Tic-tac-toe

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Tic-tac-toe
Tic Tac Toe.gif
A game of Tic-tac-toe
Genre(s)Paper-and-pencil game
Players2
Setup timeMinimal
Playing time~1 minute
Random chanceNone
Skill(s) requiredStrategy, tactics, observation
Synonym(s)Noughts and crosses
Xs and Os
 
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"Noughts and Crosses" redirects here. For the series by Malorie Blackman, see Noughts & Crosses series.
For the Ian Rankin novel, see Knots and Crosses.
"Tic tac toe" redirects here. For the band, see Tic Tac Toe (band).
Tic-tac-toe
Tic Tac Toe.gif
A game of Tic-tac-toe
Genre(s)Paper-and-pencil game
Players2
Setup timeMinimal
Playing time~1 minute
Random chanceNone
Skill(s) requiredStrategy, tactics, observation
Synonym(s)Noughts and crosses
Xs and Os

Tic-tac-toe (or Noughts and crosses, 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 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 (often referred to as cat or cat's game). Hence, Tic-tac-toe is most often played by young children.

The simplicity of Tic-tac-toe makes it ideal 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, 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.[1]

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.

History[edit]

An early variant of Tic-tac-toe was played in the Roman Empire, around the first century BC. It was called Terni Lapilli 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. However, according to Claudia Zaslavsky's book Tic Tac Toe: And Other Three-In-A Row Games from Ancient Egypt to the Modern Computer, Tic-tac-toe could originate back to ancient Egypt.[2] 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.[3]

The different names of the game are more recent. The first print reference to "Noughts and crosses", the British name, appeared in 1864. In his novel "Can You Forgive Her", 1864, Anthony Trollope refers to a clerk playing "tit-tat-toe". 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 U.S. renaming of Noughts and crosses as Tic-tac-toe occurred in the 20th century.[4]

In 1952, OXO (or Noughts and Crosses) for the EDSAC computer 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.[5] It is currently on display at the Museum of Science, Boston.

Combinatorics[edit]

The first two plies of the game tree for Tic-tac-toe. Once rotations and reflections are eliminated, there are only three opening moves – a corner, a side or the middle.

Despite its apparent simplicity, Tic-tac-toe requires detailed analysis to determine even some elementary combinatory facts, the most interesting of which are the number of possible games and the number of possible positions. A position is merely a state of the board, while a game usually refers to the way a terminal position is obtained.

Naive counting leads to 19,683 possible board layouts (39 since each of the nine spaces can be X, O or blank), and 362,880 (i.e. 9!) possible games (different sequences for placing the Xs and Os on the board). However, two matters much reduce these numbers:

The complete analysis is further complicated by the definitions used when setting the conditions, like board symmetries.

Number of terminal positions[edit]

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. Assuming that X makes the first move every time:

Number of possible games[edit]

Without taking symmetries into account, the number of possible games can be determined by hand with an exact formula that leads to 255,168 possible games (see Henry Bottomley, 2001,[6] or Steve Schaeffer, 2002[7]). Assuming that X makes the first move every time:

If board symmetries are taken into account, two games are considered the same if rotating and/or reflecting the board makes one of the games into a copy of the other game. With the use of a computer, Steve Schaeffer found in 2002[7] that the number of games in these conditions is 26,830.

Strategy[edit]

Optimal strategy for player X. In each grid, the shaded red X denotes the optimal move, and the location of O's next move gives the next subgrid to examine. Note that only two sequences of moves by O (both starting with center, top-right, left-mid) lead to a draw, with the remaining sequences leading to wins from X.
Optimal strategy for player O. Player O must take center if it is not taken by X to force a win or draw.

A player can play a perfect game of Tic-tac-toe (to win or, at best, draw) if they choose the first available move from the following list, each turn, as used in Newell and Simon's 1972 tic-tac-toe program.[8]

  1. Win: If the player has two in a row, they can place a third to get three in a row.
  2. Block: If the opponent has two in a row, the player must play the third themself to block the opponent.
  3. Fork: Create an opportunity where the player has two threats to win (two non-blocked lines of 2).
  4. Blocking an opponent's fork:
    • Option 1: The player should create two in a row to force the opponent into defending, as long as it doesn't result in them creating a fork. 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, the player should block that fork.
  5. Center: A player marks the center. (If it is the first move of the game, playing on a corner gives "O" more opportunities to make a mistake and may therefore be the better choice; however, it makes no difference between perfect players.)
  6. Opposite corner: If the opponent is in the corner, the player plays the opposite corner.
  7. Empty corner: The player plays in a corner square.
  8. Empty side: The player plays in a middle square on any of the 4 sides.

The first player, who shall be designated "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.[9]

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.

To guarantee a draw for O, however:

Initiative in start game

Variations[edit]

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, as they do in tic-tac-toe, 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.

Other variations of tic-tac-toe include:

Magicsquareexample.svg

Another isomorphic game uses a list of nine carefully chosen words, for instance "eat", "bee", "less", "via", "bits", "lily", "soda", "boo", and "loot". Each player picks one word in turn and to win, a player must select three words with the same letter. The words may be plotted on a tic-tac-toe grid in such a way that a three in a row line wins.[13]

English names[edit]

The game has a number of English names.

In the United States, the name "Noughts and crosses" is not well understood, as the word "nought" is archaic in usage and X-shaped objects are not generally called crosses except in the case of specific named crosses like St. Patrick's Cross.

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.

In popular culture[edit]

Various game shows have been based on Tic-Tac-Toe and its variants:

See also[edit]

References[edit]

  1. ^ Steve Schaefer (January 2002). "MathRec Solutions (Tic-Tac-Toe)". MathRec. Archived from the original on 28 June 2013. 
  2. ^ Zaslavsky, Claudia (1982). Tic Tac Toe: And Other Three-In-A Row Games from Ancient Egypt to the Modern Computer. Crowell. ISBN 0-690-04316-3. 
  3. ^ Canisius College – Morris Games
  4. ^ Oxford English Dictionary entries for "Noughts and Crosses", "Tick-Tack" and "Tick-Tack-Toe", dictionary.oed.com
  5. ^ "Tinkertoys and tic-tac-toe". Retrieved 2007-09-27. 
  6. ^ How many Tic-Tac-Toe games are possible? Henry Bottomley, 2001
  7. ^ a b Mathematical Recreations Steve Schaeffer, 2002
  8. ^ Kevin Crowley, Robert S. Siegler (1993). "Flexible Strategy Use in Young Children’s Tic-Tac-Toe". Cognitive Science 17 (4): 531–561. doi:10.1016/0364-0213(93)90003-Q. 
  9. ^ Martin Gardner (1988). Hexaflexagons and Other Mathematical Diversions. University of Chicago Press. 
  10. ^ Oren Patashnik, Qubic: 4 x 4 x 4 Tic-Tac-Toe, Mathematical Magazine 53 (1980) 202–216.
  11. ^ a b Golomb, Solomon W.; Hales, Alfred W. (2002), "Hypercube tic-tac-toe", More games of no chance (Berkeley, CA, 2000), Math. Sci. Res. Inst. Publ. 42, Cambridge: Cambridge Univ. Press, pp. 167–182, MR 1973012 .
  12. ^ Averbach, Bonnie; Chein, Orin (1980), Problem Solving Through Recreational Mathematics, Dover, p. 252, ISBN 9780486131740 .
  13. ^ Schumer, Peter D. (2004), Mathematical Journeys, John Wiley & Sons, pp. 71–72, ISBN 9780471220664 .
  14. ^ Check Lines, BoardGameGeek, retrieved 2013-09-13.
  15. ^ Goff, Allan (November 2006). "Quantum tic-tac-toe: A teaching metaphor for superposition in quantum mechanics". American Journal of Physics (College Park, MD: American Association of Physics Teachers) 74 (11): 962–973. doi:10.1119/1.2213635. ISSN 0002-9505. 

External links[edit]