Tic-tac-toe

From Academic Kids

Tic-tac-toe, also called noughts and crosses and many other names, is a paper and pencil game between two players, O and X, who alternate in marking the spaces in a 3×3 board. A player wins by getting three of their own marks in a horizontal, vertical or diagonal row.

This game is won by the first player, X:

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Tic-tac-toe-game-1.png
Game of tic-tac-toe, won by X

This game is drawn:

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Tic-tac-toe-game-2.png
Game of tic-tac-toe, drawn

Players soon discover that best play leads to a draw, regardless of where the first player plays. So tic-tac-toe is most often played by very young children; when they have discovered an unbeatable strategy they move on to more sophisticated games such as dots and boxes. This reputation for ease has led to Las Vegas casinos offering gamblers the chance to play tic-tac-toe against trained chickens.

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Tic-tac-toe-game-tree.png
The first two ply of the game tree for tic-tac-toe.

The simplicity of tic-tac-toe makes it ideal as a pedagogical tool for teaching the concepts of game theory and the branch of artificial intelligence that deals with the searching of game trees. It is straightforward to write a computer program to play tic-tac-toe perfectly, to enumerate the 765 essentially different positions (the state space complexity), or the 26,830 possible games (the game tree complexity) on this space. Ignoring symmetry, there are 255,168 possible games.

The first known computer game, OXO (or Noughts and Crosses, 1952) for the EDSAC computer played perfect games of tic-tac-toe against a human opponent.

Contents

Strategy

To win or avoid losing in tic-tac-toe requires that the player consistently perform as many of the following actions as possible with each mark--listed in order of priority--without sacrificing the highest possible priority:

  1. Complete three in a row.
  2. Block their opponent from completing three in a row.
  3. Threaten a win with two possible completions in two rows.
  4. Avoid a configuration in which the opponent can force the win.
  5. Threaten a win with a possible completion (two in a row).
  6. Prevent the opponent from getting two in a row.

To be successful, the player must seek to complete as many of these objectives as possible with a single mark, without sacrificing higher priorities. The player must also think ahead to see whether a mark can be made this turn that will allow them to achieve a higher priority in the next turn.

The game is won or drawn after the first two marks are made, assuming perfect play for the remainder of the game. It is therefore very important for the serious tic-tac-toe player to study these openings (of which there are 12) in order to avoid making a mark that enables the opponent to force a win, or to recognize marks that can be capitalized upon in order to force the win.

The first player, whom we shall designate "X," has 3 possible positions to mark during the first turn. Superficially, it might seem that there are 9 possible positions, corresponding to the 9 squares in the grid. However, by rotating the board, we will find that in the first turn, every corner mark is strategically equivalent to every other corner mark. The same is true of every edge mark. For strategy purposes, there are therefore only three possible first marks: corner, edge, or center. Player X can win or force a draw from any of these starting marks. The choice of which to make will depend on the player's knowledge of their opponent's weaknesses in recognizing good answers to a particular opening. In a series of games, alternating the opening mark and its superficial position can help a player win more often against a weaker player.

The second player, whom we shall designate "O," must respond to X's opening mark in such a way as to avoid the forced win. Player O must always respond to a corner opening with a center mark, and to a center opening with a corner mark. An edge opening must be answered either with a center mark, a corner mark next to the X, or an edge mark opposite the X. Any other responses will allow X to force the win. Once the opening is completed, O's task is to follow the above list of priorities in order to force the draw, or else to gain a win if X makes a weak play.

Variations

Many games share the element of trying to be the first to get n-in-a-row: three men's morris, nine men's morris, pente, gomoku, Connect Four, Quarto, Gobblet. The m,n,k-games are a family of generalized games based on tic-tac-toe.

  • 3-dimensional tic-tac-toe on a 3×3×3 board, though the first player has an easy win by playing in the centre. Another variant is played on a 4×4×4 board (called Qubic) though it was solved by Victor Allis in 1994 (the first player can force a win). A more complex variant can be played on boards utilising higher dimensional space, most commonly 4 dimensions in a 3×3×3×3 board. In such games the aim is to fill up the board and get more rows of three in total than the other player.
  • In misre tic-tac-toe you win if the other player gets n in a row. The 3×3 game is a draw.
  • In nine board tic-tac-toe nine tic-tac-toe boards are themselves arranged in a 3×3 grid. 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). If a player can't move because the indicated board is full, the next move may go on any board. Victory is attained by getting 3 in a row on any board. This makes the game considerably longer and more involved than tic-tac-toe, with a definite opening, middle game and endgame.
  • There is a game which is isomorphic to tic-tac-toe, but on the surface appears completely different. Players take it in turn to say a number between one and nine. A particular number may not be repeated. Both players aim to say three numbers which add up to 15. Plotting these numbers on a 3×3 magic square will reveal the exact correspondence with the game of tic-tac-toe, given that three numbers will be arranged in a straight line if and only if they add up to 15.

Alternative names

Sometimes, the names of the games Tic-tac-toe (where players keep adding "pieces") and Three Men's Morris (where pieces start to move when the first four have been placed) are confused.

External links

da:Kryds og bolle de:Tic Tac Toe it:Gioco del tris he:איקס מיקס דריקס nl:Boter, kaas en eieren ja:三目並べ pl:Kółko i krzyżyk pt:Jogo da velha sv:Luffarschack

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