Tanya Khovanova's Math Blog

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 color each of them should end up wearing; this assignment is a permutation of the original colors. Each sage is allowed one swap of hats with another person per day. They have two days to rearrange the hats so that everyone ends up with the correct color. Can they do it?

Many students noticed that the permutation can be decomposed into disjoint cycles and suggested solving the problem cycle by cycle. A few of them even pushed this idea all the way to a complete solution. However, none of them connected the puzzle to a topic we had discussed in class: dihedral groups.

Here is an elegant way to finish the solution once the permutation is decomposed into cycles. A cyclic permutation on n elements can be viewed as a rotation of an n-gon. Any rotation of an n-gon can be written as a product of two reflections. Each reflection of an n-gon, viewed as a permutation, consists only of 1- and 2-cycles. Ta-da!

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 letter online, in case it might be helpful to other aspiring mathematicians.

Dear PRIMES and PRIMES-USA students,

Reading math papers can be very difficult and overwhelming. I remember trying to understand every single word of my first research paper and getting stuck on the first paragraph for a long time. That was a mistake. I regret that no one ever taught me how to read math papers. As the joke goes, “There are only two kinds of math books: those you cannot read beyond the first page, and those you cannot read beyond the first sentence.”

Math papers are not stories. They are not meant to be read linearly from beginning to end. Depending on your goal, you read different parts in different ways. Here are some examples.

Goal: Decide whether to read the paper.
Read: The abstract and parts of the introduction.

Goal: See what was accomplished.
Read: The introduction, or locate and read the main theorems.

Goal: Learn a method that might be useful.
Read: Find the relevant method and focus only on that section.

Goal: Get a general idea of the topic.
Read: First understand the structure of the paper. Then try to grasp the main statements at a high level.

Goal: Master the topic.
Read: Read the paper several times, going deeper with each iteration. Try not to get stuck on a sentence; you might understand it on another try. Here is a potential list of objectives for each iteration: you can adjust them and change their order according to your needs.

• First read: understand the structure and the big picture.
• Second read: understand the definitions and main notions.
• Third read: understand the main statements and look at small examples.
• Fourth read: understand the ideas behind the proofs.
• Fifth read: go deeper and start reading the referenced papers.
• Sixth read: try to reproduce the proofs.

Goal: Check for acknowledgments.
Read: The acknowledgments and citations.

The main rule is to keep your goal in mind while reading a paper. If you do not have a specific goal, ask your mentor to suggest exercises or questions to guide your reading. Try not to feel discouraged if you don’t understand everything: the joke at the beginning of this essay implies that everyone has trouble understanding math papers.

Tanya

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 tons of ideas about what can influence the weight.

• The Sun’s gravitational pull can influence the weight.
• The Moon’s gravitational pull can influence the weight.
• It would possibly decrease slightly due to factors such as weathering.
• During rainy days, it could absorb moisture and become heavier.
• Due to thermal expansion, the brick will be larger in the summer than in the winter, which means it displaces more air. As a result, it will weigh less in the summer.

I am not into physics. So, when I got these replies, I contacted a real physicist friend, Levy Ulanovsky. He referred me to Wikipedia: The first operational definition of weight was given by Euclid, who defined weight as: “the heaviness or lightness of one thing, compared to another, as measured by a balance.” This implies that when we talk of “weight”, we need to specify how we measure it. He continued by saying that the above ideas all make sense if we measure the weight force, e.g., by using a spring or pendulum frequency. Yet if our measurement is relative, e.g., by using a lever-like scale, then, for example, the sun’s gravitational pull is not a valid answer.

For example, if we measure the weight using a lever-like scale, with our brick on one side and a known weight combination on the other, then the weight reading on the moon will be the same as on Earth. If you use a spring, the weight will be different.

He added: People often use the words “weight” and “mass” interchangeably. But for teaching, you may wish to clarify that weight force is mg. A change in pendulum frequency shows a change in g, the acceleration due to gravity. A balance (lever-like) can show a change in m, the mass on one of its two plates relative to the mass on the other plate, with g equal at both ends. A spring, a rubber band, or a springboard can show a change in the weight force mg, whether caused by a change in m, in g, or in both m and g. Of the four student answers, the moon and sun change g; weathering and moisture change m; thermal expansion has several effects that interplay in a complicated way, so we’re better off forgetting about it.

I also asked Levy which effect is the strongest. His reply was: assuming a spring, a pendulum, or the like, the strongest effect is due to the Moon.

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.

Once I wrote a blog essay titled Seven, Ace, Queen, Two, Eight, Three, Jack, Four, Nine, Five, King, Six, Ten. It was about a “magic” card trick. Magic trick. Take a deck of cards face down. Move the top card to the bottom, then deal the new top card face-up on the table. Repeat this process until all the cards are dealt. And — abracadabra — the cards come out in perfect order!

If you want to perform this trick with one suit, the title of that earlier post tells you exactly how to stack your deck.

In the fall of 2023, I gave this trick as a homework problem to my STEP students. The result? We ended up writing a 40-page paper, Card Dealing Math, now available on the arXiv. At one point, we seriously considered calling it The Art of the Deal, but decided against it.

In the homework version, the deck consisted of cards from a single suit, but we generalized it to a deck of N cards labeled 1 through N. The dealing process we studied is called under–down dealing: you alternate between placing one card under the deck and then dealing the next one face-up. It’s very similar to down–under dealing, where you start by dealing the first card instead. These two patterns are often, unsurprisingly, called the Australian dealings.

The under-down dealing turns out to be mathematically equivalent to the Josephus problem. In that famous ancient problem, people are arranged in a circle, and you repeatedly skip one person and execute the next (much grimmer than playing with cards). The classic question asks: given N people, who survives? In our card context, this corresponds to asking where the card labeled N ends up in the prepared deck.

More generally, the Josephus problem can ask the following question. If we number the people in a circle 1 through N, in what order are they eliminated? In our research, we flipped the question around: how should we number the people in the circle so that they’re eliminated in increasing order?

Naturally, we couldn’t stop there. We explored several other dealing patterns, discovered delightful mathematical properties, and along the way added 44 new sequences to the OEIS. The funnest part? We also invented a few brand-new card tricks.

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.