Kolmogorov Student Olympiad in Probability

There are too many Olympiads. Now there is even a special undergraduate Olympiad in probability, called Kolmogorov Student Olympiad in Probability. It is run by the Department of Probability Theory of Moscow State University. I just discovered this tiny Olympiad, though it has been around for 13 years.

A small portion of the problems are accessible for high school students. These are the problems that I liked. I edited them slightly for clarity.

Second Olympiad. Eight boys and seven girls went to movies and sat in the same row of 15 seats. Assuming that all the 15! permutations of their seating arrangements are equally probable, compute the expected number of pairs of neighbors of different genders. (For example, the seating BBBBBBBGBGGGGGG has three pairs.)

Third Olympiad. One hundred passengers bought assigned tickets for a 100-passenger railroad car. The first 99 passengers to enter the car get seated randomly so that all the 100! possible permutations of their seating arrangements are equally probable. However, the last passenger decides to take his reserved seat. So he arrives at his seat and if it is taken he asks the passenger in his seat to move elsewhere. That passenger does the same thing: she arrives at her own seat and if it is taken, she asks the person to move, and so on. Find the expected number of moved passengers.

Third Olympiad. There are two 6-sided dice with numbers 1 through 6 on their faces. Is it possible to “load” the dice so that when the two dice are thrown the sum of the numbers on the dice are distributed uniformly on the set {2,…,12}? By loading the dice we mean assigning probabilities to each side of the dice. You do not have to “load” both dice the same way.

Sixth Olympiad. There are M green and N red apples in a basket. We take apples out randomly one by one until all the apples left in the basket are red. What is the probability that at the moment we stop the basket is empty?

Seventh Olympiad. Prove that there exists a square matrix A of order 11 such that all its elements are equal to 1 or −1, and det A > 4000.

Twelfth Olympiad. In a segment [0,1] n points are chosen randomly. For every point one of the two directions (left or right) is chosen randomly and independently. At the same moment in time all n points start moving in the chosen direction with speed 1. The collisions of all points are elastic. That means, after two points bump into each other, they start moving in the opposite directions with the same speed of 1. When a point reaches an end of the segment it sticks to it and stops moving. Find the expected time when the last point sticks to the end of the segment.

Thirteenth Olympiad. Students who are trying to solve a problem are seated on one side of an infinite table. The probability that a student can solve the problem independently is 1/2. In addition, each student will be able to peek into the work of his or her right and left neighbor with a probability of 1/4 for each. All these events are independent. Assume that if student X gets a solution by solving or copying, then the students who had been able to peek into the work of student X will also get the solution. Find the probability that student Vasya gets the solution.

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IQ Migration

The Russian website problems.ru has a big collection of math problems. I use it a lot in my work as a math Olympiad coach. Recently I was giving a statistics lesson. While there was only one statistics problem on the website, it was a good one.

Assume that every person in every country was tested for IQ. A country’s IQ rating is the average IQ of the population. We also assume that for the duration of this puzzle no one is born and no one dies.

  • A group of citizens of country A emigrated to B. Show that the rating of both countries can go up.
  • After that a group of citizens of B (which may include former citizens of A) emigrated to A. Is it possible that the ratings of both countries go up again?
  • A group of citizens of A emigrated to B, and a group of citizens of B emigrated to C. As a result, the ratings of each country increased. After that the migration went the opposite way: some citizens of C moved to B, and some citizens from B moved to A. As a result, the ratings of all three counties went up once more. Is this possible? If yes, then how? If no, then why not?
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Math Kangaroo’s Logic Puzzle

My AMSA students loved the following puzzle from the 2003 Math Kangaroo contest for grades 7-8:

The children A, B, C and D made the following assertions.

  • A: B, C and D are girls.
  • B: A, C and D are boys.
  • C: A and B are lying.
  • D: A, B and C are telling the truth.

How many of the children were telling the truth?
A) 0   B) 1   C) 2   D) 3   E) Impossible to determine

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Express 6

I was given this puzzle at the last Gathering for Gardner.

Use arithmetic operations to express 6 using three identical digits. For each digit from 0 to 9 find at least one way to express 6.

For example, I can express 6 using three twos in many ways: 2 + 2 + 2, or 2 · 2 + 2, or 22 + 2. But the problem doesn’t ask for many ways. One way is enough, but you need to do it for every digit. So nine more cases to go: 0, 1, 3, 4, 5, 6, 7, 8, and 9.

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The Virtue of Laziness

My son, Alexey Radul, is a programmer. He taught me the importance of laziness in programming.

One of his rules:

Not to write the same line of code in the same program twice.

If you need the same line of code in the same program, that means you should either use a loop or outsource the line to a function. This style of coding saves time; it makes programs shorter and more elegant. Such programs are easier to debug and understand.

I remember how I copied and pasted lines of code before he taught me this rule. Then I needed to change parameters and missed some of the lines during changing. Debugging was such a headache.

Mathematicians are way lazier than programmers. Consider the system of two equations: x+2y=3 and 4x+5y=6. There are no repeating lines here. Only letters x and y appear twice. Mathematicians invented the whole subject of linear algebra and matrices so that they would not need to rewrite variables.

Mathematicians are driven by laziness. Once ancient mathematicians first solved a quadratic equation, they didn’t want to do it again. So they invented a formula that solves all quadratic equations once and for all.

I try to keep up with tradition. I try to make my theorems as general as possible. When I write my papers, I try to make them short and simple. When I think about mathematics I try to get to the stage where the situation is so clear I can think about it without paper and pencil. I often discover new theorems while I am in bed, about to fall asleep. Sometimes I wake up with a good idea. So I do my job while I sleep.

I love my profession. I get paid for being lazy.

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Beer Jokes and Hat Puzzles

This is one of my favorite jokes:

Three logicians walk into a bar. The waitress asks, “Do you all want beer?”
The first logician answers, “I do not know.”
The second logician answers, “I do not know.”
The third logician answers, “Yes.”

This joke reminds me of hat puzzles. In the joke each logician knows whether or not s/he wants a beer, but doesn’t know what the others want to drink. In hat puzzles logicians know the colors of the hats on others’ heads, but not the color of their own hats.

This is a hat puzzle which has the same answers as in the beer joke. Three logicians walk into a bar. They know that the hats were placed on their heads from the set of hats below. The total number of available red hats was three, and the total number of available blue hats was two.

Red Hat Red Hat Red Hat Red Hat Red Hat

Three logicians walk into a bar. The waitress asks, “Do you know the color of your own hat?'”
The first logician answers, “I do not know.”
The second logician answers, “I do not know.”
The third logician answers, “Yes.”

The puzzle is, what is the color of the third logician’s hat?

This process of converting jokes to puzzles reminds me of the Langland’s Program, which tries to unite different parts of mathematics. I would like to unite jokes and puzzles. So here I announce my own program:

Tanya’s Program: Find a way to convert jokes into puzzles and puzzles into jokes.

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How Well Do You Know Your Dice?

Each time I see John Conway he teaches me something new. At the Gathering for Gardner he decided to quiz me on how well I know a regular six-sided die. I said with some pride that the opposite sides sum up to 7. He said, “This is the first level of knowledge.” So much for my pride. I immediately realized that the next level would be to know how all the numbers are located relative to each other. I vaguely remembered that in the corner where 1, 2, and 3 meet, the numbers 1, 2, and 3 are arranged in counter-clockwise order.

Here’s how John taught me to remember every corner. There are two types of corners. In the first type numbers form an arithmetic progression. John calls such numbers counters. He chose that name so that it would be easy to remember that counters are arranged in counter-clockwise order. The other numbers he calls chaos: their increasing sequence goes clockwise.

Once I grasped that, I relaxed thinking that now I know dice. “What about the third level?” he asked. “What third level?” “Now that you know which number goes on which side, you need to know how the dots are arranged.” Luckily, there are only three sides on which the dots are not placed with rotational symmetry: 2, 3, and 6. And they all meet in a corner, which John calls the home corner. The rule is that the diagonals formed by the dots on the sides with 2, 3, and 6, meet in the home corner. You might argue that 6 doesn’t have a diagonal. But if you look at 6, you can always connect the dots to form the letters N or Z, depending on the orientation of the die. When you lay the letter N on its side, it becomes the letter Z. Thus they define the same diagonal. This diagonal has to meet the diagonals from 2 and 3 in the corner.

When I came home from the conference I picked up a die and checked that the rules work. There are 8 corners. It is enough to remember one corner of numbers to recover the other numbers by using the opposite sum rule. But it is nice to have a simple rule that allows us to bypass the calculation. Four of the corners have numbers in arithmetic progression: 1:2:3, 1:3:5, 2:4:6, and 4:5:6. They are counters and they are arranged counter-clockwise. The other four corners are: 1:2:4, 1:4:5, 2:3:6, and 3:5:6, and they are arranged clockwise.

I wanted to provide a picture of a die for this post and went online to see if I could grab one. Many of the graphic images of dice, as opposed to photographs, were arranged incorrectly. Clearly these visual artists did not study dice with John Conway.

Then I decided to check my own collection of dice. Most of them are correct. The ones that are incorrect look less professional. Here is the picture. The ones on the right are correct.

Dice

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My New Yellow Road

I started my Yellow Road a year ago on February 9, 2013, when my weight was 245.2 pounds. My system worked for eight months. I lost 25 pounds. Then I went to two parties in a row and gained four pounds. According to my plan, I was supposed to eat only apples after lunch. It was too difficult to stick to that, and I got off-target. My target weight continued decreasing daily, as per my plan, while I got stuck. The growing difference between my real weight and my target weight was very discouraging, so I lost my momentum.

I decided to reset the target weight and restart the plan. I changed my plan slightly to incorporate the lessons I had learned about myself.

On February 9, 2014, I started my New Yellow Road. I weighed 223.2 pounds. So I reset my target weight to be 223.2 on February 9. Each day my target weight goes down by 0.1 pounds. I weigh myself each morning. If I am within one pound of my target weight, I am in the Yellow zone and I will eat only fruits and vegetables after 5:00 pm. If I am more than one pound over my target weight, I am in the Red Zone and will eat only apples after 5:00 pm. If I am more than one pound below my target weight, I can eat anything.

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Salary Negotiations

I want to tell you the story of my “successful” salary negotiation. The year was 2003 and I had a temporary visiting position at Princeton University. I wanted to move to Boston and my friend showed my resume to Alphatech. The interview went well and they offered me a position as Lead Analyst with a salary of $110.000. This was particularly good news considering that the tech job market was very weak in 2003.

At that time, I was working very hard on building my self-esteem. I read lots of books and was in therapy for two years. I decided to practice what I had learned to try to negotiate a bigger paycheck. While speaking to the HR guy on the phone, I was standing up, as I was taught, and projecting my voice firmly with my chest opened up. I could hardly believe it when I heard myself ask for a $10,000 increase.

Despite my wonderful posture, the human resource person refused. However, he remembered that they had forgotten to give me a moving bonus. He asked me about my living conditions. I told him that I lived in a four-bedroom house. I didn’t elaborate: it was a tiny four-bedroom house made out of a garage. I made a counter offer: forget the moving bonus, but give me my salary increase as I asked. He agreed.

For the first year, there was no real difference, because the salary increase was equal to the moving bonus. But I was planning to stay with the company for a long time, so by the second year, my clever negotiation would start to pay off. My negotiations were a success. But were they?

Things change. The boss who hired me and appreciated me stopped being my boss. The company was bought by BAE Systems who were not interested in research. To my surprise, I started getting non-glowing performance reviews. Luckily, by that time I had made a lot of friends at work, and one of them not only knew what was going on, but was willing to tell me. My salary was higher than that of other employees in the same position. Salary increases were tied to performance. They wanted to minimize my increases to bring my salary into the range of others at my level. So to justify it, they needed a negative performance review. After one negative review it is difficult to change the trend. A negative review stays on the record and affects the future reputation.

In a long run I am not sure that my salary negotiations were a success.

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The Three Light Bulbs Puzzle

There is a famous puzzle about three light bulbs, that is sometimes given at interviews.

Suppose that you are standing in a hallway next to three light switches that are all off. There is another room down the hall, where there are three incandescent light bulbs—each light bulb is operated by one of the switches in the hallway. You can’t see the light bulbs from the hallway. How would you figure out which switch operates which light bulb, if you can only go to the room with the light bulbs one and only one time?

This puzzle worked much better in the past when we only had incandescent light bulbs and so didn’t need to specify the type of bulbs. Unfortunately, the standard solution only works with incandescent bulbs and the word “incandescent” nowadays needs to be stated. But the use of “incandescent” is a big hint. Indeed, incandescent light bulbs generate heat when they are on, so the standard solution is to turn on the first light switch, to keep the second switch off, and to turn the third switch on for five minutes before turning it off. In the room, the light bulb corresponding to the first switch will be lit, and out of the two unlit bulbs, the one corresponding to the third switch will be warm.

It’s a cute solution, but there could be so many other approaches:

  • You can ask a friend to help you. You can come to the room with the light bulbs and shout to her which switch to turn on.
  • You can place mirrors from the room to the hall, so you can see bulbs through the multiple reflections. You might not need to enter the room at all.
  • Before playing with the switches, you can place a video camera in the room to transmit the scene in real time. You will not be allowed to enter the room again to get the camera back, but what the heck: the job will be done.
  • You can put timers on the switches and set them to turn on at different times.
  • You can attach strings to switches and turn them on or off from a distance.

I invite my readers to invent other methods to solve this problem. Be creative. After all, if I were to interview you for a job, I would be more impressed by a new solution than the one that is all over the Internet.

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