Tic-tac-toe

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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 , which alternately mark spaces in grid 3 Ã- 3. Players who successfully place their three marks in horizontal, vertical, or diagonal lines win the match.

Examples of the following games were won by the first player, X:

Players soon discovered that the best game from both sides resulted in a draw. Therefore, 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 to teach good sportsmanship concepts and artificial intelligence branches related to game tree searches. It is easy to write a computer program to play tic-tac-toe perfectly, to calculate 765 basically different positions (the complexity of the state space), or 26,830 possible games up to rotation and reflection (game tree complexity) in this space..

The game can be generalized to m, n, k-games where two players take turns placing stones with their own colors on the board m ÃÆ'â € n , in order to get k their own colors in a row. Tic-tac-toe is (3,3,3) -game. Harary's general tic-tac-toe is a broader generalization of tic tac toe. It can also be generalized as a game <<> << sup> d . Tic-tac-toe is a game where n equals 3 and d equals 2. If played correctly, the game will end up with a draw that makes tic-tac-toe a waste game.

Video Tic-tac-toe

Histori

The games played on a three-in-a-row board can be traced back to ancient Egypt, where such board games have been found on roof tiles dating from around 1300 BC.

The initial variations of tic-tac-toe were played in the Roman Empire, around the first century BC. It's called terni lapilli ( three pebbles at a time ) and instead of having a number of pieces, each player has only three, so they have to move them around empty space to continue playing. Grid markers of games have been found throughout Rome. Another closely-knit ancient game is Three Men's Morris which is also played on a simple grid and requires three consecutive slices to complete, and Picaria, a Puebloans game.

The different game names are newer. The first print reference for "noughts and crosses", English name, appeared in 1864. In his novel Can I Forgive it? (1864) Anthony Trollope refers to officers playing "tit-tat-toe". The first print reference for the game called "tick-tack-toe" occurred in 1884, but refers to "children's games being played on slate, consisting of trying with closed eyes to bring a pencil to one of the numbers of the set, number of hits printed ". "Tic-tac-toe" may also come from "tick-tack", the name of the old version of backgammon first described in 1558. The US name change of "noughts and crosses" as "tic-tac-toe" occurred in the century -20.

In 1952, OXO (or Noughts and Crosses ), developed by British computer scientist Alexander S. Douglas for EDSAC computers at the University of Cambridge, became one of the first known videos games. The computer player can play the perfect tic-tac-toe game against human opponents.

In 1975, tic-tac-toe was also used by MIT students to demonstrate the computing power of Tinkertoy elements. The Tinkertoy computer, made of (almost) only Tinkertoys, can play tic-tac-toe perfectly. Currently on display at the Museum of Science, Boston.

Maps Tic-tac-toe

Combinatorics

When considering only the state of the board, and after taking into account board symmetry (ie rotation and reflection), there are only 138 positions of the terminal board. A combinatoric study of the game shows that when "X" makes the first step every time, the game is won as follows:

• 91 different positions won by (X)
• 44 different positions are won by (O)
• 3 different positions are drawn (often called "cat games")

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Strategy

A player can play the perfect tic-tac-toe game (to win or, at least, draw) if each time it turns he chooses the first available step from the following list, as used in Newell and Simon 1972 tic-tac-toe.

1. Winning : If the player has two in a row, they can place a third to earn three in a row.
2. Block : If the opponent has two in a row, the player must play the third himself to block the opponent.
3. Fork : Create an opportunity where players have two threats to win (two lines 2 that are not blocked).
4. Blocking the opponent's fork : If there is only one possible fork for the opponent, the player must block it. Otherwise, the player must block any fork in any way that enables him to make two consecutively at once. Otherwise, the player must create two consecutive rows to force the opponent to survive, as long as it does not produce them make a fork. For example, if "X" has two opposite corners and "O" has a center, "O" can not play football to win. (Playing a corner in this scenario creates a fork for "X" to win.)
5. Center : The player marks the center. (If this is the first step of the game, playing in the corner gives the second player more chances to make mistakes and therefore might be a better option; however, it makes no difference between a perfect player.)
6. The opposite corner : If the opponent is in the corner, the player plays the opposite corner.
7. Empty corner : The player plays in a corner.
8. Empty side : The player plays in the middle box on one of the four sides.

The first player, to be designated "X", has 3 possible positions to mark during the first turn. Superficially, it may seem that there are 9 possible positions, corresponding to 9 boxes on the grid. However, by rotating the board, we will find that at the first corner, each corner mark is strategically equivalent to any other angle mark. The same applies to any edge mark (middle side). For strategic purposes, there are only three possible first sign: angle, edge, or middle. Player X may win or impose a draw from one of these early signs; However, playing a corner kick gives the smallest opponent the option to play to avoid defeat. This may indicate that the angle is the best opening step for X, but other studies show that if the player is not perfect, the opening step in the middle is best for X.

The second player, who will be appointed "O", must respond to the opening sign X in such a way as to avoid a forced victory. Player O must always respond to the opening of the angle with the center mark, and to the opening of the center with an angle. The edge opening should be answered either with the center mark, the corner mark next to X, or the opposite edge mark X. Any other responses will allow X to force the win. After the opening is over, O's job is to follow the priority list above to force the draw, or else to win if X makes a weak game.

In more detail, to ensure a draw, O must adopt the following strategy:

• If X plays an angle opening movement, O should be in the middle, and then edge, forcing X to block in the next step. This will stop any fork from happening. When X and O are perfect players and X chooses to start by marking the angle, O takes center, and X takes the opposite angle to the original. In this case, O is free to choose any edge as a second step. However, if X is not a perfect player and has played a corner kick and then the edge, O should not play the opposite side as the second step, because X is not forced to block in the next step and can be branched.
• If X plays an edge-opening step, O should be in the middle or one of the corners adjacent to X, then follow the priority list above, especially note the fork beam.
• If X plays the middle opening step, O should take an angle, and then follow the priority list above, especially pay attention to beam fork.

When X plays the first corner, and O is not a perfect player, the following might happen:

• If O responds with a center mark (the best step for them), the perfect X player will take the opposite angle to the original. Then O must play an advantage. However, if O plays a corner kick as the second step, the perfect X player will mark the remaining corner, blocking 3-in-a-row O and making their own fork.
• If O responds with an angle, X is guaranteed to win, simply by taking one of the other two corners and then the last one, a fork. (Because when X takes the third angle, O can only take position between two X. Then X can take the only remaining corner to win)
• If O responds with edge marks, X is guaranteed to win, by taking center, then O can only take a corner at the corner opposite the X play first. Finally, X can take a corner to make a fork and then X will win in the next step.
More details

Consider a board with the nine numbered positions as follows:

When X plays 1 as their opening step, then O must take 5. Then X takes 9 (in this situation, O should not take 3 or 7, O should take 2, 4, 6 or 8):

• X1 -> O5 -> X9 -> O2 -> X8 -> O7 -> X3 -> O6 -> X4, this game will be a series.

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 step, because X is an imperfect player can take 4, then O can take 7 for win).

• X1 -> O5 -> X6 -> O2 -> X8, then O can not take 3, or X can take 7 to win, and O can not take 4, or X can take 9 to win, O should take 7 or 9.
  • X1 -> O5 -> X6 -> O2 -> X8 -> O7 -> X3 -> O9 -> X4, this game will be a series.
  • X1 -> O5 -> X6 -> O2 -> X8 -> O9 -> X4 (7) -> O7 (4) -> X3, this game will be a series.
• X1 -> O5 -> X6 -> O3 -> X7 -> O4 -> X8 (9) -> O9 (8) -> X2, this game will be a series.
• X1 -> O5 -> X6 -> O8 -> X2 -> O3 -> X7 -> O4 -> X9, this game will be a series.
• X1 -> O5 -> X6 -> O9, then X can not take 4, or O can take 7 to win, X should take 2, 3, 7 or 8.
  • X1 -> O5 -> X6 -> O9 -> X2 -> O3 -> X7 -> O4 -> X8, this game will be a series.
  • X1 -> O5 -> X6 -> O9 -> X3 -> O2 -> X8 -> O4 (7) -> X7 (4), this game will be a series.
  • X1 -> O5 -> X6 -> O9 -> X7 -> O4 -> X2 (3) -> O3 (2) -> X8, this game will be a series.
  • X1 -> O5 -> X6 -> O9 -> X8 -> O2 (3, 4, 7) -> X4/7 (4/7, 2/3, 2/3) -> O7/4 ( 7/4, 3/2, 3/2) -> X3 (2, 7, 4), this game will be a series.

Dalam kedua situasi ini (X memerlukan 9 atau 6 sebagai langkah kedua), X memiliki ${\ displaystyle {\ frac {1} {3}}}  ÂÂ$ properti untuk menang.

If X is not a perfect player, X can take 2 or 3 as the second step. Then this game will be a series, X can not win.

• X1 -> O5 -> X2 -> O3 -> X7 -> O4 -> X6 -> O8 (9) -> X9 (8), this game will be a series.
• X1 -> O5 -> X3 -> O2 -> X8 -> O4 (6) -> X6 (4) -> O9 (7) -> X7 (9), this game will be a series.

If X plays 1 opening movement, and O is not a perfect player, the following might happen:

Although O takes the only good position (5) as the first step, but O takes a bad position as a second step:

• X1 -> O5 -> X9 -> O3 -> X7, then X can take 4 or 8 to win.
• X1 -> O5 -> X6 -> O4 -> X3, then X can take 2 or 9 to win.
• X1 -> O5 -> X6 -> O7 -> X3, then X can take 2 or 9 to win.

Although O took a good position as the first two steps, but O took the bad position as a third step:

• X1 -> O5 -> X6 -> O2 -> X8 -> O3 -> X7, then X can take 4 or 9 to win.
• X1 -> O5 -> X6 -> O2 -> X8 -> O4 -> X9, then X can take 3 or 7 to win.

O take a bad position as the first step (except 5, all other bad positions):

• X1 -> O3 -> X7 -> O4 -> X9, then X can take 5 or 8 to win.
• X1 -> O9 -> X3 -> O2 -> X7, then X can take 4 or 5 to win.
• X1 -> O2 -> X5 -> O9 -> X7, then X can take 3 or 4 to win.
• X1 -> O6 -> X5 -> O9 -> X3, then X can take 2 or 7 to win.



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Variations

Many board-sharing games try to be the first to get n -in-the-line, including Three Morris Men, Nine Men Morris, pente, gomoku, Qubic, Connect Four, Quarto, Gobblet, Order and Chaos , Toss Across, and Mojo. Tic-tac-toe is an example of m, n, k-game, where two players take turns on a board m ÃÆ'â € n until one of them gets < i> k in a row. Harary's general tic-tac-toe is a wider generalization.

Other variations of tic-tac-toe include:

• 3-dimensional tic-tac-toe on board 3ÃÆ' â € "3ÃÆ' â €" 3. In this game, the first player has an easy victory by playing in the middle if 2 people play.

One can play on 4x4 box boards, win in some way. Winning can include: 4 in a straight line, 4 in a diagonal line, 4 in diamond, or 4 to make a square.

Another variant, Qubic, is played on board 4ÃÆ' â € "4ÃÆ' â €" 4; it was broken by Oren Patashnik in 1980 (first player can force win). Higher dimensional variations are also possible.

• In a tic-tac-toe misÃÆ'¨re, the player wins if the opponent gets n in a row. Game 3ÃÆ' â € "3 is the series. More generally, the first player can draw or win on any board (any dimension) that is oddly side-length, by playing first in the center cell and then imitating the opponent's movements.
• In "wild" tic-tac-toe, players can choose to place X or O on every move.
• The number of Scrabble or Pick15 is isomorphic to tic-tac-toe, but on the surface it looks very different. Two players in turn say numbers between one and nine. Certain numbers can not be repeated. The game is won by a player who has said three numbers that number is 15. If all numbers are used and nobody gets three numbers that add up to 15 then the game is a draw. Mapping these numbers in square 3-3 indicates that the game is perfectly compatible with tic-tac-toe, since three numbers will be arranged in a straight line if and only if the total is 15.
• Another isomorphic game uses a carefully selected list of nine words, such as "eat", "bee", "less", "air", "bit", "lip", "soda" , "books", and "many". Each player chooses one word in turn and to win, the player must choose three words with the same letter. The words can be plotted on a tic-tac-toe grid such that three successive lines win.
• Numerical Tic Tac Toe is a variant discovered by mathematician Ronald Graham. Figures 1 through 9 are used in this game. The first player plays with odd numbers, the second player plays with the even number. All numbers can only be used once. The player who put 15 points in the win of the line (number 3 digits).
• In the 1970s, there was a two-player game made by Tri-ang Toys & The game is called Check Line , where the board consists of eleven holes arranged in a geometric pattern of twelve straight lines each containing three holes. Each player has exactly five tokens and is played in turn placing one token in one of the holes. The winner is the first player whose token is set in two rows of three (which by definition are intersecting lines). If no player is won by the 10th turn, the next turn consists of moving one of one's tokens to the remaining empty hole, with the constraint that this step can only be from an adjacent hole.
• Quantum tic tac toe lets players place a quantum superposition of numbers on the board, ie the player's movement is the "superposition" of the game in the original classic game. This variation was discovered by Allan Goff of Novatia Labs.



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English name

This game has a number of English names.

• Tick-tack-toe, tic-tac-toe, tik-tat-toe, or tit-tat-toe (United States, Canada)
• Noughts and crosses or naughts and crosses (UK, Republic of Ireland, Australia, New Zealand, South Africa, Zimbabwe, India)
• Exy-ozys, xsie-osies (verbal names only) (Northern Ireland)
• Xs and Os (Egypt, Republic of Ireland, Zimbabwe)

Occasionally, a tic-tac-toe game (where players keep adding "snippets") and Three Men's Morris (where pieces start moving after a certain number is placed) confuse each other.



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In popular culture

• Wordsworth alludes to tic-tac-toe in his book The Prelude (1799).
• In the movie WarGames , computers learn through the tic-tac-toe analogy that nuclear war can not be won.
• The F test card image of the BBC contains a picture of the game. In the updated version, Test Card J, X game shows the center of the right screen.
• George Cooper wrote the words and John Rogers Thomas wrote the music for the song "Tit, Tac, Toe" in 1876.
• Adventure games Sam & amp; Max Save the World has an arcade version called Tic-Tac-Doom , but its artificial intelligence is intentionally deliberate. Instead of defeating the game, the player must lose it in order to progress.
• Episode 452 of American Life tells the true story of a legal defense team that seeks to overturn a Florida state decision to execute a serial killer by raising a tic-tac-toe chicken as evidence. Arcade game with chicken playing tic-tac-toe popular in the mid-1970s; animals are trained using operant conditioning.
• In the 1957 movie version Twelve Angry Men there is a small scene two jurors are playing games to pass the time.
• In Three Short Stooges, Disorder in the Court , Moe and Larry are seen playing tic-tac-toe in several different scenes. (One with Curly plays a jack on it.)

Game events have been based on tic-tac-toe and its variants:

• In Hollywood Squares , nine celebrities fill the tic-tac-toe grid cells; players put symbols on the board by either agreeing or disagreeing with a celebrity answer to a question. Event variations include Squares Storybook and Hip Hop Squares . The English version is Celebrity Squares . Australia has various versions under the name Celebrity Squares, Personality Squares & amp; All Star Boxes.
• In Tic-Tac-Dough , players place symbols on the board by answering questions in various categories, which randomize each player's turn.
• In Beat the Teacher , the contestant answers the question to win a turn to influence the tic-tac-toe grid.
• Crossfire game shows also allow contestants to answer questions to put symbols on the board.
• At Exact Price , some national variants feature a pricing game called "Secret X", where players have to guess the price of two small prizes to win X (other than one free X) to place on the board blank. They must place X in position to guess the secret "X" titular location hidden in the center column of the board and form a transverse or diagonal tic-tac-toe line (no vertical lines are allowed). There is no Os in this game variant.
• In the Minute to Win it, game, Ping Tac Toe has one contestant who plays a game with nine glasses of water and white and orange ping-pong balls, trying to get three in a good row of colors. He must change color after every successful landing and must be careful not to block him.



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See also

• 3-D Tic-Tac-Toe
• Hales-Jewett's theorem
• m, n, k-game
• Notakto
• Choose 15
• SOS (game)
• Tic-tac-toe variant
• Ultimate tic-tac-toe



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References




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External links

• Tic-Tac-Toe in Wolfram's MathWorld
• Discussion of the term "cat game" for the tic-tac-toe game game

Source of the article : Wikipedia