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Tic-tac-toe
Tic-tac-toe (American English), noughts and crosses (Commonwealth English), or Xs and Os (Canadian or Irish English) is a paper-and-pencil game for two players who take turns marking the spaces in a three-by-three grid with ''X'' or ''O''. 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. Gameplay Tic-tac-toe is played on a three-by-three grid by two players, who alternately place the marks X and O in one of the nine spaces in the grid. In the following example, the first player (''X'') wins the game in seven steps: There is no universally-agreed rule as to who plays first, but in this article the convention that X plays first is used. Players soon discover that the best play from both parties leads to a draw. Hence, tic-tac-toe is often played by young children who may not have discovered the optimal strategy. Because of the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tic Tac Toe
Tic-tac-toe (American English), noughts and crosses (Commonwealth English), or Xs and Os (Canadian or Irish English) is a paper-and-pencil game for two players who take turns marking the spaces in a three-by-three grid with ''X'' or ''O''. 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. Gameplay Tic-tac-toe is played on a three-by-three grid by two players, who alternately place the marks X and O in one of the nine spaces in the grid. In the following example, the first player (''X'') wins the game in seven steps: There is no universally-agreed rule as to who plays first, but in this article the convention that X plays first is used. Players soon discover that the best play from both parties leads to a draw. Hence, tic-tac-toe is often played by young children who may not have discovered the optimal strategy. Because of the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Game Tree
In the context of Combinatorial game theory, which typically studies sequential games with perfect information, a game tree is a graph representing all possible game states within such a game. Such games include well-known ones such as chess, checkers, Go, and tic-tac-toe. This can be used to measure the complexity of a game, as it represents all the possible ways a game can pan out. Due to the large game trees of complex games such as chess, algorithms that are designed to play this class of games will use partial game trees, which makes computation feasible on modern computers. Various methods exist to solve game trees. If a complete game tree can be generated, a deterministic algorithm, such as backward induction or retrograde analysis can be used. Randomized algorithms and minimax algorithms such as MCTS can be used in cases where a complete game tree is not feasible. Understanding the game tree To better understand the game tree, it can be thought of as a technique for an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Three Men's Morris
Three men's morris is an abstract strategy game played on a three by three board (counting lines) that is similar to tic-tac-toe. It is also related to six men's morris and nine men's morris. A player wins by forming a mill, that is, three of their own pieces in a row. Rules 240px, A board for three men's morris. This pattern has been found carved into the roof of the temple of Kurna. Each player has three pieces. The winner is the first player to align their three pieces on a line drawn on the board. There are 3 horizontal lines, 3 vertical lines and 2 diagonal lines. The board is empty to begin the game, and players take turns placing their pieces on empty intersections. Once all pieces are placed (assuming there is no winner by then), play proceeds with each player moving one of their pieces per turn. A piece may move to any vacant point on the board, not just an adjacent one. According to ''A History of Chess'', there is an alternative version in which pieces may not mov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harary's Generalized Tic-tac-toe
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. Harary's generalization does not include tic-tac-toe itself, as diagonal constructions are not considered a win. Like many other two-player games, strategy stealing means that the second player can never win. All that is left to study is to determine whether the first player can win, on what board sizes he may do so, and in how many moves it will take. Results Square boards Let ''b'' be the smallest size square board on which the first player can win, and let ''m'' be the smallest number of moves in which the first player can force a win, assuming perfect play by both sides. *monomino: ''b'' = 1, ''m'' = 1 *domino: ''b'' = 2, ''m' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Game Tree Complexity
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. Measures of game complexity State-space complexity The state-space complexity of a game is the number of legal game positions reachable from the initial position of the game. When this is too hard to calculate, an upper bound can often be computed by also counting (some) illegal positions, meaning positions that can never arise in the course of a game. Game tree size The game tree size is the total number of possible games that can be played: the number of leaf nodes in the game tree rooted at the game's initial position. The game tree is typically vastly larger than the state space because the same positions can occur in many games by making moves in a different order (for example, in a tic-tac-toe game with two X and one O on the board, this position co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
State Space Complexity
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. Measures of game complexity State-space complexity The state-space complexity of a game is the number of legal game positions reachable from the initial position of the game. When this is too hard to calculate, an upper bound can often be computed by also counting (some) illegal positions, meaning positions that can never arise in the course of a game. Game tree size The game tree size is the total number of possible games that can be played: the number of leaf nodes in the game tree rooted at the game's initial position. The game tree is typically vastly larger than the state space because the same positions can occur in many games by making moves in a different order (for example, in a tic-tac-toe game with two X and one O on the board, this position co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nd Game
A ''n''''d'' game (or ''n''''k'' game) is a generalization of the game tic-tac-toe to higher dimensions. It is a game played on a ''n''''d'' hypercube with 2 players. If one player creates a line of length ''n'' of their symbol (X or O) they win the game. However, if all ''n''''d'' spaces are filled then the game is a draw. Tic-tac-toe is the game where ''n'' equals 3 and ''d'' equals 2 (3, 2). Qubic 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 pla ... is the game. The or games are trivially won by the first player as there is only one space ( and ). A game with and cannot be won if both players are playing well as an opponent's piece will block the one-dimensional line. There are a total of winning lines in a ''n''''d'' game. See also * References Tic-tac-toe {{g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
M,n,k-game
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''-by-''n'' board, the winner being the player who first gets ''k'' stones of their own color in a row, horizontally, vertically, or diagonally.J. W. H. M. Uiterwijk and H. J van der Herik, ''The advantage of the initiative'', Information Sciences 122 (1) (2000) 43-58.Jaap van den Herik, Jos W.H.M. Uiterwijk, Jack van Rijswijck (2002). "Games solved: Now and in the future". Artificial Intelligence. 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''-by-''n'' board. The ''m'',''n'',''k''-games are mainly of mathematical interest. One seeks to find the game-theoretic value, the result of the game with perfect play. This is known as solving the game. Strategy stealing argument A standard strategy stealing argument from combinatorial game theory shows t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solved Game
A solved game is a game whose outcome (win, lose or draw) can be correctly predicted from any position, assuming that both players play perfectly. This concept is usually applied to abstract strategy games, and especially to games with full information and no element of chance; solving such a game may use combinatorial game theory and/or computer assistance. Overview A two-player game can be solved on several levels: ;Ultra-weak : Prove whether the first player will win, lose or draw from the initial position, given perfect play on both sides. This can be a non-constructive proof (possibly involving a strategy-stealing argument) that need not actually determine any moves of the perfect play. ;Weak : Provide an algorithm that secures a win for one player, or a draw for either, against any possible moves by the opponent, from the beginning of the game. ;Strong : Provide an algorithm that can produce perfect moves from any position, even if mistakes have already been made on one or b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paper-and-pencil Game
Paper-and-pencil games or paper-and-pen games (or some variation on those terms) are games that can be played solely with paper and pencils (or other writing implements), usually without erasing. They may be played to pass the time, as icebreakers, or to end boredom. In recent times, they have been supplanted by mobile games. Some popular examples of pencil-and-paper games include Tic-tac-toe, Sprouts, Dots and Boxes, Hangman, MASH, Paper soccer, and Spellbinder. The term is unrelated to the use in role-playing games to differentiate tabletop games from role-playing video games. Board games where pieces are never moved or removed from the board once being played, particularly abstract strategy games like Gomoku and Connect Four Connect Four (also known as Connect 4, Four Up, Plot Four, Find Four, Captain's Mistress, Four in a Row, Drop Four, and Gravitrips in the Soviet Union) is a two-player connection board game, in which the players choose a color and then take tur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Picaria
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. There are two variations to Picaria. The first version is simpler with 9 spaces or intersection points, and the second version uses 13 spaces or intersection points. The rules are the same. Goal To create a three-in-a-row of one's pieces either horizontally, vertically, or diagonally. Equipment A 3×3 board is used for the first version. Three horizontal lines form the three rows. Three vertical lines form the three columns. Two diagonal lines connect the two opposite corners of the board. Additionally, there are four more diagonal lines connecting the midpoints. These four additional diagonal lines is what makes the Picaria board different ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paper-and-pencil Game
Paper-and-pencil games or paper-and-pen games (or some variation on those terms) are games that can be played solely with paper and pencils (or other writing implements), usually without erasing. They may be played to pass the time, as icebreakers, or to end boredom. In recent times, they have been supplanted by mobile games. Some popular examples of pencil-and-paper games include Tic-tac-toe, Sprouts, Dots and Boxes, Hangman, MASH, Paper soccer, and Spellbinder. The term is unrelated to the use in role-playing games to differentiate tabletop games from role-playing video games. Board games where pieces are never moved or removed from the board once being played, particularly abstract strategy games like Gomoku and Connect Four Connect Four (also known as Connect 4, Four Up, Plot Four, Find Four, Captain's Mistress, Four in a Row, Drop Four, and Gravitrips in the Soviet Union) is a two-player connection board game, in which the players choose a color and then take tur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
