Tic-tac-toe

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Tic-tac-toe, also spelled tick tack toe, and alternatively called noughts and crosses, hugs and kisses, and many other names, is a pencil-and-paper game for two players, O and X, who take turns marking the spaces in a 3×3 grid, usually X going first. The player who succeeds in placing three respective marks in a horizontal, vertical or diagonal row wins the game.

The following example game is won by the first player, X:

Game of tic-tac-toe, won by X

Players soon discover that best play from both parties leads to a draw. Hence, tic-tac-toe is most often played by young children; when they have discovered an unbeatable strategy they move on to more sophisticated games such as dots and boxes or 12-cell tic-tac-toe. This reputation for ease has led to casinos offering gamblers the chance to play tic-tac-toe against trained chickens—though the chicken is advised by a computer program.[1]

The first two plies 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 combinatorial 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 up to rotations and reflections (the game tree complexity) on this space.

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

One example of a Tic-Tac-Toe playing computer is the Tinkertoy computer, developed by MIT students, and made out of Tinker Toys[2]. It only plays Tic-Tac-Toe and has never lost a game. It is currently on display at the Museum of Science, Boston.

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[edit] Number of possible games

Despite its apparent simplicity, it requires some complex mathematics to determine the number of possible games. This is further complicated by the definitions used when setting the conditions.

Simplistically, there are 362,880 (ie. 9!) ways of placing Xs and Os on the board, without regard to winning combinations.

When winning combinations are considered, there are 255,168 possible games. Assuming that X makes the first move every time:

  • 131,184 finished games are won by (X)
  • 1,440 are won by (X) after 5 moves
  • 47,952 are won by (X) after 7 moves
  • 81,792 are won by (X) after 9 moves
  • 77,904 finished games are won by (O)
  • 5,328 are won by (O) after 6 moves
  • 72,576 are won by (O) after 8 moves
  • 46,080 finished games are drawn

Ignoring the sequence of Xs and Os, and after eliminating symmetrical outcomes (ie. rotations and/or reflections of other outcomes), there are only 138 unique outcomes. Assuming once again that X makes the first move every time:

  • 91 unique outcomes are won by (X)
  • 21 won by (X) after 5 moves
  • 58 won by (X) after 7 moves
  • 12 won by (X) after 9 moves
  • 44 unique outcomes are won by (O)
  • 21 won by (O) after 6 moves
  • 23 won by (O) after 8 moves
  • 3 unique outcomes are drawn

[edit] Strategy

A player can play perfect tic-tac-toe if they choose the move with the highest priority in the following table[3].

  1. Win: If you have two in a row, play the third to get three in a row.
  2. Block: If the opponent has two in a row, play the third to block them.
  3. Fork: Create an opportunity where you can win in two ways.
  4. Block Opponent's Fork:
    Option 1: Create two in a row to force the opponent into defending, as long as it doesn't result in them creating a fork or winning. For example, if "X" has a corner, "O" has the center, and "X" has the opposite corner as well, "O" must not play a corner in order to win. (Playing a corner in this scenario creates a fork for "X" to win.)
    Option 2: If there is a configuration where the opponent can fork, block that fork.
  5. Center: Play the center.
  6. Opposite Corner: If the opponent is in the corner, play the opposite corner.
  7. Empty Corner: Play an empty corner.
  8. Empty Side: Play an empty side.

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, however playing the corner gives the opponent the smallest choice of squares which must be played to avoid losing[4].

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.

[edit] Variations

Many board 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, Qubic, 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 if 2 people are playing. Another variant is played on a 4×4×4 board, 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 or to play with 4 people and get 1 row of 3.
  • In misère tic-tac-toe you win if the other player gets n in a row. The 3×3 game is a draw.
  • Tic Tac Tactic: A new game played on a three-dimensional board. Each player takes it in turns to send their ball at least half way round the curved board until it drops down into one of the 9 indents of the 3 x 3 grid. The player wins balls by forming a row of 3, and they can, using a rubber device, change the outcome of their ball's path and bounce their ball to where they want. Each 3-in-a-row wins a ball of the player. The winner is skilful enough to have won five balls off his opponent.
  • 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.
  • In Tic-Tac-Chess, players play a game of chess and tic-tac-toe simultaneously. When a player captures an opponent's piece, the player can make a play on the tic-tac-toe board regardless if the other player has not yet made a play. The first person to get 3 X's or O's in a row wins the game. This makes for a much more defensive game of chess.
  • There is a game that is isomorphic to tic-tac-toe, but on the surface appears completely different. Two players in turn say a number between one and nine. A particular number may not be repeated. The game is won by the player who has said three numbers whose sum is 15. Plotting these numbers on a 3×3 magic square shows that the game exactly corresponds with tic-tac-toe, since three numbers will be arranged in a straight line if and only if they total 15.
  • Two players fill out a 3×3 grid with numbers one through nine in order of priority. They then compare their grids and play tic-tac-toe by filling in the squares by the priority they listed before.
  • In the 1970s, there was a two player game made by Tri-ang Toys & Games called Check Lines, in which the board consisted of eleven holes arranged in a geometrical pattern of twelve straight lines each containing three of the holes. Each player had exactly five tokens and played in turn placing one token in any of the holes. The winner was the first player whose tokens were arranged in two lines of three (which by definition were intersecting lines). If neither player had won by the tenth turn, subsequent turns consisted of moving one of one's own tokens to the remaining empty hole, with the constraint that this move could only be from an adjacent hole.
  • Toss Across is a tic-tac-toe game where players throw bean bags at a large board to mark squares.
  • Various game shows have been based around the game:
    • On Hollywood Squares nine celebrities filled the cells of the tic-tac-toe grid.
    • In Secret "X", a pricing game on The Price is Right, contestants must get three Xs in a row by correctly pricing items, and then find the one X in the middle column to complete a line.
    • In Tic-Tac-Dough players put symbols up on the board by answering questions in various categories.
    • In Beat the Teacher contestants answer questions to win a turn to influence a tic-tac-toe grid.
  • The object of the fictional D'ni game of Gemedet is to get six balls in-a-row in a 9×9×9 cube grid.
  • The object of the fictional game Squid-Tac-Toad is to get four (or five) pieces in-a-row on a 4×4 or 5×5 checkerboard grid.
  • Some children play where getting a Y formation also counts as a win.
  • Quantum tic tac toe allows players to place a quantum superposition of numbers on the board
  • Another variation on tic-tac-toe is played on a larger grid (say 10x10) where the object is to get 5 in a row. The increased amount of space creates a greater complexity.
  • Another variation on tic-tac-toe that is popular in Vietnam that the player has to get 5 in a row to win the game. Each player takes turn to mark "x" and "o" on the board. The stategy is to not only blocking the opponent, but creating chances for yourself to form 5 in a row in any direction. The board is unlimited and has no boundary until one wins. See Go-moku
  • The game can also be varied by limiting the number of pieces and then allowing movement. The three-a-side then becomes Three Men's Morris (see Nine Men's Morris).

[edit] Alternative names

The game has a number of alternative English 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 after a certain number have been placed) are confused.

[edit] In fiction

In the 1983 film WarGames, tic-tac-toe is used as an allegory for nuclear war. In the climax of the film, the protagonist prevents an out of control military defense computer from launching nuclear missiles by making it repeatedly play tic-tac-toe against itself. After quickly learning that good strategy by both players produces no winner, the computer then plays through all known nuclear strike scenarios, again finding no winner. The computer concludes that "The only winning move is not to play."

[edit] References

  1. ^ "Columnist Susan Snyder: Defeat a chicken? Good cluck". http://www.lasvegassun.com/news/2002/jun/21/columnist-susan-snyder-defeat-a-chicken-good-cluck. Retrieved on 2008-06-09. 
  2. ^ "Tinkertoys and tic-tac-toe". http://www.rci.rutgers.edu/~cfs/472_html/Intro/TinkertoyComputer/TinkerToy.html. Retrieved on 2007-09-27. 
  3. ^ Kevin Crowley, Robert S. Siegler (1993). "Flexible Strategy Use in Young Children’s Tic-Tat-Toe". Cognitive Science 17: 531–561. doi:10.1016/0364-0213(93)90003-Q. 
  4. ^ Martin Gardner (1988). Hexaflexagons and Other Mathematical Diversions. University of Chicago Press. 

[edit] External links

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