Tanya Khovanova's Math Blog

My son Alexey Radul and I were discussing the Tuesday’s child puzzle:

You run into an old friend. He has two children, but you do not know their genders. He says, “I have a son born on a Tuesday.” What is the probability that his second child is also a son?

Here is a letter he wrote me on the subject. I liked it because unlike many other discussions, Alexey not only asserts that different interpretations of the conditions in the puzzle form different mathematical problems, but also measures how different they are.

by Alexey Radul

If you assume that boys and girls are symmetric, and that days of the week are symmetric (and if you have no information to the contrary, assuming anything else would be sheer presumption on your part), then you can be in one of at least two states.

1) You say that “at least one son born on a Tuesday” is all the information you have, in which case your distribution including this information is uniform over consistent cases, in which case your answer is 13/27 boy and your information entropy is

− ∑₂₇ (1/27) log(1/27) = − log(1/27) = 3.2958.

2) You say that the information you have is “The guy might have said any true thing of the form ‘I have at least one {boy/girl} born on a {day of the week}’, and he said ‘boy’, ‘Tuesday’.” This is a different mathematical problem with a different solution. The solution: By a symmetry argument (see below [*]) we must assign uniform probability of him making any true statement in any particular situation. Then we proceed by Bayes’ Rule: the statement we heard is d, and for each possible collection of children h, the posterior is given by p(h|d) = p(h)p(d|h)/p(d). Here, p(h) = 1/14² = 1/196; p(d) = 1/14; and p(d|h) is either 1 or 1/2 according as whether his other child is or is not another boy also born on a Tuesday (or p(d|h) = 0 if neither child is a boy born on a Tuesday). There are 1 and 26 of these situations, respectively. The answer they lead to is of course 1/2; but the entropy is

− ∑ p log p = − 1/14 log 1/14 − 26/28 log 1/28 = 3.2827

Therefore that assumed additional structure really is more information, which is only present at best implicitly in the original problem. How much more information? The difference in entropies is 3.2958 – 3.2827 = 0.0131 nats (a nat is to a bit what the natural log is to the binary log). How much information is that? Well, the best I can do is to reproduce an argument of E.T. Jaynes’, which may or may not really apply to this situation. Suppose you have some repeatable experiment with some discrete set of possible outcomes, and suppose you assign probabilities to those outcomes. Then the number of ways those probabilities can be realized as frequencies counted over N trials is proportional to e^NH, where H is the entropy of the distribution in nats. Which means that the ratio by which one distribution is easier to realize is approximately e^N(H₁-H₂). In the case of N = 1000 and H₁ – H₂ = 0.0131, that’s circa 5×10⁵. For each way to get a 1000-trial experiment to agree with version 2, there are half a million ways to get a 1000-trial experiment to agree with version 1. So that’s a nontrivial assumption.

[*] The symmetry argument: We are faced with the following probability assignment problem

Suppose our subject’s first child is a boy born on a Tuesday, and his second child is a girl born on a Friday. What probability must we assign to him asserting that he has at least one boy born on a Tuesday?

Good question. Let’s transform our coordinates: Let Tuesday’ be Friday, Friday’ be Tuesday, boy’ be girl, girl’ be boy, first’ be second and second’ be first. Then our problem becomes

Suppose our subject’s second’ child is a girl’ born on a Friday’, and his first’ child is a boy’ born on a Tuesday’. What probability must we assign to him asserting that he has at least one girl’ born on a Friday’?

Our transformation necessitates p(boy Tuesday) = p(girl’ Friday’), and likewise p(girl Friday) = p(boy’ Tuesday’). But our state of complete ignorance about what’s going on with respect to the man’s attitudes about boys, girls, Tuesdays, Fridays, and first and second children has the symmetry that question and question’ are the same question, and must, by the desideratum of consistency, have the same answer. Therefore p(boy Tuesday) = p(boy’ Tuesday’) = p(girl Friday) = 1/2.

My classmate Misha gave me a math problem. Although I liked the math part, I hated the setup. Here is the problem using the new setup:

You are hoping to get very rich one day and you base your hopes on your new invention: a diet pill. The pill works beautifully and doesn’t have side effects. Patients simply take a pill when they get hungry. Within one hour they will fall asleep and will be unable to awake for exactly two hours, at which time they will awake by themselves feeling completely sated.
You have just arrived at your lab, when you realize that one of your interns has misplaced your bottle of wonder pills. After some investigation, you come to the conclusion that your bottle is on the shelf with 239 other bottles that contain a placebo. Unfortunately, those placebo bottles were custom-designed for your statistical tests to exactly match your medicine bottle.
You need to bring the medicine bottle to your boss in two hours. While you panic, all your five interns volunteer for experiments, hoping to be mentioned in your paper. Can you find your bottle in two hours?

The problem Misha gave me had 240 barrels of wine, one of which contained deadly poison and five slaves who could be spared.

I do not like killing people even when they are imaginary. But while I was slow in inventing my own setup, the original version of the puzzle started making the rounds on the Internet. So I decided to kill the problem by writing the solution.

The strategy is to give out some pills immediately, wait for one hour and see who falls asleep. The next step is to give some other pills at the beginning of the next hour to some of the interns who are awake.

Let’s count the information you can get out of this. Each intern will experience one of three different situations: falling asleep in the first hour, in the second hour, or not falling asleep at all. Thus, you can have a total of 3⁵ = 243 different outcomes.

If you had more bottles than 243, there would be no way to distinguish between them. The fact that you have 240 bottles might mean that 243 will work too, but apparently the designer of the puzzle didn’t want to hint into powers of three and picked the largest round number below 243. These considerations should increase your willingness to look into this problem base 3.

Let us label the bottles with different 5-character strings containing three characters 0, 1, and 2. Now we can use the label as instructions. The first character will be associated with the first intern and so on. Suppose the fourth character on the bottle’s label is 0, then the fourth intern doesn’t need to struggle with digesting a pill from this particular bottle. If the fourth character is 1, then the fourth intern gets the pill in the first hour. If the fourth character is 2, the fourth intern gets the pill in the second hour.

Note this minor detail: Suppose the fourth character on a bottle is 2, but the fourth intern is asleep by the second hour. That means, the bottle doesn’t contain the medicine, and we can put it aside.

At the end of two hours you know who fell asleep and when. This data will exactly match the label on the bottle with the medicine.

My coauthor Konstantin Knop publishes cute math problems in his blog (in Russian). Recently he posted a coin weighing problem that was given at the 2010 Euler math Olympiad in Russia to eighth graders. The author of the problem is Alexander Shapovalov.

Among 100 coins exactly 4 are fake. All genuine coins weigh the same; all fake coins, too. A fake coin is lighter than a genuine coin. How would we find at least one genuine coin using two weighings on a balance scale?

It is conceivable that your two weighings may find more than one genuine coin. The more difficult question that Konstantin and his commentators discuss is the maximum number of genuine coins you can guarantee to identify in two weighings. Konstantin and the others propose 14 as the answer, but do not have a proof yet.

I wonder if one of you can find a bigger number than Konstantin or alternatively a proof that indeed 14 is the largest possible.

You might ask, considering the title of this piece, why I think that coins are scary. No, I am not afraid of coins. It scares me that this problem was given to eighth graders in Russia, because I cannot imagine that it would be given to kids that age in the USA.

By the way, ten eighth grade students in Russia solved this problem during the competition.

I recently wrote two pieces about the puzzle relating to sons born on a Tuesday: A Son born on Tuesday and Sons and Tuesdays. I also posted a beautiful essay on the subject by Peter Winkler: Conditional Probability and “He Said, She Said”. Here is the problem:

You run into an old friend. He has two children, but you do not know what their gender is. He says, “I have a son born on a Tuesday.” What is the probability that his second child is also a son?

A side note. My son Alexey explained to me that I made an English mistake in the problem in those previous posts. It is better to say “born on a Tuesday” than “born on Tuesday.” I apologize.

Despite this error, I was gratified to hear from a number of people who told me that I had converted them from their solution to my solution. To ensure that the conversion is substantial, I’ve created a new version of the puzzle on which my readers can test out their new-found understanding. Here it is:

You run into an old friend. He has two children, but you do not know what their gender is. He says, “I have a son born on a Tuesday.” What is the probability that his second child is born on a Wednesday?

John Conway has a T-shirt with his theorem on it. I couldn’t miss this picture opportunity and persuaded John to pose for pictures with his back to me. Here is the theorem:

If you continue the sides of a triangle beyond every vertex at the distances equaling to the length of the opposite side, the resulting six points lie on a circle, which is called Conway’s circle.

Poor John Conway had to stand with his back to me until I figured out the proof of the theorem and realized which point must be the center of Conway’s circle.

The following problem was sent to me by Joel Lewis.

You have 12 mice, one of which eats faster than all the others. You need to find it. You have a supply of standard cupcakes that you value very much and want to minimize how many of them you have to use. The only way you can find the mouse is to give cupcakes to several groups of mice and see which group is the fastest.

We assume that mice chew at a constant speed and all the mice in one group can attack the cake at the same time. I love this puzzle because I love coin problems. Let me restate the puzzle as a coin problem:

You have 12 coins, one of which is fake and weighs less than all the others. You have a balance scale with multiple pans, that is you can weigh several things at once and order them by weight. You do not care about the total number of weighings as in most classical coin puzzles, instead, this time using a pan is expensive and you want to find the fake coin with as few pan-uses as possible.

Spoiler warning: below I will discuss the solution for n mice.

You can, of course, give a cake to every mouse and see which one finishes first. You can save one cake by giving cakes at the same time to all but one of the mice. If everyone finishes simultaneously, the faster mouse is the unfed one.

It wastes cakes to give them to unequally-sized groups of mice. We can do better by copying the classical way to find a fake coin with the minimum number of weighings. That is, for each test, divide the mice into three groups as evenly as possible and give a cake to each of two equally-sized groups. The number of cakes you use is about 2log₃n.

I wouldn’t have written this essay if that was the solution. Sometimes you can do even better. For example, you can find the faster mouse out of 12 using only 5 cakes.

First, if you give out k cakes in one test, the test tells you which of k+1 groups the mouse is in. In the worst case, the faster mouse will be in the biggest group, so you should minimize the biggest group. Hence, your groups that get cakes should have ⌈n/(k+1)⌉ mice.

A test with one cake gives no information. I argue that giving out more than three cakes doesn’t gain anything. Indeed, suppose we use four cakes in a test. That is, we divide the mice into five groups A, B, C, D and E, of which the first four are the same size. We can simulate the test by two tests in each of which we give out two cakes. In the first test we give cakes to A+B and C+D. If one of the groups is faster, say A+B, then in the second test give cakes to A and B; if not, E has the faster mouse. I leave it as an exercise to simulate a test with more than four cakes.

Thus, in an optimal strategy we can use two or three cakes per test. Also, if you give one test with k − 1 cakes and the next one with m − 1 cakes, you can switch them with the same effect. The largest group after either order of tests will have at most ⌈n/km⌉ mice.

I don’t need two tests of three cakes each, which would give me a group of size at least ⌈n/16⌉. I can achieve the same result with three tests of two cakes each, with the faster mouse restricted to a group of size at most ⌈n/27⌉.

That means all my tests use two cakes, except I might use three cakes once. It doesn’t matter in what order I conduct the tests, so I can wait until the end to use three cakes. I leave it as an exercise to the reader that the only small number of mice for which we would prefer three cakes is four. From this it follows quickly that for numbers of mice between 3 * 3ⁱ + 1 and 4 * 3ⁱ, the number of cakes is 2i + 1. For numbers between 4 * 3ⁱ + 1 and 3ⁱ⁺² the answer is 2i + 2.

A week ago I chatted with my son Sergei about memorable math problems. He mentioned problem 5 from USAMO 2007. The problem can be reduced to the following:

Prove that (x⁷ + 1)/(x + 1) is composite for x = 7⁷ⁿ, where n is a non-negative integer.

Perhaps Sergei remembered this problem because as far as he knew, he was the only one in that competition to solve it. That made me curious as to how he solved it. His solution is available as solution 2 at the Art of Problem Solving website. His solution seemed mysterious and impossible to invent on the spot. I became even more curious to understand the thought process underlying his solution.

Here is his recollection:

We need to factor x⁶ − x⁵ + x⁴ − x³ + x² − x + 1. If such factoring existed it would have been known. Therefore, we need to somehow use the fact that x = 7⁷ⁿ. What is the simplest way to factor? We should try to represent the polynomial in question as the difference of squares. Luckily, x is an odd power of 7. We can make it a square if we multiply or divide it by 7 or another odd power of 7. So with a supply of squares on one side, we need to find a match for one of them to build the difference.

Let us simplify the problem and see what happens for (y³ + 1)/(y + 1) for y = 3³ⁿ, when n = 1. In other words we want to represent 703 as a difference of squares. This can be done: 703 = 28² − 9². Now let us see how we can express 28² and 9² through y which in this case is equal to 27. The first term is (y + 1)², and the second is 3y.

Now let’s go back to 7 and x, and check whether (x + 1)⁶ − (x⁶ − x⁵ + x⁴ − x³ + x² − x + 1) is 7x. Oops, no. The difference is 7x⁵ + 14x⁴ + 21x³ + 14x² +7x. On the plus side, it is divisible by 7x which we know is a square. The leftover factor is x⁴ + 2x³ + 3x² +2x + 1, which is a square of x² +x + 1.

The problem is solved, but the mystery remains. The problem can’t be generalized to numbers other than 3 and 7.

I am usually disappointed with American math text books. I have had an underwhelming experience with them. Often I open a book and in the next 15 minutes, I find a mistake, a typo, a misguided explanation, sloppiness, a misconception or some other annoyance.

I was pleasantly surprised when I opened the book Introduction to Algebra by Richard Rusczyk. I didn’t find any flaws in it — not in the first 15 minutes, and not even in the first hour. In fact, having used the book many times I have never found any mistakes. Not even a typo. This was disturbing. Is Richard Rusczyk human? It was such an unusual experience with an American math book, that I decided to deliberately look for a typo or a mistake. After half a year of light usage, I finally found something.

Look at problem 7.3.1.

Five chickens can lay 10 eggs in 20 days. How long does it take 18 chickens to lay 100 eggs?

There is nothing wrong with this problem. But the book is accompanied by the Introduction to Algebra Solutions Manual in which I found the following solution, that I’ve shortened for you:

The number of eggs is jointly proportional to the number of chickens and the amount of time. One chicken lays one egg in 10 days. Hence, 18 chickens will lay 100 eggs in 1000/18 days, which is slightly more than 55 and a half days.

What is wrong with this solution? Richard Rusczyk is human after all.

I like this book for its amazing accuracy and clean explanations. There are also a lot of diverse problems in terms of difficulty and ideas. Richard Rusczyk has good taste. Many of the problems are from different competitions and require inventiveness. I like that there are a lot of challenge problems that go beyond the boring parts of algebra. Also, I like that important points of algebra are chosen wisely and are emphasized.

This book might not be for everyone. It doesn’t have pretty pictures. It doesn’t have color at all. This is not a flaw for a math book. The book concentrates on ideas and problems, not entertainment. So if you’re looking for math entertainment, you’ll find it on my blog. For solid study, try Richard Rusczyk’s books.

I love Raymond Smullyan’s books , especially the trick puzzles he includes. The first time I met him in person, he played a trick on me.

This happened at the Gathering for Gardner 8. We were introduced and then later that day, the conference participants were treated to a dinner event that included a magic show. In one evening I saw more close-up magic tricks than I had in my whole life. This left me lightheaded, doubting physics and my whole scientific outlook on life.

Afterwards, Raymond Smullyan joined me in the elevator. “Do you want to see a magic trick?” he asked. “I bet I can kiss you without touching you.” I was caught off guard. At that moment I believed anything was possible. I agreed to the bet.

He asked me to close my eyes, kissed me on the cheek and laughed, “I lost.”

by Peter Winkler

As a writer of books on mathematical puzzles I am often faced with delicate issues of phrasing, none more so than when it comes to questions about conditional probability. Consider the classic “X has two children and at least one is a boy. What is the probability that the other is a boy?”

It is reasonable to interpret this puzzle as asking you “What is the probability that X has two boys, given that at least one of the children is a boy” in which case the answer is unambiguously 1/3—given the usual assumptions about no twins and equal gender frequency.

This puzzle confounds people *legitimately*, however, because most of the ways in which you are likely to find out that X has at least one boy contain an implicit bias which changes the answer. For example, if you happen to meet one of X‘s children and it’s a boy, the answer changes to 1/2.

Suppose the puzzle is phrased this way: X says “I have two children and at least one is a boy.” What is the probability that the other is a boy?

Put this way, the puzzle is highly ambiguous. Computer scientists, cryptologists and others who must deal carefully with message-passing know that what counts is not what a person says (even if she is known never to lie) but *under what circumstances would she have said it.*

Here, there is no context and thus no way to know what prompted X to make this statement. Could he instead have said “At least one is a girl”? Could he have said “Both are boys”? Could he have said nothing? If you, the one faced with solving the puzzle, are desperate to disambiguate it, you’d probably have to assume that what really happened was: X (for some reason unconnected with X‘s identity) was asked whether it was the case that he had at least one son, and, after being warned—by a judge?—that he had to give a yes-or-no answer, said “yes.” An unlikely scenario, to say the least, but necessary if you want to claim that the solution to the puzzle is 1/3.

Consider the puzzle presented (according to Alex Bellos) by Gary Foshee at the recent 9th Gathering for Gardner:

I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?

If the puzzle was indeed put exactly this way, and your life depended on defending any particular answer, God help you. You cannot answer without knowing, for example, what the speaker would have said if he had one boy and one girl, and the boy was born on Wednesday. Or if he had two boys, one born on Tuesday and one on Wednesday. Or two girls, both born on Tuesday. Et cetera.

Now, there is nothing mathematically wrong (given the usual assumptions, including X being random) about saying that “The probability that X has two sons, given that at least one of X‘s two children is a boy born on Tuesday, is 13/27.” But if that is to be turned into an unambiguous puzzle attached to a presumed situation, some serious hypothesizing is necessary. For instance: you get on the phone and start calling random people. Each is asked if he or she has two children. If so, is it the case that at least one is a boy born on a Tuesday? And if the answer is again yes, are the children both boys? Theoretically, of the times you reach the third question, the fraction of pollees who say “yes” should tend to 13/27.

Kind of takes the fun out of the puzzle, though, doesn’t it? Kudos to Gary for stirring up controversy with a quickie.