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!

The Sierpinski Valentine, Cardioids, and Möbius Hearts

Welcome to this week’s Math Munch!

With Valentine’s Day this Thursday, let’s take a look at some mathy Valentine stuff. If you’re searching for the perfect card design for your valentine, search no more. Math Munch has you covered!

Sierpinski Valentine

xkcd creator Randall Munroe

Above you can see a clever twist on the classic Sierpinski Triangle, which I found on xkcd, a wonderfully mathematical webcomic. You can read more about xkcd creator Randall Munroe in this interview from the Sept. 2012 issue of Math Horizons. (pdf version)

Ron Doerfler designed another math-insprired Valentine’s Day card, which you can check out here. The image to the left is only part of it. Don’t get it? Well it’s a reference to a mathematical curve called the cardioid (from the Greek word for “heart”). Look what happens if you follow a point on one circle as it rolls around another. You’ll have to imagine it tipped the other way so it’s oriented like a typical heart, but that curve is a cardioid. The second animation was created by the amazing and previously featured Matt Henderson. If you have a compass, then you can make the second one at home.

 A cardioid generated by one circle rolling around another A completely different way to generate a cardioid

Pop-up Sierpinski Heart Card

Really though, nothing says “I Love you” like a Möbius strip. Am I right? Here’s a quick little project you can do to make a pair of linked Möbius hearts. You can find directions here on a blog called 360, or you can watch the video below. Oh, and as if that wasn’t enough great stuff, here’s one more project from the 360 blog, a pop-up version of the Sierpinski Heart!

Happy Valentine’s Day, and bon appetit!