Tic-tac-toe

Tic-tac-toe (also spelled tictactoe, ticktacktoe, or tick-tack-toe) is one of the world's simplest games. One could wonder how there could be any mathematical interest in it whatsoever. After all, everyone who is old enough knows that, with best play, neither player can force a win in standard tic-tac-toe (i.e. on a 3x3 grid).

However, things change when we look at altering the board or the winning conditions. For example, the first player has an easy win with three-in-a-row tic-tac-toe on a 4x4 (or larger) board. Here are some interesting variations of tic-tac-toe that you may want to try out. Think about whether one player or another has a winning strategy, a plan they can follow that guarantees that they will win every time:

Your Choice Tic-Tac-Toe

Each player may put down either an X or an O on each of their turns, and may change their mind from turn to turn. The winner is the one who finishes any row, column, or diagonal of all X's or all O's.

Magic Square Tic-Tac-Toe

Instead of X's and O's, the numbers 1 through 9 are used. Each number may be used only once. The winner is the one to get the numbers in any row, column, or diagonal to add up to 15. Question: Can you determine a winning strategy for this variant?

Last One Wins

On each player's turn, that player marks as many non-empty spaces as they like, as long as they are in the same row or column (not diagonal). Whoever fills in the last space wins.

Avoidance Tic-Tac-Toe

Same order of play, only the goal is to avoid getting three in a row.

Movable Markers

Players have three counters each and take turns placing them on the grid. If neither has won after all six counters are down, players may move one of their counters somewhere else as their move.

Hot

Each of the words HOT, HEAR, TIED, FORM, WASP, BRIM, TANK, SHIP, and WOES is printed on a card. Players take turns withdrawing cards from the pile. The first to hold three cards containing the same letter wins. An alternate version of this game uses the cards FISH, SOUP, HORN, KNIT, VOTE, ARMY, CHAT, SWAN, and GIRL. Question: Why is this listed under tic-tac-toe variants? Think about it.

Multi-Player Tic-Tac-Toe

For three to six people, try a giant board, one that is at least 10x10, perhaps infinite. Four in a row wins this variant.

3-D Tic-Tac-Toe

Play on a 4x4x4 board. There are a lot of computer games of this version of tic-tac-toe.

Martin Gardner once wrote a column (see the bibliography) on "Generalized Ticktacktoe". The generalization is as follows: Choose a polyomino (called "animals" by Frank Harary, who devised this generalization) and declare its formation to be the objective of the tic-tac-toe game. Each player tries to fill in cells that will form the desired animal. Rotations and reflections are okay.

An interesting idea is to look at each of the polyominoes and find two properties of that polyomino: The length of the side of the smallest square on which the first player can force a win (b), and the number of moves required on this board (m). Here are some "animals" of 1 to 4 cells (aside: polyominoes of four cells are called tetrominoes, and should be familiar to anyone who as played Tetris), their "names", and their b and m values.

Note that the 2x2 square, called "Fatty", is a "loser". That is, the first player can never force Fatty on a board of any size. Note that any polyomino containing any smaller loser is also a loser. This makes sense, since if you can never force a 2x2 square, for example, you can't force (say) a 3x3 square either, since to construct a 3x3 square you have to create a 2x2 square. As a matter of fact, most polyominoes of size 5 or greater are losers. There are only three pentominoes and (probably) one hexomino that are winners:

Since almost every heptomino (size 7) or larger will contain at least two different hexominoes, and there is only one winning hexomino, it is not too hard to show that all 107 order-7 animals contain a smaller loser and thus are losers themselves.

Another question: Is there ever a winning strategy for the second player? This is a pretty easy question to answer. Assume that, for some shape, the second player does have a strategy. Then the first player could win by starting with some irrelevant move and thereafter following the second player's winning strategy. Making an extra move is never a liability in tic-tac-toe, even generalized tic-tac-toe. We have reached a contradiction, so our assumption that the second player can ever have a winning strategy is false. The moral of the story is: go first!

One last tic-tac-toe item: If you play a normal game of tic-tac-toe with someone else and let him/her go first, what is the probability that s/he will start with an X? It is certainly greater than 50%. As a matter of fact, it's probably close to 100%. Interesting, eh?

In contrast, I have seen a tic-tac-toe set in the toy section of a department store that contained five O's and four X's, which would require O to go first. Maybe at least one toy designer thinks differently than most people...