Tanya Khovanova's Math Blog

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—What do you get when a bunch of sheep hang out in a circle?
—Shepherd’s pi.

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—What do you call a metric cookie?
—A gram cracker.

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—What state has the most math teachers?
—Math-achusetts.

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—What does a hungry math teacher like to eat?
—A square meal.

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—What is the mathematician’s favorite season?
—Sum-mer.

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—What adds, subtracts, multiplies, divides, and bumps into light bulbs?
—A moth-ematician.

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—What tools do you use for math?
—Multi-pliers.

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—Why didn’t the quarter roll down the hill with the nickel?
—Because it had more cents!

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—Which snakes are good at math?
—Adders.

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—What is the butterfly’s favorite subject in school?
—Moth-ematics.

The Ural Math Battle in 2016 had several mafia-themed problems of various difficulty with the same initial setup.

Puzzle Setup. Among 100 residents of Saint-San, m are mafiosi, and the rest are civilians. A commissioner arrived to the town after getting this information. In an attempt to expose the mafia, this commissioner asked each of the residents to name s mafia suspects from among the other 99 residents. The commissioner knows that none of the mafiosi would name other mafiosi, but each civilian would name at least k mafia members. What is the maximum number of mafia members the commissioner can definitively identify after his survey?

1. The most difficult case was m = s = 3 and k = 2.
2. In the next case, where m = 3 and s = k = 2, the puzzle had a different task: prove that the commissioner can find at least one mafioso.
3. In the third case, where m = s = 10 and k = 6, the question was whether the commissioner can find at least three mafiosi.
4. In the fourth case, where m = s = 10 and k = 7, the question was whether the commissioner can find all the mafiosi.
5. The last case was for younger students with m = 6, s = 10, and k = 6. The question was whether the commissioner can find all the mafiosi.

When I asked ChatGPT to translate the first and the most difficult case of this puzzle from Russian, ChatGPT decided to solve it too. At the end of its ridiculous solution, it concluded that the commissioner could identify all 21 mafiosi out of the given 3. So, if you comment on this blog that the answer to the first case is 21, I will know that you are a bot.

I recently saw another puzzle on Facebook, a generalization of a problem from the 2002 Belarus Olympiad. In the problem, there are red and white boxes. Given how Russia and Belarus are filled with propaganda, my first question was whether the Belarusian flag was red and white. But in fact, the official flag is red and green; however, the opposition uses a red and white one. It could either be a coincidence or a sneaky way of protesting. Anyway, here is the problem.

Puzzle. There are two boxes filled with candy. The red box has R candies, and the white box has W candies. Alice and Bob are playing a game where Alice starts, and both players have the same options each turn: Either move one candy from the red box to the white box or take two candies from any box and eat them. The player who can’t move loses. For which values of R and W is each of the following true?

• Alice, following her optimal strategy, wins but might lose if she makes a mistake.
• Alice wins no matter what.
• Bob, following his optimal strategy, wins but might lose if he makes a mistake.
• Bob wins no matter what.

The list of options is weird, but I decided to keep it to emphasize …. Oops, I do not want to spoil it. You can decide for yourself what I wanted to emphasize.

Recently, I decided to stop avoiding puns and childish math jokes. So I am posting a lot of famous jokes I have heard before but never added to my collection.

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—What are ten things you can always count on?
—Your fingers.

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—Why should you never mention the number 288?
—Because it’s two gross.

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—Why did the two 4’s skip lunch?
—They already 8!

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—How do you make one vanish?
—Add a “g” to the beginning.

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—Why was 6 afraid of 7?
—Because 7 8 9.

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—How do deaf mathematicians communicate?
—With sine language.

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—Why don’t math majors throw house parties?
—Because it’s dangerous to drink and derive.

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—What’s the official animal of Pi day?
—The Pi-thon!

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—What do you get when you take the sun and divide its circumference by its diameter?
—Pi in the sky.

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—Who’s the king of the pencil case?
—The ruler.

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—Why was the inchworm angry?
—He had to convert to the metric system.

I’ve been collecting math jokes for many years, but I was avoiding puns, mostly because of my English. Puns are among the most difficult things to master in a language. Here are some punny jokes that I finally understand.

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—Who invented the Round Table?
—Sir Cumference.

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—Why didn’t the hyperbola feel sick?
—It was asymptote-matic.

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—Which triangles are the coldest?
—Ice-sosceles triangles.

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—What’s the one shape you should avoid at all costs?
—A TRAP-ezoid.

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—What do you call more than one L?
—Parallel.

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—What do you call a number that can’t keep still?
—A Roamin’ Numeral.

I was browsing my analog archives, my high-school notes, and stumbled upon a couple of problems from a math battle we had.

Puzzle. A square is divided into 100 pieces of the same area, in two ways. Prove that you can find 100 points such that each piece in the first division has a point inside, and each piece in the second division also has a point inside.

Puzzle. There is a finite number of points on a plane. All distances between any pairs of points are distinct. Each point is connected to its closest neighboring point. Prove that each point is connected to no more than 5 points.

Why would I complicate my life by crocheting colored borders onto different strips?

Answer: I wanted to emphasize their borders.

Do you recognize the objects in the picture? The leftmost one is a Möbius strip. I made it by crocheting a long rectangle. Then, instead of connecting the short sides to form a cylinder, I twisted one side 180 degrees before stitching them together. For the other two objects, I made 360 and 540 degree twists, respectively.

I used green yarn for the internal part of the strips. When the twist in the strip is a multiple of 360 degrees, the resulting surface is orientable and has two borders. I used two different colors to emphasize this fact. In other cases, the resulting surface is not orientable and has only one border, so I only used one color for the border.

The point of using extra colors for the borders is to make them more prominent. For example, it is easy to see that the Möbius strip’s border is a circle. The border of the piece in the middle consists of two loops, and the different colors make it obvious that the two borders are linked. The last object has one border, and the color helps you notice that its border is a trefoil knot!

What would happen with the borders if we increase the number of degrees in a twist? Can you figure it out? Are you willing to take up crocheting to solve this puzzle?

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A math problem is the only place where a person buys 7744 watermelons for dinner, but no one knows why!

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Today I saw a tweet from someone I knew in middle school. He tweeted, “I turned my life around 360 degrees!” Now do you see why it is important to study math?

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Looking for energy? Multiply time by power!

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The mom of a third grader calls her friend, “Lucy, did you do your son’s math homework?”
“I did.”
“Can I copy your answers?”

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If money is measured in piles, then I have a pit.

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My girlfriend is the square root of −100. She’s a perfect 10, but purely imaginary.

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A mathematical collapse: while cutting a worm, you divide it by 2 and multiply it by 2, simultaneously!

I used to love math problems about weighing things. But then I got distracted by my own personal scale with its slowly rising numbers. However, having recently lost a few pounds, I want to get back to other scales!

Puzzle. You have a balance scale that is broken in a consistent way: if you put two objects on its two pans, the scale will show you that the left pan is heavier, lighter, or the same weight as the right pan, but it may be wrong. However, it will give the same answer each time you repeat this test with the same two weights. You have a bag of flour and a 1-kilo weight. How can you use this scale to measure out 1 kilo of flour?

Puzzle. This time, your scale is not broken, and, moreover, it is not a balance scale but a digital one that tells you the weight of the objects you put on it. The scale does have a quirk. It can only measure two objects at a time. You have 13 coins of potentially different weights. How can you figure out the total weight of the 13 coins in 8 measurements?

The next puzzle was sent to me by Konstantin Knop, a coin puzzle master. This time there is no physical scale involved; rather, some sort of god answers your questions.

Puzzle. 26 identical-looking coins are arranged in a circle. Two of the coins which are next to each other are fake. You are allowed to pick any set of coins and ask how many fake coins are in the set. What is the smallest number of questions you need to find both fake coins if you only get the answers after you have posed all your questions?

A link is defined as two closed curves in three-dimensional space. The first picture shows an example of a link with one yellow curve and one blue. The linking number is a simple numerical invariant of a link. Intuitively, it represents the number of times that each curve winds around the other. For example, if it is possible to pull the two curves apart, the linking number is zero.

When I studied the linking number, I would look at a picture of a link trying to calculate this number. It was confusing. It only became easy after I started crocheting. For example, the second picture shows the same link as the first but slightly rearranged. I simply slid the yellow loop along the blue one until I could clearly see a piece of the blue loop as a straight segment and the yellow loop circling around it. Now, it is easy to see that the yellow loop winds around the blue one 3 times, making the linking number 3.

The only thing to remember is that while counting the number of windings, I need to consider the direction. It is possible for a loop to wind clockwise and then counterclockwise. In this case, the linking number is the difference between the two.

I crocheted a lot of links, and now my students and I have no problem calculating the linking numbers.