Tanya Khovanova's Math Blog

Let me begin with labeled chip-firing on an infinite directed binary tree, where the root is at the top. Place 2n chips, labeled from 0 to 2n − 1, at the root. A move consists of choosing any two chips at the same vertex and firing them: the smaller chip goes to the left child, […]

For the last homework of the year, I gave my students a killer puzzle—literally. Puzzle. A mysterious man kidnaps people, takes them to his cabin, and offers each victim two identical-looking pills. He claims that one pill is poisonous and the other is harmless. The victim chooses one pill, swallows it with water, and dies, […]

Here is another puzzle posted on Facebook by Konstantin Knop. Puzzle. One day, three inhabitants of the Island of Knights and Liars invited a foreign correspondent to visit so that they could tell him about their island. Knights always tell the truth, while liars always lie. - The first inhabitant said, “The island has at most AT inhabitants. All the islanders are liars.” - The second added, “The…

I asked ChatGPT to tell me some jokes about itself. Here we go. * * * ChatGPT never loses an argument.It simply generates a more confident version of the same mistake. * * * I asked ChatGPT for a brief answer.It said, “Absolutely!” and then divided the answer into twelve sections. * * * ChatGPT […]

Konstantin Knop wrote to me about a weighing puzzle. Apparently, the balance scale keeps inserting itself into our conversation. Puzzle. There are several coins, each weighing either 10 grams or 11 grams. Determine the parity of the number of coins of each weight using a balance scale. Given the total number of coins, it is […]

I gave the following problem as part of the entrance test for my STEP program. Puzzle. What word would you use to describe a man who does not have all his fingers on one hand? The test had 17 questions, and this one was the only trick question. My goal was to check whether the […]

Here is a new puzzle by Nikolai Chernyatiev. Puzzle. Five sages, who all know one another, are blindfolded, seated in a row in a dimly lit hall, and then have their blindfolds removed. Each sage can see both of their immediate neighbors, but no farther; the sages at the ends know that they are at […]

Alexander Karabegov sends me new puzzles from time to time. This time, however, it is not a puzzle but a math joke. Joke. If a woman gives birth to a child at the age of 30, then 60 years earlier, her child was twice as old as she was. Whatever that means. Share:

I’ve been staring at my icosahedron, trying to solve the following puzzle by Konstantin Knop. Puzzle. One face of an icosahedron is special. The numbers 2, 3, and 5 are written at its vertices in some order. All other vertices of the icosahedron are labeled with 1. In one query, we may ask an oracle […]

Puzzle. A ball rolls forever on a frictionless billiard table with no pockets. Can you find a finite convex shape of the table for which no trajectory of the ball ever covers the entire surface? Share:

I gave the following puzzle to my students. Puzzle. A cowboy rides into town on Friday, stays for three days, then leaves on Friday. How come? Most of them submitted the standard answer: his horse is named Friday. One student suggested that the town was named Onfriday. This solution works well when the puzzle is […]

I am so happy that I met Ivo David and Yogev Shpilman. Together, wrote the book Mathematical Puzzles and Curiosities, and had so much fun on the way. I have already mentioned one of the puzzles from the book on my blog—Two Points on a Cube. Here is another one. Puzzle. Consider the sequence: O, […]

Puzzle. Two fathers gave money to their sons. The first father gave $200, and the second father gave $100. Yet the total amount received by the sons was only $200. How come? Standard answer: There were three people: a son, his father, and his grandfather. The grandfather gave the father $200, and the father gave […]

In the homework for my STEP program, I gave the following challenge problem. Puzzle. My sages each wear a hat of a different color. As in standard hat puzzles, they can see everyone else’s hat color. Unlike in many other hat puzzles, they know the color of their own hat as well. I announce which […]

In a recent post, Each Point has Three Closest Neighbors, I mentioned the following conjecture. Karabegov’s Conjecture. Any finite planar point configuration in which every point has exactly 3 closest neighbors must contain at least 16 points. The conjecture was proposed by my dear friend Alexander Karabegov, whom I met in 1974. Wait. What?! I […]

Every year, after the PRIMES program begins, I send a letter to our students about how to read a math paper. The students in my group are juniors just starting their research. They are often required to read advanced math papers—frequently the first research papers they have ever encountered. This year, I decided to post […]

My goal is to expand my students’ minds. So, though my STEP program is about mathematics, I sometimes give problems from other areas for homework. Here is a recent physics one. Puzzle. You have a brick of 1 kilogram. How does the weight of the brick change during the year? As always, my students generated […]

Here is a probability puzzle I heard from my son Sergei. We even included this puzzle in our book Mathematical Puzzles and Curiosities. Our book includes the answer but omits the details. So, this blog post is devoted to said details. Puzzle. Alice rolls a die until she gets 6. Then Bob observes that she […]

I wrote a book. This is my first book, so I am very proud. I wrote it together with two brilliant puzzle lovers, Ivo Fagundes David de Oliveira and Yogev Shpilman. The book is published by World Scientific and is available for pre-order: Mathematical Puzzles and Curiosities. Here is one sample puzzle from the book. […]