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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 tell you that there exists a winning strategy, the proof gives you 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 disa ... [...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. Study has been largely confined to two-player games that have a ''position'' that the players take turns changing in defined ways or ''moves'' to achieve a defined winning condition. Combinatorial game theory has not traditionally studied games of chance or those that use imperfect or incomplete information, favoring games that offer perfect information in which the state of the game and the set of available moves is always known by both players. However, as mathematical techniques advance, the types of game that can be mathematically analyzed expands, thus the boundaries of the field are ever changing. Scholars will generally define what they mean by a "game" at the beginning of a paper, and these definitions often vary as they are specific to the game being analyzed and are not meant to represent the entire scope of the field. C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sylver Coinage
Sylver coinage is a mathematical game for two players, invented by John H. Conway. It is discussed in chapter 18 of '' Winning Ways for Your Mathematical Plays''. This article summarizes that chapter. The two players take turns naming positive integers greater than 1 that are not the sum of nonnegative multiples of previously named integers. The player who cannot name such a number loses. For instance, if player A opens with 2, B can win by naming 3. Sylver coinage is named after James Joseph Sylvester, who proved that if ''a'' and ''b'' are relatively prime positive integers, then (''a'' − 1)(''b'' − 1) − 1 is the largest number that is not a sum of nonnegative multiples of ''a'' and ''b''. Thus, if ''a'' and ''b'' are the first two moves in a game of sylver coinage, this formula gives the largest number that can still be played. More generally, if the greatest common divisor of the moves played so far is ''g'', then only finitely many ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimum Poset Game
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the ''local'' or ''relative'' extrema), or on the entire domain (the ''global'' or ''absolute'' extrema). 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. 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 for all ''x'' in ''X''. The value of the function at a m ... [...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 (polynomial space) and if every other problem that can be solved in polynomial space can be 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 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 of memory (it belongs to PSPACE) and every problem in PSPACE can be tr ... [...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 and e ... [...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, Exclusive or, either this proposition or its negation is Truth value, true. It is one of the so-called Law of thought#The three traditional laws, 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 Rule of inference, inference rules, such as modus ponens or De Morgan's laws. The law is also known as the law (or principle) of the excluded third, in Latin ''principium tertii exclusi''. Another Latin designation for this law is ''tertium non datur'': "no third [possibility] is given". It is a tautology (logic), tautology. The principle should not be confused with 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 i ... [...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 s ... [...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 its ... [...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 () 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 made a broad generalisation of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Go (game)
Go is an abstract strategy board game for two players in which the aim is to surround more territory than the opponent. The game was invented in China more than 2,500 years ago and is believed to be the oldest board game continuously played to the present day. A 2016 survey by the International Go Federation's 75 member nations found that there are over 46 million people worldwide who know how to play Go and over 20 million current players, the majority of whom live in East Asia. The playing pieces are called stones. One player uses the white stones and the other, black. The players take turns placing the stones on the vacant intersections (''points'') of a board. Once placed on the board, stones may not be moved, but stones are removed from the board if the stone (or group of stones) is surrounded by opposing stones on all orthogonally adjacent points, in which case the stone or group is ''captured''. The game proceeds until neither player wishes to make another move. Wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
_-_exemple_de_triple_ko.png "Click for more on -> Ko Fight")
