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Strategy-stealing Argument
In combinatorial game theory, the strategy-stealing argument is a general argument that shows, for many two-player games, that the second player cannot have a guaranteed winning strategy. The strategy-stealing argument applies to any symmetric game (one in which either player has the same set of available moves with the same results, so that the first player can "use" the second player's strategy) in which an extra move can never be a disadvantage. A key property of a strategy-stealing argument is that it proves that the first player can win (or possibly draw) the game without actually constructing such a strategy. So, although it might prove the existence of a winning strategy, the proof gives no information about what that strategy is. The argument works by obtaining a contradiction. A winning strategy is assumed to exist for the second player, who is using it. But then, roughly speaking, after making an arbitrary first move – which by the conditions above is not a disadvantag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Game Theory
Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Research in this field has primarily focused on two-player games in which a ''position'' evolves through alternating ''moves'', each governed by well-defined rules, with the aim of achieving a specific winning condition. Unlike game theory, economic game theory, combinatorial game theory generally avoids the study of games of chance or games involving imperfect information, preferring instead games in which the current state and the full set of available moves are always known to both players. However, as mathematical techniques develop, the scope of analyzable games expands, and the boundaries of the field continue to evolve. Authors typically define the term "game" at the outset of academic papers, with definitions tailored to the specific game under analysis rather than reflecting the field’s full scope. Combinatorics, Comb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong Positional Game
A strong positional game (also called Maker-Maker game) is a kind of positional game. Like most positional games, it is described by its set of ''positions'' (X) and its family of ''winning-sets'' (\mathcal- a family of subsets of X). It is played by two players, called First and Second, who alternately take previously untaken positions. In a strong positional game, the winner is the first player who holds all the elements of a winning-set. If all positions are taken and no player wins, then it is a draw. Classic Tic-tac-toe is an example of a strong positional game. First player advantage In a strong positional game, Second cannot have a winning strategy. This can be proved by a strategy-stealing argument: if Second had a winning strategy, then First could have stolen it and win too, but this is impossible since there is only one winner. Therefore, for every strong-positional game there are only two options: either First has a winning strategy, or Second has a drawing strategy ... [...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 mathematics, 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, arithmet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimum Poset Game
In mathematical analysis, the maximum and minimum of a function are, respectively, the greatest and least value taken by the function. Known generically as extremum, they may be defined either within a given range (the ''local'' or ''relative'' extrema) or on the entire domain (the ''global'' or ''absolute'' extrema) of a function. Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions. As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum. In statistics, the corresponding concept is the sample maximum and minimum. Definition A real-valued function ''f'' defined on a domain ''X'' has a global (or absolute) maximum point at ''x''∗, if for all ''x'' in ''X''. Similarly, the function has a global (or absolute) minimum point at ''x''∗, if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PSPACE-complete
In computational complexity theory, a decision problem is PSPACE-complete if it can be solved using an amount of memory that is polynomial in the input length (PSPACE, polynomial space) and if every other problem that can be solved in polynomial space can be Polynomial-time reduction, transformed to it in polynomial time. The problems that are PSPACE-complete can be thought of as the hardest problems in PSPACE, the class of decision problems solvable in polynomial space, because a solution to any one such problem could easily be used to solve any other problem in PSPACE. Problems known to be PSPACE-complete include determining properties of regular expressions and context-sensitive grammars, determining the truth of quantified Boolean formula problem, quantified Boolean formulas, step-by-step changes between solutions of combinatorial optimization problems, and many puzzles and games. Theory A problem is defined to be PSPACE-complete if it can be solved using a polynomial amount o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chomp
Chomp is a two-player strategy game played on a rectangular grid made up of smaller square cells, which can be thought of as the blocks of a chocolate bar. The players take it in turns to choose one block and "eat it" (remove from the board), together with those that are below it and to its right. The top left block is "poisoned" and the player who eats this loses. The chocolate-bar formulation of Chomp is due to David Gale, but an equivalent game expressed in terms of choosing divisors of a fixed integer was published earlier by Frederik Schuh. Chomp is a special case of a poset game where the partially ordered set on which the game is played is a product of total orders with the minimal element (poisonous block) removed. Example game Below shows the sequence of moves in a typical game starting with a 5 × 4 bar: Player A eats two blocks from the bottom right corner; Player B eats three from the bottom row; Player A picks the block to the right of the poisoned block an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of The Excluded Middle
In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true. It is one of the three laws of thought, along with the law of noncontradiction and the law of identity; however, no system of logic is built on just these laws, and none of these laws provides inference rules, such as modus ponens or De Morgan's laws. The law is also known as the law/principle of the excluded third, in Latin ''principium tertii exclusi''. Another Latin designation for this law is ''tertium non datur'' or "no third ossibilityis given". In classical logic, the law is a tautology. In contemporary logic the principle is distinguished from the semantical principle of bivalence, which states that every proposition is either true or false. The principle of bivalence always implies the law of excluded middle, while the converse is not always true. A commonly cited counterexample uses statements unprovable now, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ko Fight
A ''ko'' ( Japanese: コウ, 劫, ''kō'', from the translation of the Sanskrit term kalpa) fight is a tactical and strategic phase that can arise in the game of Go. ''Ko'' threats and ''ko'' fights The existence of ''ko'' fights is implied by the rule of ko, a special rule of the game that prevents immediate repetition of position, by a short 'loop' in which a single stone is captured, and another single stone immediately taken back. The rule states that the immediate recapture is forbidden, for one turn only. This gives rise to the following procedure: the 'banned' player makes a play, which may have no particular good qualities, but which demands an instant reply. Then the ban has come to its end, and recapture is possible. This kind of distracting play is termed a ''ko threat''. If White, say, chooses to play a ko threat, and Black responds to the threat instead of ending the ko in some fashion, then White can recapture the stone that began the ko. This places Black in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ladder (Go)
In the game of Go, a ,() is a basic sequence of moves in which an attacker pursues a group in atari in a zig-zag pattern across the board. If there are no intervening stones, the group will hit the edge of the board and be captured. The sequence is so basic that there is a Go proverb saying "''if you don't know ladders, don't play Go.''" The ladder tactic fails if there are stones supporting those being chased close enough to the diagonal path of the ladder. Such a failing ladder is called a broken ladder. Secondary double threat tactics around ladders, involving playing a stone in such a way as to break the ladder and also create some other possibility, are potentially very complex. Such a play is called a ''ladder breaker''. A ladder can require reading 50 or more moves ahead, which even amateur players can do, as most of the moves are forced. Although ladders are one of the first techniques which human players learn, AlphaGo Zero was only able to handle them much later in it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Komidashi
in the game of Go are points added to the score of the player with the white stones as compensation for playing second. The value of Black's first-move advantage is generally considered to be between 5 and 7 points by the end of the game. Standard is 6.5 points under the Japanese and Korean rules; under Chinese, Ing and AGA rules standard is 7.5 points; under New Zealand rules standard is 7 points. typically applies only to games where both players are evenly ranked. In the case of a one-rank difference, the stronger player will typically play with the white stones and players often agree on a simple 0.5-point to break a tie ( ) in favour of white, or no at all. is the more complete Japanese language term. The Chinese term is tiē mù ( zh, t=貼目, s=贴目) and the Korean term is deom (). Efforts have been made to determine the value of for boards much smaller than the standard 19x19 grid for go, such as 7x7. When introducing Environmental Go, Elwyn Berlekamp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
