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Cooling And Heating (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\\ G_t = m & \text t > \tau \end . The amount t by which G is cooled is known as the ''temperature''; the minimum \ ... [...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] |
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. Cold games have values in the surreal numbers and so can be ordered by value, while hot games can have other values. 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 token. Obviously, victory will go to the player who starts off with more tokens, or to the second player if the number of red and blue tokens are equal. Removing a token of one's own color leaves the position slightly worse for the player who made the move, since that player now has fewer tokens on the table. Thus each token represents a "cold" component o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cold 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. Cold games have values in the surreal numbers and so can be ordered by value, while hot games can have other values. 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 token. Obviously, victory will go to the player who starts off with more tokens, or to the second player if the number of red and blue tokens are equal. Removing a token of one's own color leaves the position slightly worse for the player who made the move, since that player now has fewer tokens on the table. Thus each token represents a "cold" component o ... [...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.Contributors, ''IEEE Transactions on Information Theory'' 42, #3 (May 1996), p. 1048. DO10.1109/TIT.1996.490574Elwyn Berlekamp listing at the Department of Mathematics, . Berlekamp was widely known for his work in computer science, and |
Blockbusting (game)
Blockbusting is a solved combinatorial game introduced in 1987 by Elwyn Berlekamp illustrating a generalisation of overheating. The analysis of Blockbusting may be used as the basis of a strategy for the combinatorial game of Domineering. Blockbusting is a partisan game for two players known as Red and Blue (or Right and Left) played 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 the study of combinatorial game theory, Berlekamp provides an interpretation alluding to the practice of blockbusting by real estate agents: the players may be seen as rival agents buying up all the parcels on a ... [...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] |
Surreal Number
In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. 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 numbers (including the hyperreal numbers) can be realized as subfields of the surreals. The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations. It has also been shown (in von Neumann–Bernays–Gödel set theory) that the maximal class hyperreal field is isomorp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral
In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ..., an integral assigns numbers to functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with Derivative, differentiation, integration is a fundamental, essential operation of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others. The integrals enumerated here are those termed definite integrals, which can be int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division. The result of a multiplication operation is called a ''product''. 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''. :a\times b = \underbrace_ For example, 4 multiplied by 3, often written as 3 \times 4 and spoken as "3 times 4", can be calculated by adding 3 copies of 4 together: :3 \times 4 = 4 + 4 + 4 = 12 Here, 3 (the ''multiplier'') and 4 (the ''multiplicand'') are the ''factors'', and 12 is the ''product''. One of the main properties of multiplication is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solved Game
A solved game is a game whose outcome (win, lose or 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 and/or computer assistance. Overview A two-player game can be solved on several levels: ;Ultra-weak : 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 moves of the perfect play. ;Weak : Provide an algorithm that secures a win for one player, or a draw for either, against any possible moves by the opponent, from the beginning of the game. ;Strong : Provide an algorithm that can produce perfect moves from any position, even if mistakes have already been made on one or b ... [...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 deployed 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 side, 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] |
