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Warming (combinatorial Game Theory)
In combinatorial game theory, cooling, heating, and overheating are operations on hot games to make them more amenable to the traditional methods of the theory, which was originally devised for cold games in which the winner is the last player to have a legal move. Overheating was generalised by Elwyn Berlekamp for the analysis of Blockbusting. Chilling (or unheating) and warming are variants used in the analysis of the endgame of Go. Cooling and chilling may be thought of as a tax on the player who moves, making them pay for the privilege of doing so, while heating, warming and overheating are operations that more or less reverse cooling and chilling. Basic operations: cooling, heating The cooled game G_t (" G cooled by t ") for a game G and a (surreal) number t is defined by :: G_t = \begin \ & \text t \leq \text \tau \text G_\tau \text m \text\\ m & \text t > \tau \end . The amount t by which G is cooled is known as the ''temperature''; the minimum \tau ... [...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] |
Hot Game
__NOTOC__ In combinatorial game theory, a branch of mathematics, a hot game is one in which each player can improve their position by making the next move. By contrast, a cold game is one where each player can only worsen their position by making the next move. The class of cold games are equivalent to the class of surreal numbers and so can be ordered by value, while hot games can have other values. There are also tepid games, which are games with a temperature of exactly zero. Tepid games are formed by the class of strictly numerish games: that is, games that are equivalent to a number plus an infinitesimal. Hackenbush can only represent tepid and cold games (by its decomposition into a purple mountain and a green jungle). Example For example, consider a game in which players alternately remove tokens of their own color from a table, the Blue player removing only blue tokens and the Red player removing only red tokens, with the winner being the last player to remove a t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elwyn Berlekamp
Elwyn Ralph Berlekamp (September 6, 1940 – April 9, 2019) was a professor of mathematics and computer science at the University of California, Berkeley.Elwyn Berlekamp listing at the Department of Mathematics, . Berlekamp was widely known for his work in computer science, and . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blockbusting (game)
Blockbusting is a two-player game in which players alternate choosing squares from a line of squares, with one player aiming to choose as many pairs of adjacent squares as possible and the other player aiming to thwart this goal. Elwyn Berlekamp introduced it in 1987, as an example for a theoretical construction in combinatorial game theory. Rules Blockbusting is a partisan game for two players, meaning that the roles of the two players are not symmetric. These two players are often known as Red and Blue (or Right and Left); they play the game on an n \times 1 strip of squares called "parcels". Each player, in turn, claims and colors one previously unclaimed parcel until all parcels have been claimed. At the end, Left's score is the number of pairs of neighboring parcels both of which he has claimed. Left therefore tries to maximize that number while Right tries to minimize it. Adjacent Right-Right pairs do not affect the score. Although the purpose of the game is to further th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Go (game)
# Go is an abstract strategy game, abstract strategy board game for two players in which the aim is to fence off 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 Game piece (board game), playing pieces are called ''Go equipment#Stones, stones''. One player uses the white stones and the other black stones. The players take turns placing their stones on the vacant intersections (''points'') on the #Boards, board. Once placed, stones may not be moved, but ''captured stones'' are immediately removed from the board. A single stone (or connected group of stones) is ''captured'' when surrounded by the opponent's stones on all Orthogona ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surreal Number
In mathematics, the surreal number system is a total order, totally ordered proper class containing not only the real numbers but also Infinity, infinite and infinitesimal, infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. Research on the Go endgame by John Horton Conway led to the original definition and construction of surreal numbers. Conway's construction was introduced in Donald Knuth's 1974 book ''Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness''. The surreals share many properties with the reals, including the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field. If formulated in von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal ordered field in the sense that all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. the other being Derivative, differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter. A definite integral computes the signed area of the region in the plane that is bounded by the Graph of a function, graph of a given Function (mathematics), function between two points in the real line. Conventionally, areas above the horizontal Coordinate axis, axis of the plane are positive while areas below are n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplication
Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division (mathematics), division. The result of a multiplication operation is called a ''Product (mathematics), product''. Multiplication is often denoted by the cross symbol, , by the mid-line dot operator, , by juxtaposition, or, in programming languages, by an asterisk, . The multiplication of whole numbers may be thought of as repeated addition; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the ''multiplicand'', as the quantity of the other one, the ''multiplier''; both numbers can be referred to as ''factors''. This is to be distinguished from term (arithmetic), ''terms'', which are added. :a\times b = \underbrace_ . Whether the first factor is the multiplier or the multiplicand may be ambiguous or depend upon context. For example, the expression 3 \times 4 , can be phrased as "3 ti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solved Game
A solved game is a game whose outcome (win, lose or tie (draw), 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 or computer assistance. Overview A two-player game can be solved on several levels: Ultra-weak solution : 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 details of the perfect play. Weak solution : Provide one algorithm for each of the two players, such that the player using it can achieve at least the optimal outcome, regardless of the opponent's moves, from the start of the game, using reasonable computational resources. Strong solution : Provide an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Go Endgame
The game of Go has simple rules that can be learned very quickly but, as with chess and similar board games, complex strategies may be employed by experienced players. Go opening theory The whole board opening is called fuseki. An important principle to follow in early play is "corner, side, center." In other words, the corners are the easiest places to take territory, because two sides of the board can be used as boundaries. Once the corners are occupied, the next most valuable points are along the sides, aiming to use the edge as a territorial boundary. Capturing territory in the middle, where it must be surrounded on all four sides, is extremely difficult. The same is true for founding a living group: Easiest in the corner, most difficult in the center. The first moves are usually played on or near the 4-4 star points in the corners, because in those places it is easiest to gain territory or influence. In order to be totally secure alone, a corner stone must be placed on t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
