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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] |
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] |
Surreal Numbers
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 isomorphic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Snort (game)
Col is a pencil and paper game, specifically a map-coloring game, involving the shading of areas in a line drawing according to the rules of graph coloring. With each move, the graph must remain proper (no two areas of the same colour may touch), and a player who cannot make a legal move loses. The game was described and analysed by John Conway, who attributed it to Colin Vout, in ''On Numbers and Games''. Example game In the following game, the first of the two players is using red, and the second is using blue. The last move in each image is shown brighter than the other areas. The starting graph: image:ColAndSnortGraph blank.png The first player may colour any of the areas to begin. However, the region around the outside of the graph is not included as an area for this game. After the first move: image:ColAndSnortGraph C1.png The second player now colours a white cell. As no areas are currently blue, any white cell is allowed. Two moves in: image:ColAndSnortGraph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Col (game)
Col is a pencil and paper game, specifically a map-coloring games, map-coloring game, involving the shading of areas in a line drawing according to the rules of graph coloring. With each move, the graph must remain Graph coloring#Vertex coloring, proper (no two areas of the same colour may touch), and a player who cannot make a legal move loses. The game was described and analysed by John Horton Conway, John Conway, who attributed it to Colin Vout, in ''On Numbers and Games''. Example game In the following game, the first of the two players is using red, and the second is using blue. The last move in each image is shown brighter than the other areas. The starting graph: image:ColAndSnortGraph blank.png The first player may colour any of the areas to begin. However, the region around the outside of the graph is not included as an area for this game. After the first move: image:ColAndSnortGraph C1.png The second player now colours a white cell. As no areas are currently blu ... [...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] |
Domineering
Domineering (also called Stop-Gate or Crosscram) is a mathematical game that can be played on any collection of squares on a sheet of graph paper. For example, it can be played on a 6×6 square, a rectangle, an entirely irregular polyomino, or a combination of any number of such components. Two players have a collection of dominoes which they place on the grid in turn, covering up squares. One player places tiles vertically, while the other places them horizontally. (Traditionally, these players are called "Left" and "Right", respectively, or "V" and "H". Both conventions are used in this article.) As in most games in combinatorial game theory, the first player who cannot move loses. Domineering is a partisan game, in that players use different pieces: the impartial version of the game is Cram. Basic examples Single box Other than the empty game, where there is no grid, the simplest game is a single box. In this game, clearly, neither player can move. Since it is a second-pl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
New York City
New York, often called New York City or NYC, is the List of United States cities by population, most populous city in the United States. With a 2020 population of 8,804,190 distributed over , New York City is also the List of United States cities by population density, most densely populated major city in the United States, and is more than twice as populous as second-place Los Angeles. New York City lies at the southern tip of New York (state), New York State, and constitutes the geographical and demographic center of both the Northeast megalopolis and the New York metropolitan area, the largest metropolitan area in the world by urban area, urban landmass. With over 20.1 million people in its metropolitan statistical area and 23.5 million in its combined statistical area as of 2020, New York is one of the world's most populous Megacity, megacities, and over 58 million people live within of the city. New York City is a global city, global Culture of New ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Academic Press
Academic Press (AP) is an academic book publisher founded in 1941. It was acquired by Harcourt, Brace & World in 1969. Reed Elsevier bought Harcourt in 2000, and Academic Press is now an imprint of Elsevier. Academic Press publishes reference books, serials and online products in the subject areas of: * Communications engineering * Economics * Environmental science * Finance * Food science and nutrition * Geophysics * Life sciences * Mathematics and statistics * Neuroscience * Physical sciences * Psychology Well-known products include the ''Methods in Enzymology'' series and encyclopedias such as ''The International Encyclopedia of Public Health'' and the ''Encyclopedia of Neuroscience''. See also * Akademische Verlagsgesellschaft (AVG) — the German predecessor, founded in 1906 by Leo Jolowicz (1868–1940), the father of Walter Jolowicz Walter may refer to: People * Walter (name), both a surname and a given name * Little Walter, American blues harmonica player Marion Wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
A K Peters, Ltd
A K Peters, Ltd. was a publisher of scientific and technical books, specializing in mathematics and in computer graphics, robotics, and other fields of computer science. They published the journals ''Experimental Mathematics'' and the '' Journal of Graphics Tools'', as well as mathematics books geared to children. Background Klaus Peters wrote a doctoral dissertation on complex manifolds at the University of Erlangen in 1962, supervised by Reinhold Remmert. He then joined Springer Verlag, becoming their first specialist mathematics editor. As a Springer director from 1971, he hired Alice Merker for Springer New York: they were married that year, and moved to Heidelberg. Leaving Springer, they founded Birkhäuser Boston in 1979; Birkhäuser ran into financial difficulties, and was taken over by Springer. Klaus and Alice then spent a period running a Boston office for Harcourt Brace Jovanovich and their imprint Academic Press. With the takeover of Harcourt Brace Jovanovich by Gener ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
