# Tic, Tac, and Toe

Who moved first?

Welcome to this week’s Math Munch!  We’re taking a look at several Tic-Toe-Toe related items.

To the right you can see a little Tic-Tac-Toe puzzle I found here.  If the board below shows a real game of Tic-Tac-Toe, then which player moved first?  Think. Think!!

Now let’s talk about the basic game itself.  Tic-Tac-Toe is fun for new players, but at some point, we can all get really good at it.  How good? Well, there’s a strategy, which if you follow without making mistakes, you will never lose!  Amazing, right?  So what’s the strategy?  The picture below shows half of it.  Here’s how to play if you’re X and get to move first. (instructions below.)

### Strategy for X (1st player)

Randall Munroe

“Your move is given by the position of the largest red symbol on the grid. When your opponent picks a move, zoom in on the region of the grid where they went. Repeat.”  Now find a friend and try it out!

This image comes from xkcd, a sometimes mathematical webcomic by Randall Munroe.  (We featured his Sierpinski Heart last Valentine’s Day.) Randall talks about his Tic-Tac-Toe strategy guide and several other mathy comics in this interview with Math Horizons Magazine, which is certainly worth a read.

The undefeated Tic-Tac-Toe player, a Tinkertoy computer

The existence of strategies like the one above mean that a computer can be perfect at Tic-Tac-Toe.  In fact, in Boston’s Museum of Science, there is a computer made entirely of Tinkertoy (a construction system for kids like LEGO) that has never lost a game of tic-tac-toe. It was designed and built by a team of college students in the 1980’s. For more on this impeccable toy computer, read this article by computer scientist A.K. Dewdney.

Finally, I stumbled across a wonderful Tic-Tac-Toe variation game, sometimes called “Ultimate Tic-Tac-Toe,” but here called TicTacToe10.  Here’s a video explaining, but basically in this version, you have a Tic-Tac-Toe board of Tic-Tac-Toe boards.  That is, you have the 9 little boards, and the one big board that they make together. On your turn you make a move on one of the small boards.  Where you decide to go decides which of the nine small boards the next player gets to play in.  If you win a small board, it counts as your shape on the big board.  Crazy, right!?!?  If that’s confusing you’ll have to watch the video tutorial or just start playing.

Here’s a link to a 2-player version of Ultimate Tic-Tac-Toe so that you can play with a friend, although you could also do it on paper, you just have to remember where the last move was.

I hope you found something tasty this week.  Bon appetit!

# Partial Cubes, Open Cubes, and Spidrons

Welcome to this week’s Math Munch!

Recently the videos that Paul and I made about the Yoshimoto Cube got shared around a bit on the web. That got me to thinking again about splitting cubes apart, because the Yoshimoto Cube is made up of two pieces that are each half of a cube.

A part of Wall Drawing #601
by Sol LeWitt

A friend of mine once shared with me some drawings of cubes by the artist Sol LeWitt. The cubes were drawn as solid objects, but parts of them were cut away and removed. It was fun trying to figure out what fraction of a cube remained.

On the web, I found a beautiful image that Sol made called Wall Drawing #601. In the clipping of it to the left, I see 7/8 of a cube and 3/4 of a cube. Do you? You can view the whole of this piece by Sol on the website of the Greater Des Moines Public Art Foundation.

The Cube Vinco by Vaclav Obsivac.

There are other kinds of objects that break a cube into pieces in this way, like this tricky puzzle by Vaclav Obsivac and this “shaved” Rubik’s cube modification. Maybe you’ll design a cube dissection of your own!

As I further researched Sol LeWitt’s art, I found that he had investigated partial cubes in other ways, too. My favorite of Sol’s tinkerings is the sculpture installation called “Variations of Incomplete Cubes“. You can check out this piece of artwork on the SFMOMA site, as well as in the video below.

In the video, a diagram appears that Sol made of all of the incomplete open cubes. He carefully listed out and arranged these pictures to make sure that he had found them all—a very mathematical task. It reminds me of the list of rectangle subdivisions I wrote about in this post.

Sol’s diagram got me to thinking and making: what other shapes might have interesting “incomplete open” variations? I started working on tetrahedra. I think I might try to find and make them all. How about you?

Two open tetrahedra I made. Can you find some more?

Finally, as I browsed Google Images for “half cube”, one image in particular jumped out at me.

What are those?!?!

Dániel’s original spidron from 1979

These lovely rose-shaped objects are called spidrons—or more precisely, they appear to be half-cubes built out of fold-up spidrons. What are spidrons? I had never heard of them, but there’s one pictured to the right and they have their own Wikipedia article.

The first person who modeled a spidron was Dániel Erdély, a Hungarian designer and artist. Dániel started to work with spidrons as a part of a homework assignment from Ernő Rubik—that’s right, the man who invented the Rubik’s cube.

 A cube with spidron faces. Two halves of an icosahedron.

A hornflake.

Here are two how-to videos that can help you to make a 3D spidron—the first step to making lovely shapes like those pictured above. The first video shows how to get set up with a template, and the second is brought to you by Dániel himself! Watching these folded spidrons spiral and spring is amazing. There’s more to see and read about spidrons in this Science News article and on Dániel’s website.

And how about a sphidron? Or a hornflake—perhaps a cousin to the flowsnake? So many cool shapes!

To my delight, I found that Dániel has created a video called Yoshimoto Spidronised—bringing my cube splitting adventure back around full circle. You’ll find it below. Bon appetit!

Reflection Sheet – Partial Cubes, Open Cubes, and Spidrons

# Polyominoes, Clock Calculator, and Nine Bells

Welcome to this week’s Math Munch!

The first thing I have to share with you comes with a story. One day several years ago, I discovered these cool little shapes made of five squares. Maybe you’ve seen these guys before, but I’d never thought about how many different shapes I could make out of five squares. I was trying to decide if I had all the possible shapes made with five squares and what to call them, when along came Justin. He said, “Oh yeah, pentominoes. There’s so much stuff about those.”

Justin proceeded to show me that I wasn’t alone in discovering pentominoes – or any of their cousins, the polyominoes, made of any number of squares. I spent four happy years learning lots of things about polyominoes. Until one day… one of my students asked an unexpected question. Why squares? What if we used triangles? Or hexagons?

We drew what we called polyhexes (using hexagons) and polygles (using triangles). We were so excited about our discoveries! But were we alone in discovering them? I thought so, until…

A square made with all polyominoes up to heptominoes (seven), involving as many internal squares as possible.

… I found the Poly Pages. This is the polyform site to end all polyform sites. You’ll find information about all kinds of polyforms — whether it be a run-of-the-mill polyomino or an exotic polybolo — on this site. Want to know how many polyominoes have a perimeter of 14? You can find the answer here. Were you wondering if polyominoes made from half-squares are interesting? Read all about polyares.

I’m so excited to have found this site. Even though I have to share credit for my discovery with other people, now I can use my new knowledge to ask even more interesting questions.

Next up, check out this clock arithmetic calculator. This calculator does addition, subtraction, multiplication, and division, and even more exotic things like square roots, on a clock.

What does that mean? Well, a clock only uses the whole numbers 1 through 12. Saying “15 o’clock” doesn’t make a lot of sense (unless you use military time) – but you can figure out what time “15 o’clock” is by determining how much more 15 is than 12. 15 o’clock is 3 hours after 12 – so 15 o’clock is actually 3 o’clock. You can use a similar process to figure out the value of any positive or negative counting number on a 12 clock, or on a clock of any size. This process (called modular arithmetic) can get a bit time consuming (pun time!) – so, give this clock calculator a try!

Finally, here is some wonderful mathemusic by composer Tom Johnson. Tom writes music with underlying mathematics. In this piece (which is almost a dance as well as a piece of music), Tom explores the possible paths between nine bells, hung in a three-by-three square. I think this is an example of mathematical art at its best – it’s interesting both mathematically and artistically. Observe him traveling all of the different paths while listening to the way he uses rhythm and pauses between the phrases to shape the music. Enjoy!

Bon appetit!

# Andrew Hoyer, Cameron Browne, & Sphere Inversion

Welcome to this week’s Math Munch! Fractals, origami, math art, games, and a mind-bending video are all ahead, so let’s get into it.

Andrew Hoyer

First up, let’s take a look at the work of Andrew Hoyer.  According to his website, he’s a “software engineer in his mid-twenties living it up in sometimes sunny San Francisco.” I came across his work when I found his beautiful and completely engaging introduction to simple fractals.  (Go on! Click. Then read, experiment and play!)

A Cantor set

At the bottom of that page, Andrew links to a wonderful, long list of fractals, arranged by Hausdorff dimension, which is a way of measuring fractals as being something like 2.5 dimensions.  A line is 1 dimensional.  A plane is 2D, and you can find many fractals with dimension in between!!  Weird, right?

I was also really pleased to find Andrew’s Instagram feed, which features some of his beautiful origami creations.  Andrew’s agreed to answer your questions for an upcoming Q&A, so ask away!

Cameron Browne

Up next, meet Cameron Browne. He’s an accomplished researcher who designs and studies games. Take a look at the many many games Cameron has created. The rules and descriptions are there, and Cameron sent along links to playable versions of a few, which you can find by clicking the pictures below.  For the third one, you’ll need to search for “Margo” or “Spargo.” For his research, Cameron investigates the possibilities of artificial intelligence, and how a computer can be used to generate games and puzzles.

 Akrondescription Yavalathdescription Margo and Spargodescriptiondescription

Cameron is also an artist, and he has a page full of his graphic designs.  I found Cameron through his page of Truchet curves.  I love the way his pages are full of diagrams and just enough information to start making sense of things, even if it’s not perfectly clear.  Cameron also has MANY pages of wonderful fractal-ish graphics: Impossible Fractals, Cantor Knots, Fractal Board Games, Woven Horns, Efficient Trees, and on and on…  And he has agreed to do a Q&A with us, so please, submit a question. What are you wondering?

 A Cantor Knot A Truchet curve “Mona Lisa” An “impossible” fractal

And, as if that wasn’t enough mathy awesomeness, check out this video about turning a sphere inside out.  A bit of personal history, I actually used this video  (though it was only on VHS back then, checked out from the library) as part of the research for my independent research project during my senior year of college.  It gets pretty tricky, but if you watch it all the way through it starts to make some sense.

Have a great week.  Bon appetit!

Reflection sheet – Andrew Hoyer, Cameron Browne, & Sphere Inversion