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First-player-win
In combinatorial game theory, a two-player deterministic perfect information turn-based game is a first-player-win if with perfect play the first player to move can always force a win. Similarly, a game is second-player-win if with perfect play the second player to move can always force a win. With perfect play, if neither side can force a win, the game is a draw. Some games with relatively small game trees have been proven to be first or second-player wins. For example, the game of nim with the classic 3–4–5 starting position is a first-player-win game. However, Nim with the 1-3-5-7 starting position is a second-player-win. The classic game of Connect Four has been mathematically proven to be first-player-win. With perfect play, checkers has been determined to be a draw; neither player can force a win. Another example of a game which leads to a draw with perfect play is tic-tac-toe, and this includes play from any opening move. Significant theory has been completed in the ef ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 in the Soviet Union, Gravitrips) is a game in which the players choose a color and then take turns dropping colored tokens into a six-row, seven-column vertically suspended grid. The pieces fall straight down, occupying the lowest available space within the column. The objective of the game is to be the first to form a horizontal, vertical, or diagonal line of four of one's own tokens. It is therefore a type of ''m'',''n'',''k''-game (7, 6, 4) with restricted piece placement. Connect Four is a solved game. The first player can always win by playing the right moves. The game was created by Howard Wexler, and first sold under the ''Connect Four'' trademark by Milton Bradley in February 1974. Gameplay A gameplay example (right), shows the first player starting Connect Four by dropping one of their yellow discs into the center column of an empty game board. The t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First-move Advantage In Chess
In chess, there is a consensus among players and chess theory, theorists that the player who makes the first move (White and Black in chess, White) has an inherent advantage, albeit not one large enough to win with Solved game#Perfect play, perfect play. This has been the consensus since at least 1889, when the first World Chess Championship, World Chess Champion, Wilhelm Steinitz, addressed the issue, although chess has not been Solving chess, solved. Since 1851, compiled statistics support this view; White consistently slightly more often than White and Black in chess, Black, usually achieving a between 52 and 56 percent. White's advantage is less significant in Fast chess, blitz games and games between lower-level players, and becomes greater as the level of play rises; however, raising the level of play also increases the percentage of draw (chess), draws. As the standard of play rises, all the way up to top engine level, the number of decisive games approaches zero, an ... [...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] |
Determinacy
Determinacy is a subfield of game theory and set theory 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 Gale–St ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zugzwang
Zugzwang (; ) is a situation found in chess and other turn-based games wherein one player is put at a disadvantage because of their obligation to make a move; a player is said to be "in zugzwang" when any legal move will worsen their position. Although the term is used less precisely in games such as chess, it is used specifically in combinatorial game theory to denote a move that directly changes the outcome of the game from a win to a loss. Putting the opponent in zugzwang is a common way to help the superior side win a game, and in some cases it is necessary in order to make the win possible. More generally, the term can also be used to describe a situation where none of the available options lead to a good outcome. The term ''zugzwang'' was used in German chess literature in 1858 or earlier, and the first known use of the term in English was by World Champion Emanuel Lasker in 1905. The concept of zugzwang was known to chess players many centuries before the term was coine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
University Of Limburg
Maastricht University (abbreviated as UM; ) is a public research university in Maastricht, Netherlands. Founded in 1976, it is the second youngest of the thirteen Dutch universities. In 2023, 22,976 students studied at Maastricht University, 59% of whom were students from outside of the Netherlands, with over 4,000 employees. About half of the bachelor's programmes are fully offered in English, while the other half is taught wholly or partly in Dutch. Most of the master's and doctoral programmes are in English. Besides traditional programmes, Maastricht University also has three honours liberal arts colleges: University College Maastricht and the Maastricht Science Programme in the same liberal arts tradition. The satellite University College Venlo opened in 2015. History Maastricht University was officially established in 1976. Faced with a shortage of medical professionals, the Dutch government decided in the late 1960s that a new public institution of higher education was ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philosophical Magazine
The ''Philosophical Magazine'' is one of the oldest scientific journals published in English. It was established by Alexander Tilloch in 1798;John Burnett"Tilloch, Alexander (1759–1825)" Dictionary of National Biography#Oxford Dictionary of National Biography, Oxford Dictionary of National Biography, Oxford University Press, Sept 2004; online edn, May 2006, accessed 17 Feb 2010 in 1822 Richard Taylor (editor), Richard Taylor became joint editor and it has been published continuously by Taylor & Francis ever since. Early history The name of the journal dates from a period when "natural philosophy" embraced all aspects of science. The very first paper published in the journal carried the title "Account of Mr Cartwright's Patent Steam Engine". Other articles in the first volume include "Methods of discovering whether Wine has been adulterated with any Metals prejudicial to Health" and "Description of the Apparatus used by Lavoisier to produce Water from its component Parts, Oxyg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chess Engine
In computer chess, a chess engine is a computer program that analyzes chess or List of chess variants, chess variant positions, and generates a move or list of moves that it regards as strongest. A chess software engine, engine is usually a Front and back ends, back end with a command-line interface with no graphics or windowing system, windowing. Engines are usually used with a front end, a windowed graphical user interface such as Chessbase or WinBoard that the user can interact with via a keyboard, mouse or touchscreen. This allows the user to play against multiple engines without learning a new user interface for each, and allows different engines to play against each other. Many chess engines are now available for mobile phones and tablets, making them even more accessible. History The meaning of the term "chess engine" has evolved over time. In 1986, Linda and Tony Scherzer entered their program Bebe into the 4th World Computer Chess Championship, running it on "Chess E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solving Chess
Solving chess consists of finding an optimal strategy for the game of chess; that is, one by which one of the players ( White or Black) can always force either a victory or a draw (see solved game). It is also related to more generally solving ''chess-like'' games (i.e. combinatorial games of perfect information) such as Capablanca chess and infinite chess. In a weaker sense, ''solving chess'' may refer to proving which one of the three possible outcomes (White wins; Black wins; draw) is the result of two perfect players, without necessarily revealing the optimal strategy itself (see indirect proof). No complete solution for chess in either of the two senses is known, nor is it expected that chess will be solved in the near future (if ever). Progress to date is extremely limited; there are tablebases of perfect endgame play with a small number of pieces (up to seven), and some chess variants have been solved at least weakly. Calculated estimates of game-tree complexity and st ... [...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] |
