easy to see that new fractions must have an odd denominator. Thus unlike the standard case, not every fraction will appear. The question is: will every rational number between 0 and 1, which in reduced form has an odd denominator appear?
I started working on this project with Dhroova Aiylam when he was a high-school junior. We didn't finish this project during the PRIMES program. But Dhroova finished another project I already wrote about: he analyzed the case of the standard mediants with any starting points. He showed that any rational number in between the starting points appears.
Dhroova became an undergraduate student at MIT and we decided to come back to the initial PRIMES project of weighted mediants. In our paper Stern-Brocot Trees from Weighted Mediants we prove that indeed every fraction with an odd denominator between 0 and 1 appears during the recursive procedure. We also analyzed what happens if we start with any two rational numbers.
My brother, Mikhail Khovanov, has invented a new game: Ringiana. It is now available for iPhone, and soon should be available for Android.
In the starting position you see four multi-colored quadrants of a ring. For example, the first picture shows the starting position of level 33.
You can either swipe or touch between the quadrants. A swipe expands one quadrant into two quadrants and compresses two other quadrants into one. You can swipe clockwise or counterclockwise. The second figure shows the result of the clockwise swipe of the North opening. The NW quadrant that was half-red and half-yellow has expanded into two quadrants. The red piece now occupies the entire NW quadrant and the yellow piece—the entire NE quadrant. Two East quadrants got contracted into one SE quadrant. The former blue NE quadrant became the top blue half of the SE quadrant. The former SE half-blue half-green quadrant became the bottom half of the SE quadrant. In general the quadrant where the swiping movement starts expands in the direction of the swipe and the quadrant where the swiping movements ends together with the next quadrant compresses.
You can also touch an opening between quadrants. In this case the neighboring quadrants exchange places. The third figure shows the result of touching the East opening.
The goal of the game is to reach the final position: have each quadrant in one color. The next image shows the end of this particular level. As you can see the game was finished in 7 moves. In this particular case this is the smallest number of moves possible. To tell you a secret, it wasn't me who finished the game in the smallest number of moves: it was my brother.
There is also a tutorial for this game on youtube.
I was at the 1976 International Mathematics Olympiad as part of the USSR team. There were eight people on the team and I decided to find out what they have achieved in the last 40 years. Here is the picture of our team. From left to right: Sergey Finashin, Yuri Burov, Nikita Netsvetaev, Boris Solomyak, Alexander Goncharov, Tanya Khovanova, Sergei Mironov, our chaperone Z.I. Moiseeva, our team leader A.P. Savin, no clue who this is (probably a translator), Piotr Grinevich.
The list below is ordered by the number of points received at the Olympiad.
Tanya Khovanova, 39 points: a lecturer at MIT. I am interested in a wide range of topics, mostly recreational mathematics and have written 60 papers.
Sergey Finashin, 37 points: a full professor at Middle East Technical University in Turkey, who is interested in topology. He wrote 40+ papers.
Alexander Goncharov, 37 points: a full professor at Yale University. He is the highest achiever on the team. He won the EMS Prize in 1992 and was an Invited Speaker at the 1994 International Congress of Mathematicians. He is interested in geometry, representation theory, and mathematical physics. He published 75 papers in refereed journals.
Nikita Netsvetaev, 34 points: a full professor at Saint-Petersburg University in Russia. He is interested in topology and algebraic geometry and wrote 20 papers.
Boris Solomyak, 31 points: a full professor at Bar-Ilan University. He is interested in fractal geometry and dynamical systems and wrote 90 papers.
Piotr Grinevich, 26 points: a leading scientific researcher at the Landau Institute for Theoretical Physics. He also teaches at Moscow State University. He is interested in integrable systems and wrote 80 papers.
Sergei Mironov, 24 points. Sergey became very ill while he was an undergrad. He stopped doing mathematics.
Yuri Burov, 22 points. Yuri wrote two papers, but quit mathematics after graduate school. He died several years ago from multiple sclerosis.
Six out of eight people on our team became mathematicians. Or more precisely five an a half. I consider myself a mathematician and am grateful for my position at MIT, where I run innovative programs for young mathematicians. But in the research world, a lecturer is a nobody. This makes me sad. I had to take breaks from research in order to raise my two children. And then I had to work in the private sector in order to support them. I was the best on my team and now I am the only one who is not a full professor. If you are looking for an example of how gender affects a career in academia, this is it.
For many years I tried to lose weight using the idea that most appealed to me: intuitive eating. By that I mean that you eat only when you are hungry. Looking back at frustrating years of trying, I wonder what took me so long to realize that this doesn't work.
I am most hungry at the end of a meal. The only time I want to kill people is when they get between me and my future third helping of cheesecake. I never get the same ravenous feeling if I haven't eaten for some time, or have just started eating.
I am least hungry in the morning. I can work on my computer for hours before I even remember that I need to eat.
When my stomach signals that it needs more food, it is always wrong. Recently, I discovered what is right. I am using myfitnesspal app for my phone that counts my calories. From time to time I know that I haven't eaten enough. At these times, I don't experience my hunger in my gut. I feel it in my head: I feel dizzy.
I decided to drop the idea of intuitive eating and outsource the decision of when and what to eat to a higher power: my smart-phone.
After graduating from MIT, my son Sergei joined Two Sigma, a hedge-fund. Last year he came to MIT representing Two Sigma at a career fair. I visited him in his booth. He was just standing there next to himself on the poster.
I should stop worrying about him. If math does not work out, he has a chance at modelling.
I have written ad nauseam about the ambiguity of problems with children. Usually a problem with two children is formulated as follows:
Mr. Smith has two children and at least one of them is a boy. What is the probability that he has two boys?
I don't want to repeat my arguments for why this problem is ambiguous. Today I want to discuss other problematic assumptions about these problems.
Assumption 1: The probability of a child being a boy is 1/2. We know that this is not the case. Usually boys are born more often than girls. In addition to that, when policy interferes, the numbers can change. When China had their one-child policy, 118 boys were born per 100 girls. That makes a probability of a boy 0.54.
Assumption 2: The gender of one child in a family is independent of the gender of the other children. I am not sure where this assumption comes from, but I easily came up with a list of possible influences on this situation.
I would like to discuss how the last bullet point changes the probabilities in two children problems. Let us consider China. Up to now China had a one-child policy with some exceptions. In some cases if the first child was a girl, the family was allowed a second child. For the sake of argument, imagine a county where people are allowed to have a second child only if the first one is a girl. A family with two boys wouldn't exist in this county. Thus the probability of having two boys is zero.
I tried to find the data about the distribution of children by gender in multi-children families. I couldn't find any. I would be curious to know what happens in real life, especially in China.
A long time ago, before anyone ever heard about ultrasound, there was a psychic who could predict the gender of a future child. No one ever filed a complaint against him.
The psychic had a journal in which he wrote the client's name and the gender of the future child. The beauty of the scam was that what he wrote in the journal was the opposite gender that he had predicted. Whenever a client complained that the gender was wrong, he would show the journal and argue that the client had misunderstood.
Happy clients don't return to complain.
Oh, the power of conditional probability! It is useful to understand it to run scams or to expose them.
Last revised November 2021