Tanya Khovanova's Math Blog

Here is one of my all-time favorite problems.

Puzzle. Four glasses are placed on the four corners of a square rotating table; each glass is either right-side up or upside down. You need to turn them all in the same direction, either all facing up or all down. You may do so by grasping any two glasses and turning either none, one of them, or both over.
There are two catches: 1) You are blindfolded, and 2) the table is spun after each round. Assuming a bell rings when you have them all facing the same way, how do you do it?

When I first heard this problem, the person who tortured me with it forgot to mention the bell. The problem was impossible: there was no feedback. Then, the bell was mentioned. It was an aha moment: you get information, but only a confirmation that the problem is solved. I tried to think about the puzzle backwards. What could be my last step? Assume that I know that the glasses alternate around the table: up, down, up, down. Then, as my last step, I can reach for the diagonal glasses and turn them over.

This is how you might start thinking about this puzzle. You can find the rest of the solution on the puzzle’s Wikipedia page.

Here is a wonderful variation, proposed by Michael Hotiner from Ukraine, that appeared ten years ago at a puzzle competition. This variation is not cheap; the players must pay with their lives, albeit virtual ones.

Puzzle setup. Consider a computer game where at level M, there is an M-sided rotating table with glasses at each corner. The setup is similar to the four-glasses puzzle above. The player is blindfolded, and the table rotates between rounds. As soon as level M is over, if all the glasses face one direction, the bell rings, and the player moves to the next level, M+1.
At each round, the player can decide on the number N of glasses to touch. Upon deciding, they have to pay N! lives for that round. Then, the player can touch the glasses one by one, choosing the next glass depending on how the previous glasses are oriented. After all the glasses are touched, the player can decide which ones to turn upside down.

Let’s see what happens at the very beginning of this game. The game starts at level 1, with only one glass. The round is solved before it starts. The cost is 0. The bell rings, and the game immediately goes to level 2.

Consider level 2. If the bell doesn’t ring immediately, the glasses face different directions. The cheapest way to proceed is to choose N = 1 and turn any glass. The cost is 1.

Consider level 3. A player can solve it in one round by touching all three glasses and turning them the same way. The cost is 3! = 6. There is a cheaper method that costs 4 lives in two rounds. In the first round, the player can touch any two glasses and turn both of them up. If the bell doesn’t ring, then the third glass is upside down. In the next round, the player can touch two glasses. If one is upside down, that glass should be turned up. If both glasses are right side up, then the player should turn both of them over to finish the round.

Puzzle tasks.

1. Find a way to pass level 3 using only three lives.
2. Find the cheapest way to pass level 4.
3. Find the cheapest way for level 6.
4. Find a way to pass level 5 using only 30 lives.

One of my all-time favorite puzzles is about tiling a chessboard with 1-by-2 dominoes. But the chessboard is special: two cells at the opposite corners are removed. A similar elegant chessboard puzzle recently appeared on Facebook.

Puzzle. Our special chessboard has one corner cell removed. On each of the remaining cells, there is an ant. At a signal, each ant moves two cells horizontally or vertically (for example, an ant can move from b3 to b5). Is it possible that after all the ants perform their move, each of the 63 cells will have an ant?

This puzzle is a variation of another puzzle, where all the setup is the same, but each ant moves only one cell horizontally or vertically. You can start with this variation, as it is easier.

This puzzle was in a homework assignment I gave to my students.

Puzzle. In the given configuration of matches, move two matches to form three squares.

I assume that the intended solution is the one below. Its appeal is that it has three squares of the same size and no dangling matches.

As usual, my students were inventive and suggested many alternative solutions.

• The first picture shows a solution with three squares of different sizes: two small ones and one twice as big.
• The next solution has two big squares and one smaller one.
• The following picture shows three squares of different sizes.
• Since the problem doesn’t forbid forming more than three squares, there are, of course, many solutions with more than three squares. The last picture has six squares.

A Facebook Challenge

I recently stumbled upon the following challenge on Facebook.

Puzzle. Type the number of seconds in a week using the smallest number of characters.

I lied. The challenge was to type the same number using the smallest number of clicks on a computer keyboard. There is a slight difference, and I like my version better.

I agreed to review the book, The Raven’s Hat, because of the hats. I love hat puzzles. When I give them to my students, I bring hats to class to reenact the solutions.

The book contains eight awesome puzzles as well as ideas for playing with them. I both loved and hated this book. I loved it because it is great, and hated it because it isn’t perfect. Let me start with three places I didn’t like.

Consider a famous hat puzzle when there are hats of N colors. The sages are in a line, and hats are put on their heads. As usual, they are not allowed to give each other signals. Each of them has to announce their hat’s color, and they want to minimize the number of mistakes.

The big idea is to number the colors. The book suggests that the last sage in line calculates the total number of colors they see modulo N and announces the result to the rest. Then the others, starting from the end of the line, one by one, can calculate and name their hat colors. With this strategy, only the last sage in line might be mistaken.

This is a correct solution, but this is the first place I didn’t like. I prefer a different strategy, where everyone assumes that the total sum of the hat colors is 0 modulo N. In this case, every sage makes the same calculation: each sage sums up everything they see or hear and subtract the result from 0 modulo N. This solution is more elegant, since all the sages follow the same rule.

Then the book extends the same puzzle to an infinite number of sages. My second point of contention is that the authors think that, in this case, two sages might be mistaken. No. The answer is still the same, there is a strategy where not more than one sage is mistaken. See my blog post for the solution.

My third pet peeve happened when the authors introduced ballroom dancing in the puzzle on picture hanging. What is the connection between picture hanging and ballroom dancing? I’ll keep the book’s secret. My beef is with how the roles in ballroom dancing are described. Ballroom dancing is usually danced in pairs with asymmetric roles, which, in the past, were designated for males and females. Gender doesn’t play such a big role anymore; anyone can dance any role.

The authors are afraid to be politically incorrect by calling the dancers male and female. Instead, they say that the dancers dance male and female parts. Though formally, this choice of words might be politically correct, it still sounds awkward and draws attention to gender. If the authors ever talked to any person who has ever danced, they would have known that there is a much simpler way to describe dance roles. The dancers are divided into leaders and followers.

Did I ever tell you that reviewing my students’ writing is part of my job? So I am good at it and like critiquing other people’s writing. Now that my complaints are out, the issues with the book are actually minor.

The book is great. I even bought a second inflatable globe because of this book. The game, described in the book, is to rotate two globes randomly and then find a point on the globes in the same relative position towards the center. The game helped me teach my students that any movement of a sphere is a rotation.

My main goal in this post is to describe the only puzzle in the book that I haven’t seen before.

Puzzle. In a group of opera singers, there are two stars who are either friends or enemies. Surprisingly, only the host, who is not an opera singer, knows who the stars are and the nature of their relationship (the stars do not know that they are stars and whether or not they are friends). The group’s common goal is to identify the stars and to determine whether they are friends or enemies. To do so, they send a few of the singers to sing opera on a stage, which is divided into two halves: left and right. During the opera, the singers do not move between the halves. After the opera is over, the host classifies the opera. If there were no stars or only one star on stage, he classifies it as “neutral”. If both stars were on stage, the opera is a big success or a disaster. If both stars are friends and sing on the same half of the stage, or if they are enemies and sing on different halves, then the opera is a big success. Otherwise, it is a disaster.
What is the best strategy for a group of five singers to determine who are the stars and what is their relationship? What is the smallest number of operas they have to sing to guarantee that they can figure everything out?

It is weird that two people do not know whether they are friends. But sacrifices are needed for mathematics. I am excited that there is a nontrivial puzzle related to information theory, and it is ternary based. All other such puzzles I know are about weighing coins on a balance scale. I wrote too many papers about coin weighing. Now I can switch to opera singers with passionate relationships, secret from themselves.

Who will be Champion?

I stumbled on this puzzle on Facebook the day of the World Cup finals. How timely!

Puzzle. Sixteen teams play a single-elimination tournament, where the losing team is immediately out. The teams have different rankings, and a higher-ranking team always wins against a lower-ranking team. The initial seeding is random.
In the semifinals, A won against B, and C won against D. Given that B is stronger than D, what is the probability that A will become the champion?

Consider a group of people in which some are friends. We assume that friendship is symmetric: if Alice is Bob’s friend, then Bob is Alice’s friend. That means we can build a friendship graph where vertices are people and edges correspond to friendships. Let’s assume that every person has at least one friend, so the friendship graph doesn’t have isolated vertices.

Darla needs to conduct research by surveying random pairs of friends. But first, she has to find those pairs. To ensure that the pairs are randomly selected, she must pick two random people from the group, contact them, and ask them whether or not they are friends. If they are, she gives them her questionnaire. If not, Darla wasted tons of time and had to keep looking.

The group she is surveying is enormous. So, when she picks two random people, they typically have never even heard of each other. Bother!

Darla decides to speed up the process. She would pick a random person, ask them for a list of friends, and then randomly pick one person from the list. Since every person has at least one friend, Darla always ends up with a filled questionnaire.

Puzzle question. Why is Darla’s method wrong? Can you describe the pairs of friends her method favors?

As my readers know, I run a PRIMES STEP program, where we conduct mathematical research with students in grades 6 through 9. Last year, our junior group wrote the paper, The Struggles of Chessland, which is now posted on the arXiv.

This is the funniest paper ever written at PRIMES STEP. In the paper, the King, with his self-centered Queen and their minions, try to protect their lands in the Bermuda triangle, called the Chessland, which consists of square islands of different sizes.

In the first part of their fairy tale, the King, his lazy Queen, and other chess pieces are trying to survey their islands. The first volunteer for this surveying job is the Knight. The Knight starts somewhere on an island and walks around it by using the knight’s moves in chess. The goal is to see every cell, and a Knight can see all cells that are a knight’s move away from where he is standing. In other words, every cell is either visited by the Knight or is one knight’s move away from a visited cell. In mathematical terms, the Knight walks a path that is a domination set on a knight’s graph.

The students found a lot of such paths. The picture shows a surveying path for the Knight for the island of size 7. The students called this pattern a shoelace pattern. They used similar shoelaces to survey any island of size 7 or larger. However, when I look at the picture, I see a cat.

Other chess pieces survey the land too. Funnily, the King surveys better when he is drunk. Do you know why? Take a look at the paper.

After all the islands had been surveyed, enemy spies started to appear in Chessland, and our band of chess pieces tried to trap them. My students invented the rules for trapping enemies. An example of the black Queen being trapped is in the picture below, and the rules are the following. Wherever the black Queen might move, should be under attack by a white queen. Also, white queens can’t directly attack the black Queen, or each other.

Oh! I forgot to mention: the trapping should use as few white queens as possible. Also, if the enemy is not a queen but another kind of chess piece, it can only be trapped by its own kind.

The trapping can be described in terms of graph theory. The black Queen is located at a vertex of the queen’s graph. The white queens should be positioned at vertices in such a way that they are not neighbors of the black Queen or each other. In addition, any vertex that is a neighbor of the black Queen, has to be a neighbor of at least one white queen.

This year’s PRIMES STEP project was a real chess adventure!

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.

1. Suppose the first nine people said, “I don’t know”, and the tenth person said, “Yes”. How many of them wanted coffee?
2. 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.

1. Adds to one of the numbers one of its digits.
2. Shuffles the digits of one of the numbers.

Every time they do it, they erase the old number and write the new one. The rule is that the new number has to be greater than the old one. If a composite number gets written on the wall of this cave, then the world collapses into nothingness. Can the monks save the world for eternity?

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?