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Zero Game
In combinatorial game theory, the zero game is the game where neither player has any legal options. Therefore, under the normal play convention, the first player automatically loses, and it is a second-player win. The zero game has a Sprague–Grundy value of zero. The combinatorial notation of the zero game is: .. A zero game should be contrasted with the star game , which is a first-player win since either player must (if first to move in the game) move to a zero game, and therefore win. Examples Simple examples of zero games include Nim with no piles or a Hackenbush diagram with nothing drawn on it. Sprague-Grundy value The Sprague–Grundy theorem applies to impartial game In combinatorial game theory, an impartial game is a game in which the allowable moves depend only on the position and not on which of the two players is currently moving, and where the payoffs are symmetric. In other words, the only difference bet ...s (in which each move may be played by either pl ... [...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. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Play Convention
A normal play convention in a game is the method of determining the winner that is generally regarded as standard. For example: *Preventing the other player from being able to move *Being the first player to achieve a target *Holding the highest value hand *Taking the most card tricks In combinatorial game theory, the normal play convention of an impartial game In combinatorial game theory, an impartial game is a game in which the allowable moves depend only on the position and not on which of the two players is currently moving, and where the payoffs are symmetric. In other words, the only difference betw ... is that the last player able to move is the winner. By contrast " misère games" involve upsetting the convention and declaring a winner the individual who would normally be considered the loser. Gaming {{Game-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star (game Theory)
In combinatorial game theory, star, written as * or *1, is the value given to the game where both players have only the option of moving to the zero game. Star may also be denoted as the surreal form . This game is an unconditional first-player win. Star, as defined by John Conway in '' Winning Ways for your Mathematical Plays'', is a value, but not a number in the traditional sense. Star is not zero, but neither positive nor negative, and is therefore said to be ''fuzzy'' and ''confused with'' (a fourth alternative that means neither "less than", "equal to", nor "greater than") 0. It is less than all positive rational numbers, and greater than all negative rationals. Games other than may have value *. For example, the game *2 + *3, where the values are nimbers, has value * despite each player having more options than simply moving to 0. Why * ≠ 0 A combinatorial game has a positive and negative player; which player moves first is left ambiguous. The combinatorial game& ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hackenbush
Hackenbush is a two-player game invented by mathematician John Horton Conway. It may be played on any configuration of colored line segments connected to one another by their endpoints and to a "ground" line. Gameplay The game starts with the players drawing a "ground" line (conventionally, but not necessarily, a horizontal line at the bottom of the paper or other playing area) and several line segments such that each line segment is connected to the ground, either directly at an endpoint, or indirectly, via a chain of other segments connected by endpoints. Any number of segments may meet at a point and thus there may be multiple paths to ground. On their turn, a player "cuts" (erases) any line segment of their choice. Every line segment no longer connected to the ground by any path "falls" (i.e., gets erased). According to the normal play convention of combinatorial game theory, the first player who is unable to move loses. Hackenbush boards can consist of finitely many (in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sprague–Grundy Theorem
In combinatorial game theory, the Sprague–Grundy theorem states that every impartial game under the normal play convention is equivalent to a one-heap game of nim, or to an infinite generalization of nim. It can therefore be represented as a natural number, the size of the heap in its equivalent game of nim, as an ordinal number in the infinite generalization, or alternatively as a nimber, the value of that one-heap game in an algebraic system whose addition operation combines multiple heaps to form a single equivalent heap in nim. The Grundy value or nim-value of any impartial game is the unique nimber that the game is equivalent to. In the case of a game whose positions are indexed by the natural numbers (like nim itself, which is indexed by its heap sizes), the sequence of nimbers for successive positions of the game is called the nim-sequence of the game. The Sprague–Grundy theorem and its proof encapsulate the main results of a theory discovered independently by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Impartial Game
In combinatorial game theory, an impartial game is a game in which the allowable moves depend only on the position and not on which of the two players is currently moving, and where the payoffs are symmetric. In other words, the only difference between player 1 and player 2 is that player 1 goes first. The game is played until a terminal position is reached. A terminal position is one from which no moves are possible. Then one of the players is declared the winner and the other the loser. Furthermore, impartial games are played with perfect information and no chance moves, meaning all information about the game and operations for both players are visible to both players. Impartial games include Nim, Sprouts, Kayles, Quarto, Cram, Chomp, Subtract a square, Notakto, and poset games. Go and chess are not impartial, as each player can only place or move pieces of their own color. Games such as poker, dice Dice (singular die or dice) are small, throwable objects with marked si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fuzzy Game
In combinatorial game theory, a fuzzy game is a game which is ''incomparable'' with the zero game: it is not greater than 0, which would be a win for Left; nor less than 0 which would be a win for Right; nor equal to 0 which would be a win for the second player to move. It is therefore a first-player win. Classification of games In combinatorial game theory, there are four types of game. If we denote players as Left and Right, and G be a game with some value, we have the following types of game: 1. Left win: G > 0 :No matter which player goes first, Left wins. 2. Right win: G < 0 :No matter which player goes first, Right wins. 3. Second player win: G = 0 :The first player (Left or Right) has no moves, and thus loses. 4. First player win: G ║ 0 (G is fuzzy with 0) :The first player (Left or Right) wins. Using standard Dedekind-section game notation, , where L is the list of undominated move ... [...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. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
