## Saturday, September 21, 2013

### Anti Tic-Tac-Toe

We've all heard of the game tic-tac-toe. It's easy to set up, simple to understand, and fun to play. All you have to do is get three in a row.

There is some mathematics behind correct play of tic-tac-toe, but the real goal is to see where the other player has a chance of forcing a win and make sure that doesn't happen.

However, there is an interesting mathematical fact in the game "anti tic-tac-toe." This is where you have to avoid getting three in a row.

As X, what would be a good starting move in this game? In regular tic-tac-toe, most people start in the center because it is part of 4 of the 8 winning possibilities. So in this instance, it is part of 4 of the 8 losing possibilities. So, I don't think the center would be an appealing move to most players.

Let's start in the left corner square and see what happens.

 X

Since O wants to see X get three in a row, they will get as out of the way as possible.

 X O

Now, where are X's safest moves? Well, if X moves in any of the squares with asterisks, then it would give them two in a row, which would put them closer to a loss. So, X's best move is in the bottom side square.

 X * * * * O * X *

Now, O would probably go in the top side square to keep out of having 2 in a row.

 X O O X

Now, X's best move is in the top right corner, since all other squares would have an asterisk.

 X O X O X

In this instance, the only technically "safe" move for O is in the bottom left corner, but you can see that this move eliminates 3 possible three in a rows for X. Since the bottom right corner is pretty safe too, O's best move would be there.

 X O X O X O

Now, X is forced to go in an unsafe square. In fact, O can force a win wherever X goes. If X goes in the left side square, O will go in the center. If X goes in the center square, O will go in the left side square. If X goes in the bottom left corner square, O can go in either square. There is no way for X to win.

You might notice that in this game, X has a huge disadvantage. O has one less letter to put down, so it is impossible for X to win against a rational player. However, X can force a tie.

Eight of the squares on the board are a guaranteed loss for X. If X moves there on the first turn, O should be able to succeed. However, this ninth square is a square that X can move to and O will not be able to force three in a row.

Which square is it? It is the one that would never be suspected: the center square. Earlier, we said that the center square has so many losing possibilities that nobody would consider it. But, it is actually the right move. Let's look at it.

 X

Currently, there is no specific square that O would have an advantage in. Let's say they went in the top left corner square.

 O X

Then, X would play the bottom right corner square. This would be the symmetrical move.

 O X X

X would continue playing the symmetrical move through the whole game. With any logical player, this would result in a draw.

Though anti tic-tac-toe wouldn't be a popular game to play, there are much more fun games that use symmetric properties to force ties/wins like nim or cram. Here is another game with this type of strategy called napkin chess that I learned on the show Scam School which can also be turned into a fun game with friends: