Tic-tac-toe

"Tic tac toe" and "Noughts and Crosses" redirect here. For other uses, see Tic Tac Toe (disambiguation) and Noughts and Crosses (disambiguation).
Tic-tac-toe

A completed game of Tic-tac-toe
Genre(s) Paper-and-pencil game
Players 2
Setup time Minimal
Playing time ~1 minute
Random chance None
Skill(s) required Strategy, tactics, observation
Synonym(s) Noughts and crosses
Xs and Os

Tic-tac-toe (also known as noughts and crosses 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 wins the game.

The following example game is won by the first player, 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.

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, 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

An early variation 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.[2] 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 be traced back to ancient Egypt.[3] 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.[4]

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.[5]

In 1952, OXO (or Noughts and Crosses), developed by British computer scientist Alexander S. Douglas for the EDSAC computer at the University of Cambridge, became one of the first known video games.[6][7] The computer player could play perfect games of tic-tac-toe against a human opponent.[6]

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.[8] It is currently on display at the Museum of Science, Boston.

Combinatorics

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

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:

Strategy

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 can always force a win or draw by taking center. If it is taken by X, then O must take a corner

A player can play a perfect game of tic-tac-toe (to win or, at least, 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.[9]

  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 themselves 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 (side middle) 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.[10] This makes the corner the best opening move for X, when the opponent is not a perfect player.

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 detailedly, to guarantee a draw, O should adopt the following strategies:

When X plays corner first (best move for them), and O is not a perfect player, the following may happen:

Further details

Let's label the nine positions on the board as follows:

1 2 3
4 5 6
7 8 9

When X plays 1 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 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):

Variations

Main article: Tic-tac-toe variants

Tic tac toe has several variants.

English names

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:

Rules

Since the goal is to get 3 in a row, each person must switch taking turns, first X, then O. Players must use the board given to them, they cannot add extra sides on to the board. In order to win, the 3 letters must all connect in a straight line in one direction, up or down, left or right, or diagonally.

See also

References

  1. ↑ Schaefer, Steve (2002). "MathRec Solutions (Tic-Tac-Toe)". Retrieved 2015-09-18.
  2. ↑ "Sweetooth Design Company | Food & Design | Oakland, USA". Sweetooth Design Company | Food & Design | Oakland, USA. Retrieved 2016-12-04.
  3. ↑ 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.
  4. ↑ Canisius College – Morris Games
  5. ↑ Oxford English Dictionary entries for "Noughts and Crosses", "Tick-Tack" and "Tick-Tack-Toe", dictionary.oed.com
  6. 1 2 Wolf, Mark J. P. (2012-08-16). Encyclopedia of Video Games: The Culture, Technology, and Art of Gaming. Greenwood Publishing Group. pp. 3–7. ISBN 978-0-313-37936-9.
  7. ↑ Cohen, D. S. (2014-09-20). "OXO aka Noughts and Crosses - The First Video Game". About.com. IAC. Archived from the original on 2015-12-22. Retrieved 2015-12-18.
  8. ↑ "Tinkertoys and tic-tac-toe". Archived from the original on August 24, 2007. Retrieved 2007-09-27.
  9. ↑ 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.
  10. ↑ Martin Gardner (1988). Hexaflexagons and Other Mathematical Diversions. University of Chicago Press.
  11. ↑ "452: Poultry Slam 2011". This American Life. Retrieved 28 May 2016.
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