Tic-tac-toe

Tic-Tac-Toe, also called Noughts and Crosses, is a classic game between two players, "O" and "X", that alternate in marking the spaces in a 3x3 board, trying to put three of their own marks in a horizontal, vertical or diagonal row.

For instance, the following game was won by the first player, "X":

  | |X   O| |X   O| |X   O| |X   O| |X   O| |X   O| |X
 -+-+-   -+-+-   -+-+-   -+-+-   -+-+-   -+-+-   -+-+-
  | |     | |     | |     |O|     |O|     |O|O    |O|O
 -+-+-   -+-+-   -+-+-   -+-+-   -+-+-   -+-+-   -+-+-
  | |     | |    X| |    X| |    X| |X   X| |X   X|X|X

In a normal 3x3 tic-tac-toe game, both players have a strategy to draw the game. In fact, any move by the first player leads to a draw with best play.

Statistically the best opening move is in one of the corners, after this move has been made if the opponent takes any square other than the centre one, then the first player can play in such a way that a win is certain, as shown in the above game.

Variations

There are some variations: 4x4, NxN, 3D tic-tac-toe, 9-board tic-tac-toe and inverse tic-tac-toe. There are several games with an element of being the first to get n-in-a-row: Nine Men's Morris and relatives, Pente, Connect Four (http://www.irt.org/games/js/connect/), and Score Four (http://www.abstractstrategy.com/score-four).

In 3D tic-tac-toe, when played on a 3x3x3 and 4x4x4 board, the first player can force a win.

On a 3x3x3 board it is relatively trivial to win, the first move should be in the centre, the third move should form a link of two in such a way that a blocking move will not allow the opponent to create two in a row, now a forking move will be possible such that on the players next move there will be two places they can go to win. The opponent will be able to block only one of these and thus the first player can take the other and win the game. The same applies for any number of dimensions (i.e 3x3x3x3, 3x3x3x3x3, etc).

The 4x4x4 board game has been solved. Victor Allis[?] showed in 1994 that there are 76 different possible winning lines.

An interesting variant is inverse tic-tac-toe, in which each player tries to force the other to get N in a row.

Another interesting variant is 9 board tic-tac-toe. The nine boards are themselves arranged like a tic-tac-toe board. The first player's move may go on any board; all moves afterwards are placed in the empty spaces on the board corresponding to the square of the previous move(that is, if a move were in the upper-left square of a board, the next move would take place on the upper-left board) - should all nine squares get taken on a board and the last square taken points to the full board(unlikely), the next move may go on any board. Victory is attained by winning 3 in a row on any board. This makes the game considerably longer and more involved, with a definite beginning, middle and endgame.

An interesting extension of this idea is the commercial game Quarto (http://ssel.vub.ac.be/Members/LucGoossens/quarto/quartotext.htm) which is played on a 4x4 board. There are 16 pieces each with four aspects:

• Height: short or tall
• Color: dark or light
• Cross-section: round or square
• Texture: smooth or grooved

Play proceeds in turn by choosing a piece and handing it to the opponent. The opponent plays the piece. If the play results in four pieces which all share the value of one of the attributes, the person who plays the piece wins.

Alternative names

• Noughts and Crosses (Britain)
• Tick tat toe (USA)
• Boter, kaas en eieren (The Netherlands)
• Tripp trapp trull (Sweden)