Welcome to this week’s Math Munch!

Here’s a little game with a big name: tsoro yematatu. If you enjoyed Paul’s recent post about tic-tac-toe, I think you’ll like tsoro yematatu a lot.

I ran across this game on a website called Behind the Glass. The site is run by the Cincinnati Art Museum. (What is it with me and art museums lately?) The museum uses Behind the Glass to curate many pieces of African art and culture, including four mathematical games that are played in Africa.

The simplest of these is tsoro yematatu. It has its origin in Zimbabwe. Like tic-tac-toe, the goal is to get three of your pieces in a row, but the board is “pinched” and you can move your pieces. Here’s an applet where you can play a modified version of the game against a computer opponent. While the game still feels similar to tic-tac-toe, there are brand-new elements of strategy.

Tsoro yematatu reminds me of one that I played as a kid called Nine Men’s Morris. I learned about it and many other games—including go—from a delightful book called *The Book of Classic Board Games*. Kat Mangione—a teacher, mom, and game-lover who lives in Tennessee—has compiled a wonderful collection of in-a-row games. And wouldn’t you know, she includes Nine Men’s Morris, tsoro yematatu, tic-tac-toe, and dara—another of the African games from Behind the Glass.

The board for tsoro yematatu also reminds me of the Fano plane. This mathematical object is very symmetric—even more than meets the eye. Notice that each point is on three lines and that each line passes through three points. The Fano plane is one of many projective planes—mathematical objects that are “pinched” in the sense that they have vanishing points. They are close cousins of perspective drawings, which you can check out in these videos.

Can you invent a game that can be played on the Fano plane?

Closely related to the Fano plane is an object called the Klein quartic. They have the same symmetries—168 of them. Felix Klein discovered not only the Klein quartic and the famous Klein bottle, but also the gorgeous Kleinian groups and the Beltrami-Klein model. He’s one of my biggest mathematical heroes.

This article about the Klein quartic by mathematician John Baez contains some wonderful images. The math gets plenty tough as the article goes on, but in a thoughtfully-written article there is something for everyone. One good way to learn about new mathematics is to read as far as you can into a piece of writing and then to do a little research on the part where you get stuck.

If you’ve enjoyed the animation of the Klein quartic, then I bet my last find this week will be up your alley, too. It’s a Tumblr by David Whyte and Brian Fitzpatrick called Bees & Bombs. David and Brian create some fantastic GIFs that can expand your mathematical imagination.

This one is called Pass ‘Em On. I find it entrancing—there’s so much to see. You can follow individual dots, or hexagons, or triangles. What do you see?

This one is called Blue Tiles. It makes me wonder what kind of game could be played on a shape-shifting checkerboard. It also reminds me of parquet deformations.

A few of my other favorites are Spacedots and Dancing Squares. Some of David and Brian’s animations are interactive, like Pointers. They have even made some GIFs that are inspired by Tilman Zitzmann’s work over at Geometry Daily (previously).

I hope you enjoy checking out all of these new variations on some familiar mathematical objects. Bon appetit!

On the Pass ’em On board I saw multiple hexagons that after rotating once fed one of its points to the next hexagon and so forth. This was interesting because I noticed that the one point that is passed on will always stay on one of the three hexagons near it because the point will be pushed down, then to the side, then back up to its original hexagon.

Hi Matt,

Great job tracking that pattern. It’s really tricky–it sure looks like there’s more shuffling around–but you’re spot on with your description. Hope you enjoyed the GIFs!

Justin

I also find the GIF’s entrancing as well. For the Blue Tiles GIF I can see it as pairs of blue squares in a checkered formation (diagonally across from each other). They rotate around the vertex at which they meet. It was confusing at first, but now I can only see it this way.

Both of the GIF’s were really cool. I found the Pass ’em On confusing at first, but once I started following one of the smaller red dots, I saw that the dot travels between three of the hexagons- always the same ones, in the same order. I found this very interesting. On the Blue Tiles one, it was cool because I saw that after a square was made, it split into 2. One of those pieces would go with another to make a square. Also, when I followed the one of the pieces that went to the left, I saw that it went up and to the left. The opposite goes when you follow a piece that goes to the right. Those go down and to the right.

Hi Alisha,

Pass ’em On sure looks more complicated that it actually is, doesn’t it? I appreciate your keen observations of these GIFs. I hope you had the chance to explore some of Bees & Bombs’s site.

Bon appetit!

Justin

I loved this game because it is like a tic tac toe but in a better, and more complex way. I like this one better than the regular tic tac toe because in tic tac toe, you could go for as many games without anyone winning, whereas Tsoro Yematatu you cant quit the game in a tie, one person will win. Also I like how you can move your pieces unlike tic tac toe. I cant wait untill your guys next post. Thank so much! 😀

Hi Daniel!

I agree that Tsoro Yematatu is a really interesting game. Unlike tic-tac-toe, I definitely haven’t figured it out yet.

Do you think there could be a tie in this game? Maybe there’s an arrangement of pieces where the two players would just shuffle their pieces back and forth, avoiding losing but never winning…

Also, here are some Tsoro Yematatu puzzle that I made up recently. Try them out and let me know what you think!

Click to access mm-reflection-tsoro-yematatu-fanos-plane-and-gifs.pdf

Justin

Bees and bombs is very interesting… If you look at it from your chair to your desk it resembeles a red dot circling a black one. But if you look closer It resembeles a red dot circling not one but 3 black dots.

Hi Takafumi,

I see what you mean. Yes, the way the black dots move is very tricky. I wonder if other motion patterns would also look tricky?

Thanks for the food for thought,

Justin

The rotating dots is very interesting. I saw many things. But the one thing that i thought was really weird is that i saw a flower made out hexagons. I know that`s unusual but i thought that was really cool.

Hi Arianna!

I’m glad you enjoyed the GIFs. Your flower idea sounds interesting. Maybe you’ll do a drawing, and share it in our new Readers’ Gallery?

Justin

I have looked at the red dots circling the black dots before and I couldn’t really find a certain pattern but then I looked at it again and watched just a single red dot and I noticed that if you continue to follow the same dot that the it rotates around the same 3 black dots again and again

The Bees and Bombs patterns were very cool. I thought the “pass em on” pattern was interesting. I followed one red dot, and found it goes into multiple hexagons and creates the illusion. Everywhere you look there’s hexagons! Were the black dots supposed to represent the center of the red hexagons, or are they there just to look as hexagons also?

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I thought the pattern of the board was very different from the regular tic tac toe and I thought it was just a bit easier

That’s fair, Desirae. I like to think of it as a “pinched” tic-tac-toe board, but I admit that tsoro yematatu has a different feel to it than tic-tac-toe. Thanks for reading, playing, and commenting!