Tanya Khovanova's Math Blog

I recently gave my STEP students a question from our old 2014 PRIMES entrance test.

Puzzle. John’s secret number is between 1 and 2¹⁶ inclusive, and you can ask him yes-or-no questions, but he may lie in response to one of the questions. Explain how to determine his number in 21 questions.

Here is the standard solution. We start by asking John to convert his number into binary and add zeros at the beginning if needed to make the result a binary string of length 16. For the first 15 questions, we do the following. For question i, we ask: “Is the i-th digit of your string zero?” For question 16, we ask, “Have you lied in response to a previous question?” If he lied on a previous question, he must say YES. If he didn’t, he might lie on question 16 and also say YES. In any case, if the answer is NO, he didn’t lie on the first 15 questions and we know the first 15 digits of the number. Then, we ask about the last digit three times, and the answer given at least twice is correct, so we know the number.

If the answer to question 16 is YES, then he lied on one of the questions 1–16. From now on, he has to tell the truth since he already lied. We use binary search (4 questions) to determine on which question he lied. This will tell us the first 15 digits, and we can use the 21st question to find the last digit.

One of my students, Tanish, invented an out-of-the-box solution that uses 18 questions. The idea is to force John to lie in the first two questions, and then safely proceed with the binary search.

He suggested asking the following two questions: “Are you going to answer NO in response to the next question?” and “Did you respond YES to the previous question?” The reader can check that whatever John replies, he is forced to lie exactly once.

Another student, Vivek, had a similar idea but used only one question to force John to lie: “Will you say NO to this question?”

My team, Death and Mayhem, organized the 2025 MIT Mystery Hunt. The hunt was a great success. Many people commented that it was the best mystery hunt ever.

This year, we added a new and interesting feature. Not only were teams allowed to choose which puzzles to unlock, but they were also given a short description of each puzzle in addition to its title. So, small teams who liked crosswords could choose to work only on crosswords.

As usual, I will list the mathy puzzles, including our official puzzle descriptions. All the puzzles can be found at the hunt’s All puzzles page.

We had a special round called Stakeout, with easy puzzles. My team isn’t too nerdy, so we didn’t have too many mathematical puzzles overall, and just two puzzles with a math flavor in the Stakeout round, incidentally coauthored by me. Somehow, I like designing easy puzzles. There were two additional puzzles in this round that I enjoyed during testing. I loved the popsicle puzzle so much that I brought it to my grandchildren to solve.

• A Math Quiz: A list of math questions.
• A Sudoku?: Grids with numbers.
• Anything Is Popsicle: Physical Puzzle — popsicle sticks.
• Some Assembly Required: A list of phrases and a word bank.

The first round wasn’t too difficult either. Several people praised the ChatGPT puzzle, though it’s not mathy.

• ChatGPT: A blank textbox with a text entry field below it.

Now, moving to more difficult puzzles, Denis Auroux is famous for designing fantastic logic puzzles. His puzzles below aren’t easy, but many people loved them. I even heard magnificent as praise.

• Maze of Lies: A text adventure exploring a maze.
• Bermuda Triangle: Triangular battleship puzzles.
• This is Just a Test: A bunch of grid logic puzzles.
• Good Fences Make Otherwise Incompatible Neighbors: A hexagonal logic puzzle.

Here are two puzzles I test-solved and enjoyed. The first one is a logic puzzle, while the second one isn’t math-related.

• Unreal Islands: A grid filled with letters and numbers.
• To Do: Tile That Rectangle: Rectangles of different lengths with words on the center and edges.

Here are two puzzles that I edited and highly recommend. The first puzzle was initially called Gin and Tonic; I wonder if anyone can guess why.

• Follow The Rules: An interactive interface with a grid of toggle switches and a grid of lights.
• Incognito: Cryptic crossword.

These are math-related puzzles that people liked.

• The Inspectre: Physical Puzzle — A bag of acrylic pieces and a sheet of paper.
• Knights of the Square Table: A 9×9 grid logic puzzle.
• Do The Manual Calculations (Don’t Try Monte Carlo): Some unusual chessboards and graphs with rules.

I asked only a few people for recommendations. These are math-related puzzles that weren’t mentioned but seem cool. The fourth puzzle was an invitation to the Mystery Hunt, which, not surprisingly, was a puzzle.

• The 10000-Sheet Excel File: An Excel file with 10000 sheets.
• Eponymous Forensic Accountant: Physical Puzzle — Bring your own bag (The PDF with receipts is available).
• _land: A thin, multicolored horizontal line.
• Engagements and Other Crimes: A printed invitation.
• A Map and a Shade (or Four): A chart with letters, colors, ticks, crosses, and a series of pie charts below.
• Garden Anecdotes: Post-it notes and some gibberish text.

I also got a recommendation for a non-math puzzle, which I would definitely have enjoyed watching solved. I’m not sure I’d enjoy solving it alone.

• Men’s at My Nose: A series of quotes.

Finally, here is the list of non-math puzzles that seem cool. A warning about the first puzzle: It’s rated R. The first three puzzles are relatively easy; they are from the Stakeout round.

• A Recipe For Success: Several completely normal and reasonable pickup lines.
• Charged: Interactive interface with text entry boxes and lines between them.
• Magic i: Two columns of text.
• Star Crossed: A large isometric grid of letters in the shape of a circle.
• Beyond a Shadow of a Doubt: A series of colorful dropquote puzzles.

Here is a video from Cracking the Cryptic, joined in this episode by Matt Parker, titled Matt Parker Sets Us A Challenge!. The video is devoted to the second part of the puzzle Maze of Lies, mentioned above, by Denis Auroux and Becca Chang.

I am a proud member of the Death and Mayhem team, which participates in the MIT Mystery Hunt every year. This year, our team had the honor of running the hunt.

Here is a puzzle I contributed, titled A Math Quiz. It consists of a list of math problems. I am especially happy that I was able to turn a collection of cute math puzzles into a puzzle-hunt challenge with a word or phrase as its final answer.

I met Alexander Karabegov during the All-Soviet Math Olympiad in Yerevan. He was one year older than me. By then, when I was still competing in 1976, he was already a freshman at Moscow State University. He proposed the following two related puzzles for the Moscow Olympiad, which I had to solve.

Puzzle 1. You are given a finite number of points on a plane. Prove that there exists a point with not more than 3 closest neighbors.

Just in case, by closest neighbors I mean all points at the minimal distance from a given point. I am sure I solved both puzzles at the time. I leave the solution to the first one to the reader.

Puzzle 2. Can you place a finite number of points on the plane in such a way that each point has exactly 3 closest neighbors?

The last problem has an elegant solution with 24 points chosen from a triangular grid. The story continued almost 40 years later, when Alexander sent me an image (below) of such a configuration with 16 points. He conjectures that this is the minimal configuration.

Karabegov’s Conjecture. Any finite planar point configuration in which every point has exactly 3 closest neighbors must contain at least 16 points.

Can you prove it?

Initially, I didn’t want to give the 24-points solution, but the image above is a big hint, so here you go.

Both constructions reveal the same underlying pattern. The constructions consist of rhombuses formed by two equilateral triangles, and the rhombuses are connected to each other. The 24-point construction consists of 6 rhombuses, while the 16-point construction consists of 4 rhombuses. What will happen if we try the construction with 3 rhombuses? The image below shows such a configuration, which now has extra edges with the shortest distance. We now see 3 points with more than three closest neighbors each, violating the condition. So the conjecture doesn’t break.

So far, every smaller attempt failed — can you prove that 16 is minimal?

Recently, I gave my STEP students the following discussion question.

Puzzle. A long time ago, before anyone had ever heard of ultrasound, there was a psychic who could predict the gender of a future child. No one ever filed a complaint against her. Why?

I based this puzzle on a story I once read. In the story, the psychic kept a neat little journal where she wrote down each client’s name and the predicted gender — except she secretly wrote down the opposite of what she told them. When someone came back complaining that she was wrong, she would calmly open her journal and say, “Oh, you must have misheard”.

This scam demonstrates conditional probability. The satisfied customers never came back; only the unhappy ones did — and those she could ‘prove’ wrong. Understanding probability can help my students detect and expose scams.

My students, of course, had their own theories. The most mathematical one was a pay-on-delivery scheme: if the psychic was right, she got paid; if not, she didn’t. Another innocent idea was for the psychic to keep moving. By the time the babies were born, she’d be long gone predicting future children’s genders somewhere far away.

ChatGPT offered a different explanation: the psychic never said whose future child she was predicting. If the prediction failed, she could always clarify that she meant someone else’s child. After some prodding, the idea evolved and became even sneakier: If the prediction failed, she could always clarify that she meant the couple’s next child, or, if they weren’t planning more children, a grandchild. Another brilliant, but unrealistic idea was to never charge anyone. Hard to sue someone who never took your money.

One student suggested that the psychic wasn’t wrong at all — she was predicting the baby’s true inner gender. In today’s world, rather than in the world before ultrasound, that one almost sounds plausible!

And finally, I’ll leave you to guess one more explanation — proposed, surprisingly, by several students. (Hint: they were disturbingly creative.)

To conclude: I enjoy teaching my students. Understanding probability won’t let them predict the future, but it might make them less gullible.

Puzzle. How can you make the following equation correct without changing it: 8 + 8 = 91?

The intended answer: turn the paper over! When flipped upside down, the equation becomes 16 = 8 + 8.

As you might expect, my blog post doesn’t stop there. I’d like to share some creative ideas my students came up with when they tackled this puzzle as part of their homework.

The most common suggestion was to interpret the equation modulo some number. For example, it works modulo 75. By extension, it also works modulo any divisor of 75: 3, 5, 15, or 25.

They also suggested interpreting the equation in base 5/3.

One far-fetched but imaginative submission proposed the following: Suppose the equation is written in an alien language whose symbols look identical to ours but have different meanings. In this alien base-10 system, the symbols + and = mean the same as on Earth, but an 8 represents 6, a 9 represents 1, and a 1 represents 2. Then the alien equation 8 + 8 = 91 translates to 6 + 6 = 12 in human, which is perfectly true.

But my favorite answer was the following:

• Interpret the question mark as a variable and solve the equation. This gives ? = 16/91. We didn’t change the equation — just solved it!

The title sounds like a list of healthy foods. However, this list is from the homework I gave to my students.

Puzzle. Which one doesn’t belong: egg, banana, apple, walnut, tangerine, or avocado?

The book answer was apple as the only one which we can eat without peeling.

Other students suggested a lot of reasons why egg is the odd one out.

• Egg is the only one not grown from a plant.
• Egg is the only one without a letter a.
• Egg is the only one you can’t eat without cooking.

Overall, the students found reasons for each of them. In addition to the above, we have the following.

• Banana is the only one not in a spherical or ellipsoidal shape.
• Walnut is the only word without repeated letters.
• Tangerine is the only word with a square number of letters, and it is also the only citrus.
• Avocado is the only word with more vowels than consonants.

I start my homework with warm-up puzzles.

Puzzle. Two friends went for a walk and found $20. How much money would they have found if there were four of them?

The answer, of course, is $20. The number of people doesn’t change the amount of money lying around. Even ChatGPT gave this answer. Duh!

My hope was to catch them not paying attention and mindlessly multiply to get $40.

To my surprise, some of them answered $80. The ‘them’ in the problem is not specified. It appears that they read the puzzle as if they found one 20-dollar bill, and them was referring to bills.

One student wrote a thoughtful reply: Having more friends most likely wouldn’t change the amount of money found, considering the amount of money is independent of the number of people, meaning the friends would still find $20. However, with double the people, they may find more money in other locations. There is also a chance that the 2 extra friends would make the group walk a different path, meaning they wouldn’t find money at all.

I recently posted the following puzzle about identical triplets.

Puzzle. Three brothers who are identical triplets live on the seventh, eighth, and ninth floors of the same apartment building. Their apartments are identical and vertically stacked. One day, all three step onto their balconies, standing in the same upright posture. The brother on the eighth floor shouts, “AAAA!” Which of the other two will hear him first?

Most readers got it right: our mouths sit lower than our ears. That means the distance from the mouth of the brother on the eighth floor to the ears of the brother on the seventh floor is shorter than the distance to the ears of the brother on the ninth floor. So the seventh-floor brother hears it first.

However, one reader, Ivan, taught me something I didn’t know: identical twins aren’t always identical. He even sent a photo of Mark and Scott Kelly — identical twins of different heights.

Of course, as a first approximation, we can assume identical triplets are identical. But mathematicians are nitpicky and like precision. Ivan (clearly a mathematician at heart) also noted that even identical twins might wear shoes with different heel heights, which could tweak the distances.

Here’s another reader submission that made me smile:

• The seventh-floor brother will hear it first, because the eighth-floor brother has fallen off the balcony and is screaming as he plummets towards the earth.

Nitpicking again: that’s a stretch, since the problem says they’re standing — but it’s still funny.

Here’s a problem from our 2025 STEP entrance test, taken by nearly a hundred students.

Problem. Pavel likes pets. All his pets except two are dogs. All his pets except two are cats. All his pets except two are parrots. The rest of the pets are cockroaches. How many pets of each kind does Pavel have?

Here is a solution from one student: one cat, one dog, and one parrot. No cockroaches—phew. Most students (and ChatGPT) found this one. By the way, I ran my whole test through ChatGPT, and this was the only mistake it made. ChatGPT, along with many students, missed the second solution: Pavel has two cockroaches.

Two more students’ answers made me smile:

• His pet cockroach is named Two. It follows that Pavel has zero cats, zero dogs, zero parrots, and one cockroach named Two.
• The parrots would eat the cockroaches, the cats would eat the parrots, and the dogs would eat the cats. Whatever he has now, he’ll be left with only dogs.