Icosahedron’s Resistance

I rarely post physics puzzles, but this one is too good to pass on.

Puzzle. A wireframe icosahedron is assembled so that each of its edges has a resistance of 1. What is the total resistance between opposite vertices of the icosahedron?

While we are at it, another interesting question would be the following.

Puzzle. A wireframe cube is assembled so that each of its edges has a resistance of 1. What is the total resistance between opposite vertices of the cube?

And this reminds me of a question I heard when I was preparing for an IMO many years ago.

Puzzle. A wireframe infinite square grid is assembled so that each of its edges has a resistance of 1. What is the total resistance between two neighboring vertices?


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Ikigai

Have you ever heard of ikigai? The Japanese concept which gives four simple requirements for a happy career:

  • Do what you love.
  • Do what you are good at.
  • Do what impacts the world.
  • Do what you can get paid for.

I often think about it for myself and for my students. Is this good advice for finding a career path?

I like that ikigai separates the first two requirements: passion and gift. Many of my students do not see the difference, as passion and gift are highly correlated. When you love something, you practice it and become better at it. When you are good at something, it becomes easy and enjoyable.

Nonetheless, passion and gift are different. Unfortunately, I’ve seen students who are good at math only because their parents push them, but they do not love it. Some of them already found their passion but are afraid to tell their parents. Some haven’t yet found their passion, but it is perfectly clear that math is not it. So, a gift doesn’t imply passion.

What about the other way around? My programs are too selective, so I haven’t seen students who are not gifted in math. I will use myself as an example. I have always passionately loved dancing, but it is obvious that my dancing career would have been a disaster. I am very happy I closed that career path in fifth grade.

Anyway, the first two ikigai requirements are not the same, and both are necessary.

The third ikigai requirement is about doing what the world needs. Impacting the world is a great motivator and makes you feel good. And yet, I see happy and successful mathematicians who only care about the beauty of what they are doing and nothing else. This requirement is important but might not be a deal breaker for everyone.

The last ikigai requirement is crucial. If you are not being paid for your efforts, it is not a career; it is a hobby. I got attracted to it because it includes an important caveat: you need to find people who want to pay you for what you can offer. I recently wrote an essay Follow Your Heart? about many young aspiring opera singers who ignored this last requirement and ended up changing careers.

Nevertheless, the whole concept of ikigai bugs me. People who find their dream job might agree to work for much less pay than they are worth. It opens them up to potential exploitation by greedy employers.

Have I reached my ikigai? Judging by my low pay, I am close.


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Mini Stupidity

My grandkids like playing a game while I drive. They look out the window to spot the cars they like and score points. A Jeep is 1, a convertible is 10, a Mini Cooper is 40, and a Bug is 100. If we are lucky and see a convertible Mini or Bug, we get 10 extra points for convertibility. I play with them, of course. As a result, I can recognize minis and bugs from hundreds of miles away (I am exaggerating).

Recently, a Mini annoyed me. I was driving behind one, warmly thinking about my grandchildren, when its right turn signal started flashing. The signal looked like an arrow pointing to the left. I got so confused that my grandkids flew from my mind.

When I came home, I started googling and discovered that Mini designers wanted the British symbolism on their cars. The right signal is reminiscent of the right half of the British flag.

UK flag
Mini Cooper Right Turn Signal

Here is the picture from Reddit with the left turn signal on.

Mini Cooper Turn Signal

I am writing this essay but afraid to show my grandkids these pictures. They would be maxi-disappointed with Minis.


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The Angry Wife

Here is the homework problem I gave to my PRIMES STEP students.

Puzzle. A man called his wife from the office to say that he would be home at around eight o’clock. He got in at two minutes past eight. His wife was extremely angry at this lateness. Why?

The expected answer is that she thought he would be home at 8 in the evening, while he arrived at 8 in the morning. However, my students had more ideas.

For example, one student extended the time frame.

  • The man was one year late.

Another student found the words “got in” ambiguous.

  • He didn’t get into his house two minutes past eight. He got into his car.

A student realized that the puzzle never directly stated why she got angry.

  • The wife already got angry when he said he would be home around eight, as she needed him home earlier.

The students found alternative meanings to “called his wife from the office” and “minutes.”

  • He had an office wife whom he called. But the wife at home was a different wife, and she was angry.
  • “Two minutes past eight” could be a latitude.
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My Unique Christmas Card

One of the perks of being a teacher is receiving congratulations not only from family and friends, but also from students. By the way, I do not like physical gifts — I prefer just congratulations. Luckily, MIT has a policy that doesn’t allow accepting gifts of any monetary value from minors and their parents.

Thus, my students are limited to emails and greeting cards.

One of my former students, Evin Liang, got really creative. He programmed the Game of Life to generate a Christmas card for me. You can see it for yourself on YouTube at: Conway Game of Life by Evin Liang.

This is one of my favorite congratulations ever.

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A Probability Puzzle from Facebook

Puzzle. There are 100 cards with integers from 1 to 100. You have three possible scenarios: you pick 18, 19, or 20 cards at random. For each scenario, you need to estimate the probability that the sum of the cards is even. You do not need to do the exact calculation; you just need to say whether the probability is less than, equal to, or more than 1/2.

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Four Sheep

I like including warm-up puzzles with every homework.

Puzzle. Farmer Giles has four sheep. One day, he notices that they are all standing the same distance away from each other. How can this be so?

The expected answer: The configuration is impossible in 2D. So, one of the sheep is on a hill or in a pit.

Some students thought big: The sheep could be placed at different locations around the Earth, forming a really big tetrahedron. In this case, we need to explain what it means for the farmer to “notice”, but this minor issue could be resolved in many ways.

Some of the students questioned the meaning of the word distance. They argued that if sheep are all touching each other, they are the same distance 0 from each other. One way this could happen is if the tails of the four sheep were entangled.

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Back to Coins

I loved coin puzzles, but after several research projects related to those shiny discs, I got tired of them. The fatigue was temporary, as confirmed by the following Facebook puzzle that reignited my interest.

Puzzle. There are 30 coins in a circle that look the same. However, 20 of them are fake, and the rest are real. Fake coins weigh the same, and real coins weigh the same but heavier than fake ones. You need to find as many fake coins as possible using a balance scale once, given that the fake coins are positioned consecutively. What is your strategy?

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SOS

My PRIMES STEP program consists of two groups of ten students each: the senior group and the junior group. The senior group is usually stronger, and they were especially productive last academic year. We wrote four papers, which I described in the post EvenQuads at PRIMES STEP. The junior group wrote one paper related to the game SOS. The game was introduced in the following 1999 USAMO problem.

Problem. The game is played on a 1-by-2000 grid. Two players take turns writing an S or an O in an empty square. The first player who produces three consecutive squares that spell SOS wins. The game is a draw if all squares are filled without producing SOS. Prove that the second player has a winning strategy.

The solution is quite pretty, so I do not want to spoil it. If my readers want it, the solution for this grid, and, more generally, for any grid of size 1-by-n, is posted in many places.

My students studied generalizations of this game, and the results are posted at the arXiv: SOS. We tried different target strings and showed that:

  • The SOO game is always a draw.
  • The SSS game is always a draw.
  • The SOSO game is always a draw.

Then, we tried a version where the winner needed to spell one of two target strings. We showed that:

  • The SSSS-OOOO game is always a draw.

We tried several more elaborate variations, but I want to keep this post short.

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A Quadrilateral in a Rectangle Solution

I recently posted A Quadrilateral in a Rectangle puzzle.

Puzzle. A convex quadrilateral is inscribed in a rectangle with exactly one quadrilateral’s vertex on each side of the rectangle. Prove that the area of the rectangle is twice the area of the quadrilateral if and only if a diagonal of the quadrilateral is parallel to two parallel sides of the rectangle.

Now it is time for a solution where I use the sample rectangle pictured below. We draw lines parallel to the sides of the rectangle from every vertex of the quadrilateral. Now, we can find four pairs of congruent triangles where one triangle is inside the quadrilateral and the other is outside. In the picture below, the pairs are colored the same color. We see that green and red rectangles overlap, creating a brown rectangle. The fact that they overlap means that the quadrilateral’s area is less than half of the rectangle’s area.

Polyomino Cutting
Polyomino Cutting

It could go the other way, as the next picture shows. Here, the quadrilateral’s area is more than half of the rectangle’s area. In this case, we have an “underlap” as opposed to an overlap.

Therefore, the quadrilateral’s area is exactly half of the rectangle’s if and only if there is no overlap/underlap, implying that the thickness of the overlap/underlap rectangle is zero. This means that one of the diagonals of the quadrilateral has to be parallel to two sides of the rectangle.

Polyomino Cutting

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