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Ultimate Tic-tac-toe
Ultimate tic-tac-toe (also known as super tic-tac-toe, meta tic-tac-toe or (tic-tac-toe)²) is a board game composed of nine tic-tac-toe boards arranged in a 3 × 3 grid. Players take turns playing on the smaller tic-tac-toe boards until one of them wins on the larger board. Compared to traditional tic-tac-toe, strategy in this game is conceptually more difficult and has proven more challenging for computers. Rules Just like in regular tic-tac-toe, the two players (X and O) take turns, starting with X. The game starts with X playing wherever they want in any of the 81 empty spots. Next the opponent plays, however they are forced to play in the small board indicated by the relative location of the previous move. For example, if X plays in the top right square of a small (3 × 3) board, then O has to play in the small board located at the top right of the larger board. Playing any of the available spots decides in which small board the next player plays. If a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Super Tic-tac-toe Rules Example
Super may refer to: Computing * SUPER (computer program), or Simplified Universal Player Encoder & Renderer, a video converter / player * Super (computer science), a keyword in object-oriented programming languages * Super key (keyboard button) Film and television * Super (2005 film), ''Super'' (2005 film), a Telugu film starring Nagarjuna, Anushka Shetty and Ayesha Takia * Super (2010 Indian film), ''Super'' (2010 Indian film), a Kannada language film starring Upendra and Nayantara * Super (2010 American film), ''Super'' (2010 American film), a film written and directed by James Gunn, and starring Rainn Wilson and Elliot Page * Super (Person of Interest), "Super" (''Person of Interest''), an episode of the TV series ''Person of Interest'' Music * Super (Cordae song), "Super" (Cordae song), a 2021 song by American rapper Cordae * Super (Neu! song), "Super" (Neu! song), a 1972 song by German band Neu! * "Super (1, 2, 3)", a 2000 song by Italian DJ Gigi D'Agostino * Super ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Winning Strategy
Determinacy is a subfield of set theory, a branch of mathematics, that examines the conditions under which one or the other player of a game has a winning strategy, and the consequences of the existence of such strategies. Alternatively and similarly, "determinacy" is the property of a game whereby such a strategy exists. Determinacy was introduced by Gale and Stewart in 1950, under the name "determinateness". The games studied in set theory are usually Gale–Stewart games—two-player games of perfect information in which the players make an infinite sequence of moves and there are no draws. The field of game theory studies more general kinds of games, including games with draws such as tic-tac-toe, chess, or infinite chess, or games with imperfect information such as poker. Basic notions Games The first sort of game we shall consider is the two-player game of perfect information of length ω, in which the players play natural numbers. These games are often called Ga ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Games
A mathematical game is a game whose rules, strategies, and outcomes are defined by clear mathematical parameters. Often, such games have simple rules and match procedures, such as Tic-tac-toe and Dots and Boxes. Generally, mathematical games need not be conceptually intricate to involve deeper computational underpinnings. For example, even though the rules of Mancala are relatively basic, the game can be rigorously analyzed through the lens of combinatorial game theory. Mathematical games differ sharply from mathematical puzzles in that mathematical puzzles require specific mathematical expertise to complete, whereas mathematical games do not require a deep knowledge of mathematics to play. Often, the arithmetic core of mathematical games is not readily apparent to players untrained to note the statistical or mathematical aspects. Some mathematical games are of deep interest in the field of recreational mathematics. When studying a game's core mathematics, arithmetic theory i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Strategy Games
Abstract strategy games admit a number of definitions which distinguish these from strategy games in general, mostly involving no or minimal narrative theme, outcomes determined only by player choice (with no randomness), and perfect information. For example, Go is a pure abstract strategy game since it fulfills all three criteria; chess and related games are nearly so but feature a recognizable theme of ancient warfare; and Stratego is borderline since it is deterministic, loosely based on 19th-century Napoleonic warfare, and features concealed information. Definition Combinatorial games have no randomizers such as dice, no simultaneous movement, nor hidden information. Some games that do have these elements are sometimes classified as abstract strategy games. (Games such as '' Continuo'', Octiles, '' Can't Stop'', and Sequence, could be considered abstract strategy games, despite having a luck or bluffing element.) A smaller category of abstract strategy games manages to ... [...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] |
Tic-tac-toe Variants
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. 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. Variants of tic-tac-toe date back several millennia. Historic 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. The game's grid ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursion
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. While this apparently defines an infinite number of instances (function values), it is often done in such a way that no infinite loop or infinite chain of references ("crock recursion") can occur. Formal definitions In mathematics and computer science, a class of objects or methods exhibits recursive behavior when it can be defined by two properties: * A simple ''base case'' (or cases) — a terminating scenario that does not use recursion to produce an answer * A ''recursive step'' — a set of rules that reduces all successive cases toward the base case. For example, the following is a recursive definition of a person's ''ancestor''. One's ances ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Play
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] |
Monte Carlo Tree Search
In computer science, Monte Carlo tree search (MCTS) is a heuristic search algorithm for some kinds of decision processes, most notably those employed in software that plays board games. In that context MCTS is used to solve the game tree. MCTS was combined with neural networks in 2016 and has been used in multiple board games like Chess, Shogi, Checkers, Backgammon, Contract Bridge, Computer Go, Scrabble, and Clobber as well as in turn-based-strategy video games (such as Total War: Rome II's implementation in the high level campaign AI). History Monte Carlo method The Monte Carlo method, which uses random sampling for deterministic problems which are difficult or impossible to solve using other approaches, dates back to the 1940s. In his 1987 PhD thesis, Bruce Abramson combined minimax search with an ''expected-outcome model'' based on random game playouts to the end, instead of the usual static evaluation function. Abramson said the expected-outcome model "is shown to b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Super Tic-tac-toe Rules Example - Simple
Super may refer to: Computing * SUPER (computer program), or Simplified Universal Player Encoder & Renderer, a video converter / player * Super (computer science), a keyword in object-oriented programming languages * Super key (keyboard button) Film and television * ''Super'' (2005 film), a Telugu film starring Nagarjuna, Anushka Shetty and Ayesha Takia * ''Super'' (2010 Indian film), a Kannada language film starring Upendra and Nayantara * ''Super'' (2010 American film), a film written and directed by James Gunn, and starring Rainn Wilson and Elliot Page * "Super" (''Person of Interest''), an episode of the TV series ''Person of Interest'' Music * "Super" (Cordae song), a 2021 song by American rapper Cordae * "Super" (Neu! song), a 1972 song by German band Neu! * "Super (1, 2, 3)", a 2000 song by Italian DJ Gigi D'Agostino * ''Super'' (album), a 2016 album by Pet Shop Boys Other uses * Super!, an Italian television network * Super (gamer) (born 2000), Ameri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Evaluation Function
An evaluation function, also known as a heuristic evaluation function or static evaluation function, is a function used by game-playing computer programs to estimate the value or goodness of a position (usually at a leaf or terminal node) in a game tree. Most of the time, the value is either a real number or a quantized integer, often in ''n''ths of the value of a playing piece such as a stone in go or a pawn in chess, where ''n'' may be tenths, hundredths or other convenient fraction, but sometimes, the value is an array of three values in the unit interval, representing the win, draw, and loss percentages of the position. There do not exist analytical or theoretical models for evaluation functions for unsolved games, nor are such functions entirely ad-hoc. The composition of evaluation functions is determined empirically by inserting a candidate function into an automaton and evaluating its subsequent performance. A significant body of evidence now exists for several games l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimax
Minimax (sometimes MinMax, MM or saddle point) is a decision rule used in artificial intelligence, decision theory, game theory, statistics, and philosophy for ''mini''mizing the possible loss for a worst case (''max''imum loss) scenario. When dealing with gains, it is referred to as "maximin" – to maximize the minimum gain. Originally formulated for several-player zero-sum game theory, covering both the cases where players take alternate moves and those where they make simultaneous moves, it has also been extended to more complex games and to general decision-making in the presence of uncertainty. Game theory In general games The maximin value is the highest value that the player can be sure to get without knowing the actions of the other players; equivalently, it is the lowest value the other players can force the player to receive when they know the player's action. Its formal definition is: :\underline = \max_ \min_ Where: * is the index of the player of interest. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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