## Another Symmetry Puzzle

I recently posted a symmetry puzzle from Donald Bell. He just sent me another one.

Puzzle. Start with a 30-60-90 triangle (half of an equilateral triangle). Divide it into two 30-60-90 triangles of different sizes by dropping a perpendicular from the right-angled corner to the opposite side. Put the resulting two pieces together to form a symmetrical shape. There are two solutions.

It took me some time to find the second solution. I love this puzzle.

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## Is It Possible?

Usually, I only post puzzles to which I know the solution. However, I don’t know the solution to this exciting geometry question from Facebook, yet. But I like the puzzle so much that I’d rather post it than wait until I find time to think about it.

Puzzle. A centrally-symmetric figure is cut into two equal polygons: A and B. Is it possible that the center of symmetry is in A but not in B?

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## An Alternator Coin Puzzle

I run a program at MIT called PRIMES STEP, where we conduct mathematical research with children in grades 6 through 9. Our first research project was about a funny coin called an alternator. This coin exists only in a mathematician’s mind as it can change weight according to its own will. When you put the alternator on the scale, it can either weigh the same as a real coin or a fake coin (the fake coins are lighter than real ones). The coin strictly alternates how much it weighs each time it is put on the scale. My colleague, Konstantin Knop, recently sent me a fresh alternator puzzle.

Puzzle. There are four identical-looking coins: two real, one fake, and one alternator. How do you find the alternator using a balance scale at most three times?

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## 2024 MIT Mystery Hunt

I am not as excited about the MIT Mystery Hunt as I used to be. So, for this year’s hunt, I didn’t go through all the puzzles but present here only the puzzles that were recommended to me. I start with math, logic, and CS.

Then we have some word puzzles.

Now, the rest.

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## Calculus in Life

Here is an old joke.

A former student runs into an old calculus teacher as she is shuffling home. The student is happy to see her and says, “I recently thought about you and our classes.” “How so?” she perks up. “I was in a bit of a pickle when calculus helped me,” he says. The old lady straightens, and her face begins to glow. “Can you elaborate?”, asks the teacher in anticipation. The student proceeds with his story. “I was walking home in the pouring rain when a gust of wind snatched my hat right off my head. The hat landed in a puddle. This wasn’t just any hat; it was a gift from my dad, so I really wanted to get it back. But I didn’t exactly fancy diving headfirst into the puddle. So, I looked around and saw a piece of wire. I bent it into the shape of an integral and used it to fetch my hat.”

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## Tesseracts and Foams

Foams are a recent craze in homology theory. I want to explain what a foam is using a tesseract as an example. Specifically, the 2D skeleton of a tesseract is a foam.

We can view a tesseract as a convex hull of 16 points in 4D space with coordinates that are either 0 or 1. The edges connect two vertices with the same three out of four coordinates. Faces are squares with corners being four vertices that all share two out of four coordinates.

Foam definition. A foam is a finite 2-dimensional CW-complex. Each point’s neighborhood must be homeomorphic to one of the three objects below.

1. An open disc. Such points are called regular points.
2. The product of a tripod and an open interval. Such points are called seam points.
3. The cone over the 1D-skeleton of a tetrahedron. Such points are called singular vertices.

My favorite example of a foam is a tesseract. Or, more precisely, the set of tesseract’s vertices, edges, and faces form a foam.

1. The regular points are the insides of the tesseract’s faces. Their neighborhoods are obviously open discs.
2. The seam points are the insides of the tesseract’s edges. Each edge is incident to three faces, and the projection of its neighborhood to a plane perpendicular to the edge is a tripod.
3. The singular vertices are the tesseract’s vertices. Each vertex is incident to four edges and six faces. We can view this neighborhood as a cone cover of a tetrahedron formed by this vertex’s neighbors.

Some of the coolest foams are tricolorable. A foam is called tricolorable if we can color it using three colors in such a way that each face has its own color, and any three faces that meet at a seam are three distinct colors.

Not surprisingly, I chose a tricolorable foam for our example. Let me prove that the tesseract’s 2D skeleton is tricolorable. We start by coloring the edges in four colors depending on the direction: red, green, blue, or gray, as in the first picture (source: Wikipedia). Each face has two pairs of edges of two different colors. We can color the faces in the following manner: if none of the edges are gray, then the face color is the complementary non-gray color (For example, if the edges are red and blue, the face is green). If the edges are gray and one other color, then the face color matches the non-gray color (For example, if the edges are red and gray, the face is red). I leave it to the reader to prove that this coloring means that each edge is the meeting point of three different face colors.

Here is an interesting property of tricolorable foams. It is Proposition 2.2 in the paper Foam Evaluation and Kronheimer-Mrowka Theories, by Mikhail Khovanov and Louis-Hardien Robert.

Proposition. If we remove the regular points of one particular color from a tricolored foam, we will get a closed compact surface containing all the seam points and singular vertices.

In our example, the result is a torus, which you can recognize in the second picture. Here, I use the Schlegel diagram as a model for a tesseract, shown on the right (source: Wikipedia). The exercise for the reader is to explain where the eight green faces of the torus were before they were removed.

The following lemma from the aforementioned paper describes another cool property of a tricolorable foam.

Lemma. If a foam is tricolorable, its 1D skeleton (the graph formed by seams and singular points) is bipartite.

And, surely, I am leaving it up to the reader to check that the tesseract’s 1D skeleton forms a bipartite graph.

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## Three L-tetraminoes

Here is a cool puzzle I heard from Tiago Hirth at the last Gathering for Gardner, who in turn heard it from Donald Bell.

Puzzle. You have three L-tetrominoes. Arrange them on a plane without overlaps so that the resulting shape has a line of symmetry.

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## Why We Need Foreign Languages

Here is an old joke.

A cat is chasing a mouse, and the mouse hides into its little hole. While it’s in there, the mouse hears some barking, “Woof! Woof!” Smart mouse figures a dog scared off the cat, so it peeks out. But guess what? Cat’s still there and catches the mouse, saying, “See, that’s why it pays to know foreign languages!!”

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## Hundred Colors of Math

I recently bought a book by Evdokimov, titled Hundred Colors of Math. The book has lovely math puzzles and cute pictures. The book has answers but doesn’t explain them. Also, the English translation is decent but not perfect. For these two reasons, I am not sure I would recommend the book. However, I do like the puzzles, and here is one of them, called Runaway Cell.

Puzzle. The figure depicted in the picture (a 6-by-6 square, in which the top row is moved by one square) was cut along the grid lines into several identical parts which could be put together to form a 6-by-6 square. The parts are allowed to be turned over. What is the minimal possible number of such parts?

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## Jokes from the Audience

I gave a short talk about my favorite math jokes at G4G15. G4G stands for the Gathering for Gardner, my favorite conference. Here is a joke about Heisenberg from my talk.

* * *

Heisenberg gets pulled over on the highway.
Cop: “Do you know how fast you were going, sir?”
Heisenberg: “No, but I know exactly where I am.”

After my talk, David Albert sent me a sequel to this joke.

* * *

Heisenberg gets pulled over on the highway.
Cop: “Do you know how fast you were going, sir?”
Heisenberg: “No, but I know exactly where I am.”
Cop: “You were going 85 miles per hour”.
Heisenberg: “Oh great—now I’m lost!”

Here is another joke from the conference.

* * *

—Did you hear about the mathematician who’s afraid of negative numbers?
—He’ll stop at nothing to avoid them.

This joke, too, got an awesome sequel from Jesse Lauzon.

* * *

Did you hear about the mathematician who is afraid of negative numbers?
—He’ll stop at nothing to avoid them.
—Well, that’s only natural!

Here is the most recent addition to my collection from my friend, Alexander Karabegov.

* * *

A trigonometry professor lost his voice and had to use sine language.

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