A completed game of Tic-tac-toe
|Playing time||~1 minute|
|Skill(s) required||Strategy, tactics, observation|
|Synonym(s)||Noughts and crosses |
Xs and Os
Tic-tac-toe (American English), noughts and crosses (Commonwealth English), or Xs and Os, is a paper-and-pencil game for two players, X and O, who take turns marking the spaces in a 3×3 grid. The player who succeeds in placing three of their marks in a horizontal, vertical, or diagonal row is the winner. It is a solved game with a forced draw assuming best play from both players.
In order to win the game, a player must place three of their marks in a horizontal, vertical, or diagonal row.
The following example game is won by the first player, X:
Players soon discover that the best play from both parties leads to a draw. Hence, tic-tac-toe is most often played by young children, who often have not yet discovered the optimal strategy.
Because of the simplicity of tic-tac-toe, it is often used as a pedagogical tool for teaching the concepts of good sportsmanship 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 or 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.If played optimally by both players, the game always ends in a draw, making tic-tac-toe a futile game.
The game can be generalized to an m,n,k-game in which two players alternate placing stones of their own color on an m×n board, with the goal of getting k of their own color in a row. Tic-tac-toe is the (3,3,3)-game.Harary's generalized tic-tac-toe is an even broader generalization of tic-tac-toe. It can also be generalized as a nd game. Tic-tac-toe is the game where n equals 3 and d equals 2. It can be generalised even further by playing on an arbitrary incidence structure, where rows are lines and cells are points. Tic-tac-toe is the game given by the incidence structure shown to the right, consisting of nine points, three horizontal lines, three vertical lines, and two diagonal lines, each line consisting of at least three points.
Games played on three-in-a-row boards can be traced back to ancient Egypt,where such game boards have been found on roofing tiles dating from around 1300 BCE.
An early variation of tic-tac-toe was played in the Roman Empire, around the first century BC. It was called terni lapilli (three pebbles at a time) and instead of having any number of pieces, each player only had three, thus they had to move them around to empty spaces to keep playing.The game's grid markings have been found chalked all over Rome. Another closely related ancient game is three men's morris which is also played on a simple grid and requires three pieces in a row to finish, and Picaria, a game of the Puebloans.
The different names of the game are more recent. The first print reference to "noughts and crosses" (nought being an alternative word for zero), the British name, appeared in 1858, in an issue of Notes and Queries .The first print reference to a game called "tick-tack-toe" occurred in 1884, but referred to "a children's game played on a slate, consisting in trying with the eyes shut to bring the pencil down on one of the numbers of a set, the number hit being scored". "Tic-tac-toe" may also derive from "tick-tack", the name of an old version of backgammon first described in 1558. The US renaming of "noughts and crosses" as "tic-tac-toe" occurred in the 20th century.
In 1952, OXO (or Noughts and Crosses), developed by British computer scientist Sandy Douglas for the EDSAC computer at the University of Cambridge, became one of the first known video games.The computer player could play perfect games of tic-tac-toe against a human opponent.
In 1975, tic-tac-toe was also used by MIT students to demonstrate the computational power of Tinkertoy elements. The Tinkertoy computer, made out of (almost) only Tinkertoys, is able to play tic-tac-toe perfectly.It is currently on display at the Museum of Science, Boston.
When considering only the state of the board, and after taking into account board symmetries (i.e. rotations and reflections), there are only 138 terminal board positions. A combinatorics study of the game shows that when "X" makes the first move every time, the game outcomes are as follows:
A player can play a perfect game of tic-tac-toe (to win or at least draw) if, each time it is their turn to play, they choose the first available move from the following list, as used in Newell and Simon's 1972 tic-tac-toe program.
The first player, who shall be designated "X", has 3 possible strategically distinct 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 (side middle) mark. From a strategical point of view, 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.This might suggest that the corner is the best opening move for X, however another study shows that if the players are not perfect, an opening move in the center is best for X.
The second player, who shall be designated "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.
More detailed, to guarantee a draw, O should adopt the following strategies:
When X plays corner first, and O is not a perfect player, the following may happen:
Consider a board with the nine positions numbered as follows:
When X plays 1 as their opening move, then O should take 5. Then X takes 9 (in this situation, O should not take 3 or 7, O should take 2, 4, 6 or 8):
or 6 (in this situation, O should not take 4 or 7, O should take 2, 3, 8 or 9. In fact, taking 9 is the best move, since a non-perfect player X may take 4, then O can take 7 to win).
In both of these situations (X takes 9 or 6 as second move), X has a 1/ property to win.
If X is not a perfect player, X may take 2 or 3 as second move. Then this game will be a draw, X cannot win.
If X plays 1 opening move, and O is not a perfect player, the following may happen:
Although O takes the only good position (5) as first move, but O takes a bad position as second move:
Although O takes good positions as the first two moves, but O takes a bad position as third move:
O takes a bad position as first move (except of 5, all other positions are bad):
Many board games share the element of trying to be the first to get n-in-a-row, including three men's morris, nine men's morris, pente, gomoku, Qubic, Connect Four, Quarto, Gobblet, Order and Chaos, Toss Across, and Mojo. Tic-tac-toe is an instance of an m,n,k-game, where two players alternate taking turns on an m×n board until one of them gets k in a row. Harary's generalized tic-tac-toe is an even broader generalization. The game can be generalised even further by playing on an arbitrary hypergraph, where rows are hyperedges and cells are vertices.
Other variations of tic-tac-toe include:
One can play on a board of 4x4 squares, winning in several ways. Winning can include: 4 in a straight line, 4 in a diagonal line, 4 in a diamond, or 4 to make a square.
Another variant, Qubic, is played on a 4×4×4 board; it was solved by Oren Patashnik in 1980 (the first player can force a win).Higher dimensional variations are also possible.
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The game has a number of English names.
Sometimes, 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 with each other.
Various game shows have been based on tic-tac-toe and its variants:[ citation needed ]
Three men's morris is an abstract strategy game played on a three by three board that is similar to tic-tac-toe. It is also related to six men's morris and nine men's morris.
Combinatorial game theory has several ways of measuring game complexity. This article describes five of them: state-space complexity, game tree size, decision complexity, game-tree complexity, and computational complexity.
An m,n,k-game is an abstract board game in which two players take turns in placing a stone of their color on an m×n board, the winner being the player who first gets k stones of their own color in a row, horizontally, vertically, or diagonally. Thus, tic-tac-toe is the 3,3,3-game and free-style gomoku is the 15,15,5-game. An m,n,k-game is also called a k-in-a-row game on an m×n board.
3D tic-tac-toe, also known by the trade name Qubic, is an abstract strategy board game, generally for two players. It is similar in concept to traditional tic-tac-toe but is played in a cubical array of cells, usually 4x4x4. Players take turns placing their markers in blank cells in the array. The first player to achieve four of their own markers in a row wins. The winning row can be horizontal, vertical, or diagonal on a single board as in regular tic-tac-toe, or vertically in a column, or a diagonal line through four boards.
In mathematics, the Hales–Jewett theorem is a fundamental combinatorial result of Ramsey theory named after Alfred W. Hales and Robert I. Jewett, concerning the degree to which high-dimensional objects must necessarily exhibit some combinatorial structure; it is impossible for such objects to be "completely random".
In combinatorial game theory, the strategy-stealing argument is a general argument that shows, for many two-player games, that the second player cannot have a guaranteed winning strategy. The strategy-stealing argument applies to any symmetric game in which an extra move can never be a disadvantage.
Toss Across is a game first introduced in 1969 by the now defunct Ideal Toy Company. The game was designed by Marvin Glass and Associates and created by Hank Kramer, Larry Reiner and Walter Moe, and is now distributed by Mattel. It is a game in which participants play tic-tac-toe by lobbing small beanbags at targets in an attempt to change the targets to their desired letter. As in traditional tic-tac-toe, the first player to get three of their letters in a row wins the game. There are other similar games to Toss Across known under different names, such as Tic Tac Throw.
Criss Cross Quiz was a quiz programme that combined the game noughts and crosses with general knowledge questions and aired on the ITV network from 1957 to 1967. It was produced by Granada Television.
Quantum tic-tac-toe is a "quantum generalization" of tic-tac-toe in which the players' moves are "superpositions" of plays in the classical game. The game was invented by Allan Goff of Novatia Labs, who describes it as "a way of introducing quantum physics without mathematics", and offering "a conceptual foundation for understanding the meaning of quantum mechanics".
Harary's generalized tic-tac-toe or animal tic-tac-toe is a generalization of the game tic-tac-toe, defining the game as a race to complete a particular polyomino on a square grid of varying size, rather than being limited to "in a row" constructions. It was devised by Frank Harary in March 1977, and is a broader definition than that of an m,n,k-game.
Tant Fant is a two-player abstract strategy game from India. It is related to tic-tac-toe, but more closely related to three men's morris, Nine Holes, Achi, Shisima, and Dara, because pieces are moved on the board to create the 3 in-a-row. It is an alignment game.
Shisima is a two-player abstract strategy game from Kenya. It is related to tic-tac-toe, and even more so to three men's morris, Nine Holes, Achi, Tant Fant, and Dara, because pieces are moved on the board to create the 3-in-a-row. Unlike those other games, Shisima uses an octagonal board.
Picaria is a two-player abstract strategy game from the Zuni Native American Indians or the Pueblo Indians of the American Southwest. It is related to tic-tac-toe, but more related to three men's morris, Nine Holes, Achi, Tant Fant, and Shisima, because pieces can be moved to create the three-in-a-row. Picaria is an alignment game.
Zillions of Games is a commercial general game playing system developed by Jeff Mallett and Mark Lefler in 1998. The game rules are specified with S-expressions, Zillions rule language. It was designed to handle mostly abstract strategy board games or puzzles. After parsing the rules of the game, the system's artificial intelligence can automatically play one or more players. It treats puzzles as solitaire games and its AI can be used to solve them.
Ultimate tic-tac-toe is a board game composed of nine tic-tac-toe boards arranged in a 3 × 3 grid. Players take turns playing in the smaller tic-tac-toe boards until one of them wins in the larger tic-tac-toe board. Compared to traditional tic-tac-toe, strategy in this game is conceptually more difficult and has proven more challenging for computers.
Notakto is a tic-tac-toe variant, also known as neutral or impartial tic-tac-toe. The game is a combination of the games tic-tac-toe and Nim, played across one or several boards with both of the players playing the same piece. The game ends when all the boards contain a three-in-a-row of Xs, at which point the player to have made the last move loses the game. However, in this game, unlike tic-tac-toe, there will always be a player who wins any game of Notakto.
Number Scrabble is a mathematical game where players take turns to select numbers from 1 to 9 without repeating any numbers previously used, and the first player to amass a personal total of exactly 15 wins the game. The game is isomorphic to tic-tac-toe, as can be seen if the game is mapped onto a magic square.
Wild tic-tac-toe is a game similar to tic-tac-toe. However, in this game players can choose to place either X or O on each move. This game can also be played in its misere form where if a player creates a three-in-a-row of marks, that player loses the game.
Tic-tac-toe is an instance of an m,n,k-game, where two players alternate taking turns on an m×n board until one of them gets k in a row. Harary's generalized tic-tac-toe is an even broader generalization. The game can also be generalized as a nd game. The game can be generalised even further from the above variants by playing on an arbitrary hypergraph where rows are hyperedges and cells are vertices.
The Matchbox Educable Noughts and Crosses Engine or MENACE was an analogue computer made from 304 matchboxes designed and built by Donald Michie in 1961. It was designed to play human opponents in games of noughts and crosses by returning a move for any given state of play and to refine its strategy through reinforcement learning.