Tanya Khovanova's Math Blog

I found a new Russian Olympiad for high schools related to universities. I translated my favorite problems from last year’s final round. These are the math problems:

8th grade. In a certain family everyone likes their coffee with milk. At breakfast everyone had a full cup of coffee. Given that Alex consumed a quarter of all consumed milk and one sixth of all coffee, how many people are there in the family?

8th grade. How many negative roots does the equation x⁴ − 5x³ − 4x² − 7x + 4 = 0 have?

10th grade. Find a real-valued function f(x) that satisfies the following inequalities for any real x and y: f(x) ≤ x and f(x+y) ≤ f(x) + f(y).

I liked the physics problems even more:

8th grade. Winnie-the-Pooh weighs 1 kg. He hangs in the air with density 1.2kg/m³ next to a bee hive. He is holding a rope connected to a balloon. Estimate the smallest possible diameter of the balloon, assuming that this happens on Earth.

10th grade. Two containers shaped like vertical cylinders are connected by a pipe underneath them. Their heights are the same and they are on the same level. The cross-sectional area of the right container is twice bigger than the left’s. The containers are partially filled with water of room temperature. Someone put ice into both containers: three times more ice into the right one than into the left one. After that, the containers are closed hermetically. How will the water level will change after the ice melts completely:

• The levels will not change.
• The level on the left will be higher than on the right.
• The level on the left will be lower than on the right.
• The answer depends on the initial volume of water in the containers.

Once I talked to my friend Michael Plotkin about IQ tests, which we both do not like. Michael suggested that I run an experiment and send a standard IQ question for children to my highly-educated friends. So I sent a mass email asking:

What’s common between an apple and an orange?

I believe that the expected answer is that both are fruits.

Less than half of my friends would have passed the IQ test. They gave four types of answer. The largest group chose the expected answer.

The second group related the answer to language. For example, apples and oranges both start with a vowel and they both have the letters A and E in common.

The third group connected the answer to what was on their minds at the time:

• Apples and oranges are both healthy foods that I enjoy, but do not eat as often as I should.
• They have the same thing in common as do a saxophone and a guitar.
• You can’t shave with either one.
• They both are much worse than a cucumber in the bedroom.

And the last group were people who just tried to impress me:

• One should not decide that n apples is better than m oranges just because n > m.
• They both can provoke the discovery of gravity.
• You can’t compare apples and oranges.
• Existence.
• They both have fundamental meaning in food tongue.
• They’re topologically homeomorphic.

If my friends with high IQs have given so many different answers, I would expect children to do the same. The variety of answers is so big that no particular one should define IQ. By the way, my own well-educated kids’ answers are quoted above — and they didn’t go with the standard answer. I’m glad they never had IQ tests as children: I’m sure they would never have passed.

I am just wondering:

What is the largest integer consisting of distinct digits such that, in its English pronunciation, all the words start with the same letter?

I continue to wonder:

What is the largest integer consisting of the same digit such that, in its English pronunciation, all the words start with distinct letters?

I already gave an example of the kinds of problems that were given to Jewish people at the oral entrance exam to the math department of Moscow State University. In fact, I have a whole page with a collection of such problems, called Jewish problems or Coffins. That page was one of the first pages I created when I started my website more than ten years ago.

When my son Alexey was in high school, I asked him to help me type these problems into a file and to recover their solutions from my more than laconic notes, and solve the problems that I didn’t have notes for. He did the job, but the file was lying dormant on my computer. Recently I resurrected the file and we prepared some of the solutions for a publication.

The problems that were given during these exams were very different in flavor: some were intentionally ambiguous questions, some were just plain hard, some had impossible premises. In our joint paper “Jewish Problems” we presented problems with a special flavor. These are problems that have a short and “simple” solution, that is nonetheless very difficult to find. This way the math department of MSU was better protected from appeals and complaints.

Try the following problem from our paper:

Find all real functions of real variable F(x) such that for any x and y the following inequality holds: F(x) − F(y) ≤ (x − y)².

I will give a talk on the subject for UMA at MIT on October 18, at 5pm.

What’s “plagiarism”? It’s when you take someone else’s work and claim it’s your own. It’s basically STEALING.

Ideas improve. The meaning of words participates in the improvement. Plagiarism is necessary. Progress implies it. It embraces an author’s phrase, makes use of his expressions, erases a false idea, and replaces it with the right idea.

Perhaps the Russians have done the right thing, after all, in abolishing copyright. It is well known that conscious and unconscious appropriation, borrowing, adapting, plagiarizing, and plain stealing are variously, and always have been, part and parcel of the process of artistic creation. The attempt to make sense out of copyright reaches its limit in folk song. For here is the illustration par excellence of the law of Plagiarism. The folk song is, by definition and, as far as we can tell, by reality, entirely a product of plagiarism.

If you copy from one author, it’s plagiarism. If you copy from two, it’s research.

You know that the negation of a true statement is a false statement, and the negation of a false statement is a true statement. You also know that you can negate a sentence by preceding it with “It is not true that ….”

Now look at the following statement and its negation, invented by David Bernstein. Which one is true?

• This sentence contains five words.
• It is not true that this sentence contains five words.

How about this pair?

• This sentence contains ten words.
• It is not true that this sentence contains ten words.

by Tanya Khovanova and Alexey Radul

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 is offered a bet. She may pay $550 in which case she will get $1000 if the coin was tails. If the coin was tails, she is put back to sleep with her memory erased, and awakened on Tuesday and given the same bet again. She knows the protocol. Should she take the bet?

As we discussed in our first essay about Sleeping Beauty, she should take the bet. Indeed, if the coin was heads her loss is $550. But if the coin was tails her gain is $900.

To tell you the truth, when Beauty is offered the bet, she dreams: “It would be nice to know the day of the week. If it were Tuesday, then the coin must have been tails and I would gladly take the winning bet.”

In our next variation of the riddle her dream comes true.

Every time she is awakened she is offered to buy the knowledge of the day of the week. How much should she be willing to pay to know the day of the week?

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, then she is put back to sleep with her memory erased, and awakened on Tuesday and asked the same question again. She knows the protocol. She is awakened one morning and instead of the expected questions she is offered a bet. She may pay $600 in which case she will get $1000 if the coin was tails. Should she take the bet?

by David Wilson

We know that tripling the triangular number 1 yields the triangular number 3. The figure shows how we can use this fact to conclude that tripling the triangular number 15 yields the triangular number 45.

Using this new fact, can you modify the figure to find even larger examples of tripling triangles?

by Tanya Khovanova and Alexey Radul

This post is inspired by the following 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, then she is put back to sleep with her memory erased, and 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?

Some people argue: asleep or awake, the probability of a fair coin being heads is one half, so her probability should be one half.

Other people, including us, argue that those people didn’t study conditional probability. On the information of the setup to the problem and the information of having awakened, the three situations “Coin was heads and it is Monday”, “Coin was tails and it is Monday”, and “Coin was tails and it is Tuesday” are symmetric and therefore equiprobable; thus the probability that the coin was tails is, on this information, two thirds.

So who is right? We are, of course. A good way to visualize probability judgements is to turn them into bets. Suppose each time Beauty wakes up she is offered the following bet: She pays $600 and gets $1000 if the coin was tails. Should she take it? If her probability of the coin being tails were one half, then obviously not; if her probability of the coin being tails were two thirds, obviously yes. So which is it? Consider the situation from her perspective as of Sunday. She can either always take this bet or always refuse it. If she always refuses, she gets nothing. If she always accepts: If the coin turns up heads, she will be asked the question once and will lose $600. If the coin turns up tails, she will be asked the question twice and will gain $800. So on average she will win, so she should take the bet. By this thought experiment, her probability of tails is clearly not one half.

To make matters more interesting, let’s try another bet. Suppose she is given the above bet just once, in advance, on Sunday. She pays $600, and she gets paid $1000 on Wednesday if the coin was tails. This has nothing to do with sleeping and awakening. If she takes the bet she loses $600 with probability one half and gains $400 otherwise. So she shouldn’t take the bet. Her probability on Sunday that the coin will come up heads is, of course, one half. The point is that just as these two bets are different bets, the sets of information Beauty has on Sunday vs at awakening are different, and lead to different conclusions. On Sunday she knows that the next time she wakes up it will be Monday, but when she then wakes up, she doesn’t know that it’s Monday.

Parting thought: The phenomenon of predictably losing information leads to the phenomenon of predictably changing one’s assessments. Suppose for some reason she decided to take that unprofitable bet on Sunday. When she wakes up during the experiment, should she feel happy or sad? From her perspective during the experiment, the odds of gaining $400 vs losing $600 are two to one, so she should be happy. Given that she knows on Sunday how she will (with complete certainty!) feel about this bet on Monday, should she take it, even given her Sunday self’s assessment that it’s a bad bet?