Tanya Khovanova's Math Blog

I wrote a lot about the inventiveness of my students. Here is more proof.

Puzzle. A police officer saw a truck driver going the wrong way down a one-way street but didn’t try to stop him. Why?

Many of my students came up with the expected answer:

The truck driver was walking.

They also found some legit ways for a truck driver to not be stopped.

• The police officer was too far away.
• There was construction nearby, so the police officer directed the driver to drive the wrong way.
• The truck was a fire truck responding to an emergency.
• The driver bribed the police officer.
• The driver was a kid playing with a toy truck.

Some more ideas, rather far-fetched.

• The police officer was off duty, so he called another police officer to stop the driver.
• The truck driver was going too fast to stop.
• The police officer was responding to a bank robbery, and stopping the truck driver was not high priority.
• The police officer was driving the wrong way too, and it would be hypocritical to stop the truck driver.
• The street was a dead end, and the only way out was to go the wrong way.

Some funny ones.

• The police officer had a history of hallucinating and thought the truck driver was a figment of his imagination.
• The police officer was a ghost.
• The police officer was the truck driver.
• The police officer was busy eating a donut.
• The truck driver was the police officer’s boss.
• The truck driver was the police officer’s grandma.

I often receive letters from parents of math geniuses — “My twelve-year-old is reading an algebraic geometry book: accept him to PRIMES,” or “My ten-year-old finished her calculus course: here is her picture to post on your blog,” or “My two-year-old knows the multiplication table, can you write a research paper with him?” The last letter was a sarcastic extrapolation.

I am happy to hear that there are a lot of math geniuses out there. They are potentially our future PRIMES and PRIMES STEP students. But, it is difficult to impress me. The fact that children know things early doesn’t tell me much. I’ve seen a student who didn’t know arithmetic and managed to pass calculus. I’ve also met a student claiming the full knowledge of fusion categories, which later appeared to be from half-watching a five-minute YouTube video.

There are a lot of products catering to parents who want to bring up geniuses. My grandson received a calculus book for his first birthday: Introductory Calculus For Infants. Ten years later, he still is not ready for calculus.

Back to gifted children. Once a mom brought us her kid, who I can’t forget. The child bragged that he solved 30 thousand math problems. What do you think my first thought was? Actually, I had two first thoughts: 1) Why on Earth would anyone count all the problems they solved? 2) And, what is the difficulty of the problems he solved 30 thousand of?

From time to time, I receive an email from a parent whose child is a true math genius. My answer to this parent is the same as to any other parent: “Let your child apply to our programs. We do a great job at working with math geniuses.”

Our programs’ admissions are done by entrance tests. Surprisingly, or not surprisingly, the heavily advertised kids often do poorly on these tests. It could be that the parents overestimate their children’s abilities. But sometimes, the situation is more interesting and sad: I have seen children who sabotage the entrance tests so as not to be accepted into our programs. We also had students give us hints on their application forms that they were forced to apply.

In the first version of this essay, I wrote funny stories of what these students did. Then, I erased the stories. I do not want the parents to know how their children are trying to free themselves.

Dear parents, do not push your children into our programs. If they do not want to be mathematicians, you are decreasing their chances of getting into a good college. Imagine an admission officer who reads an essay from a student who wants to be a doctor but wastes ten hours a week on a prestigious math research program. Such a student doesn’t qualify as a potential math genius, as their passion lies elsewhere. Nor does this student qualify as a future doctor, as they didn’t do anything to pursue their claimed passion. In the end, the student is written off as a person with weak character.

On the other hand, the students who do want to be in our programs, thrive. They often start breathing mathematics and are extremely successful. Encourage your children to apply to our programs if they have BOTH: the gift for mathematics and the heart for it.

This puzzle was in last week’s homework.

Puzzle. How can an egg fly three meters and not break?

The expected answer:

• The egg flew more than 3 meters and broke afterward.

Some students tried to protect the egg:

• The egg was bubble-wrapped.
• The egg was dropped on a cushion.
• The egg was thrown up, then caught.
• The egg was thrown into water.
• My favorite: The egg used a parachute.

Other students specified qualities of an egg making it more resistant:

• The egg was hard-boiled.
• The egg was made of plastic.
• The egg was a frog egg.
• An educated answer: It could be an ostrich egg, which is extremely strong. (I checked that online, and, indeed, a human can stand on an ostrich egg without breaking it.)
• My favorite: The egg was fried.

Here are some more elaborate explanations:

• The egg flew on a plane.
• The egg was thrown on another planet with low gravity.
• The egg was thrown in space and will orbit the Earth forever.
• My favorite: The egg was not birthed yet: it flew inside a chicken.

To conclude this essay, here is a punny answer:

• The egg was confident, not easy to break by throwing around.

I am so tired of spam emails. I keep thinking about how we can fight spam, and here is an idea.

Gmail should change its system: every email you send to me would cost 1 dollar, payable to me. We can add an exception for people on my contact list. Everyone else, pay up!

I do not often contact strangers. But if I do, it is always important. So paying 1 dollar seems more than fair. On the other hand, this system will immediately discourage mass emails to strangers. Spam would go down, and I would stop receiving emails inviting me to buy a pill to increase the size of a body part I do not have.

This idea of getting paid for reading an email is not new. It was implemented by Jim Sanborn, the creator of the famous Kryptos sculpture. Kryptos is located at the CIA headquarters and has four encrypted messages. People tried to decrypt them and would send Jim their wrong solutions. Jim got tired of all the emails and administered Kryptos fees. Anyone who wants Jim to check their solution, can do so by paying him 50 dollars. I wonder if Jim would still charge the fee if someone sent him the correct solution.

Thinking about it, I would like the payable email system to be customizable, so I can charge whatever I want. After all, I do value my time.

Gmail could get a small percentage. Either Gmail, together with me, gets rich, or spam goes away. Both outcomes would make my life easier.

Puzzle. A goat was on a 10-meter leash. Yet it managed to go 300 meters away from the post. How come?

The standard answer. The leash wasn’t attached to the post.

My students scrutinized the puzzle and found some other possible ambiguities. For example, there might be two posts: the goat was leashed to one and was far away from the other. In another example, the timing is not given. It is possible that the goat was on the leash at one time and unleashed and far away from the post at another time. Here is my favorite answer.

My favorite answer. The goat ate the leash.

Puzzle. Find the largest and the smallest 4-digit numbers n such that when you erase the first two digits of n, you get the sum of the digits of n.

I recently posted a gnome puzzle by Alexander Gribalko.

Puzzle. Nine gnomes repeat the following procedure three times. They arrange themselves on a 3 by 3 chessboard with one gnome per cell and greet all of their orthogonal neighbors with handshakes. Prove that not all pairs of gnomes greet each other.

As often happens with my blog puzzles, I used this puzzle as homework for my students. They calculated that the total number of handshakes needed for all nine gnomes to greet each other is 36. On the other hand, each arrangement of gnomes creates 12 handshakes. This means that the numbers are tight: no greeting can be wasted, and every pair of gnomes need to greet each other exactly once. The students then studied different cases to prove this was impossible.

In each arrangement, a gnome can have either 2, 3, or 4 handshakes. Hence, we can distribute handshakes over three placements as 2+2+4 or 2+3+3. It follows that if a gnome is ever in the center of the grid, he has to be in a corner for the other two arrangements. Therefore, the three gnomes who end up in the center for one of the arrangements never greet each other.

However, I always love solutions that involve coloring the board in a checkerboard manner. Here is my solution.

Solution. Let’s color the board in a checkerboard manner. We assign each gnome a binary string, of length 3, describing the colors of the cells where the gnome was in each placement. There are 8 different possible strings. It follows that at least two gnomes are assigned the same string. But they can’t greet each other if they are standing on the same colors in each arrangement!

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I hate getting into debates about Möbius strips. They’re always one-sided.

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North Korea’s ballistic missile test failed due to a bug in Windows. The next missile containing a bug report has been automatically sent to Microsoft.

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4 out of 3 people have trouble with fractions.

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Do you know what seems odd to me?
Numbers that aren’t divisible by two.

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Why was algebra so easy for the Romans? X was always 10.

My readers know that I love hat puzzles. This is why I decided to turn a number trick by Konstantin Knop (in Russian) into a hat trick.

Hat Trick. The audience has a bottomless supply of hats in ten different colors. They arrange ten people in a line and put one of the hats on each person. Then the magician’s assistant comes in and removes a hat from one of the ten people. After that, the magician appears and, abracadabra, guesses the color of the removed hat. The magician and the assistant agreed on a strategy beforehand. What is it?

Keep in mind that this trick won’t work with fewer than ten colors. As a bonus, can you explain why?

Sep 18, 2022 Correction: I meant “with fewer than ten people.”

As my readers know, I am devoted to my students. When I need something I can’t buy, I try to make it. That is why I crocheted a lot of mathematical objects. One day, I resolved to have in my possession a Seifert surface bounded by Borromean rings (a two-sided surface that has Borromean rings as its border).

However, my crocheting skills were not advanced enough, so I signed up for a wet and needle felting workshop. When I showed up, Linda, our teacher, revealed her lesson plan: a felted soap with a nice pink heart on top. It looked cool to have soap inside a sponge, not to mention that wool is anti-bacterial. But I had bigger plans than soap and eagerly waited for no one else to show up.

When my dream materialized, and, as I had hoped, no one else was interested in felt, I asked Linda if we could drop the hearty soap and make my dream thingy. She agreed, but my plan didn’t survive for long. As soon as Linda saw a picture of what I wanted, she got scared. Seifert surfaces were not in the cards, so soap it was. I told her that there was no way I was going to needle-felt a pink heart onto my felted soap. I ended up with a blue Sierpiński gasket.

We had a great time. Linda was teaching me felting, and I was teaching her math. I am a good teacher, so even felters working on a farm enjoy my lessons.

After the workshop, I went online and found my dream surface on Shapeways. In the end, I was happy to just buy it and not have to make it.

But my felting workshop wasn’t a waste of time: tomorrow I will wash myself with a gasket.