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Go And Mathematics
The game of Go is one of the most popular games in the world. As a result of its elegant and simple rules, the game has long been an inspiration for mathematical research. Shen Kuo, an 11th century Chinese scholar, estimated in his ''Dream Pool Essays'' that the number of possible board positions is around 10172. In more recent years, research of the game by John H. Conway led to the invention of the surreal numbers and contributed to development of combinatorial game theory (with Go Infinitesimals being a specific example of its use in Go). Computational complexity Generalized Go is played on ''n'' × ''n'' boards, and the computational complexity of determining the winner in a given position of generalized Go depends crucially on the ko rules. Go is “almost” in PSPACE, since in normal play, moves are not reversible, and it is only through capture that there is the possibility of the repeating patterns necessary for a harder complexity. Without ko Without ko, Go is PSPACE ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Go (board 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. When ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Checkers
Checkers (American English), also known as draughts (; British English), is a group of strategy board games for two players which involve diagonal moves of uniform game pieces and mandatory captures by jumping over opponent pieces. Checkers is developed from alquerque. The term "checkers" derives from the checkered board which the game is played on, whereas "draughts" derives from the verb "to draw" or "to move". The most popular forms of checkers in Anglophone countries are American checkers (also called English draughts), which is played on an 8×8 checkerboard; Russian draughts, Turkish draughts both on an 8x8 board, and International draughts, played on a 10×10 board – the latter is widely played in many countries worldwide. There are many other variants played on 8×8 boards. Canadian checkers and Singaporean/Malaysian checkers (also locally known as ''dum'') are played on a 12×12 board. American checkers was weakly solved in 2007 by a team of Canadian computer ... [...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] |
Christos Papadimitriou
Christos Charilaos Papadimitriou ( el, Χρήστος Χαρίλαος "Χρίστος" Παπαδημητρίου; born August 16, 1949) is a Greek theoretical computer scientist and the Donovan Family Professor of Computer Science at Columbia University. Education Papadimitriou studied at the National Technical University of Athens, where in 1972 he received his Bachelor of Arts degree in electrical engineering. He then pursued graduate studies at Princeton University, where he received his Ph.D. in electrical engineering and computer science in 1976 after completing a doctoral dissertation titled "The complexity of combinatorial optimization problems." Career Papadimitriou has taught at Harvard, MIT, the National Technical University of Athens, Stanford, UCSD, University of California, Berkeley and is currently the Donovan Family Professor of Computer Science at Columbia University. Papadimitriou co-authored a paper on pancake sorting with Bill Gates, then a Harvard undergra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
New York Times
''The New York Times'' (''the Times'', ''NYT'', or the Gray Lady) is a daily newspaper based in New York City with a worldwide readership reported in 2020 to comprise a declining 840,000 paid print subscribers, and a growing 6 million paid digital media, digital subscribers. It also is a producer of popular podcasts such as ''The Daily (podcast), The Daily''. Founded in 1851 by Henry Jarvis Raymond and George Jones (publisher), George Jones, it was initially published by Raymond, Jones & Company. The ''Times'' has won List of Pulitzer Prizes awarded to The New York Times, 132 Pulitzer Prizes, the most of any newspaper, and has long been regarded as a national "newspaper of record". For print it is ranked List of newspapers by circulation, 18th in the world by circulation and List of newspapers in the United States, 3rd in the U.S. The paper is owned by the New York Times Company, which is Public company, publicly traded. It has been governed by the Sulzberger family since 189 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shannon Number
The Shannon number, named after the American mathematician Claude Shannon, is a conservative lower bound of the game-tree complexity of chess of 10120, based on an average of about 103 possibilities for a pair of moves consisting of a move for White followed by a move for Black, and a typical game lasting about 40 such pairs of moves. Shannon's calculation Shannon showed a calculation for the lower bound of the game-tree complexity of chess, resulting in about 10120 possible games, to demonstrate the impracticality of solving chess by brute force, in his 1950 paper "Programming a Computer for Playing Chess". (This influential paper introduced the field of computer chess.) Shannon also estimated the number of possible positions, "of the general order of \scriptstyle \frac, or roughly 1043". This includes some illegal positions (e.g., pawns on the first rank, both kings in check) and excludes legal positions following captures and promotions. After each player has mov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
God's Algorithm
God's algorithm is a notion originating in discussions of ways to solve the Rubik's Cube puzzle, but which can also be applied to other combinatorial puzzles and mathematical games. It refers to any algorithm which produces a solution having the fewest possible moves. The allusion to the Deity is based on an assumption that only an omniscient being would know an optimal step from any given configuration. Scope Definition The notion applies to puzzles that can assume a finite number of "configurations", with a relatively small, well-defined arsenal of "moves" that may be applicable to configurations and then lead to a new configuration. Solving the puzzle means to reach a designated "final configuration", a singular configuration, or one of a collection of configurations. To solve the puzzle a sequence of moves is applied, starting from some arbitrary initial configuration. Solution An algorithm can be considered to solve such a puzzle if it takes as input an arbitrary initi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Game Complexity
Combinatorial game theory has several ways of measuring game complexity. This article describes five of them: state-space complexity, game tree size, decision complexity, game-tree complexity, and computational complexity. Measures of game complexity State-space complexity The state-space complexity of a game is the number of legal game positions reachable from the initial position of the game. When this is too hard to calculate, an upper bound can often be computed by also counting (some) illegal positions, meaning positions that can never arise in the course of a game. Game tree size The game tree size is the total number of possible games that can be played: the number of leaf nodes in the game tree rooted at the game's initial position. The game tree is typically vastly larger than the state space because the same positions can occur in many games by making moves in a different order (for example, in a tic-tac-toe game with two X and one O on the board, this position co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Googolplex
A googolplex is the number 10, or equivalently, 10 or 1010,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 . Written out in ordinary decimal notation, it is 1 followed by 10100 zeroes; that is, a 1 followed by a googol of zeroes. History In 1920, Edward Kasner's nine-year-old nephew, Milton Sirotta, coined the term ''googol'', which is 10, and then proposed the further term ''googolplex'' to be "one, followed by writing zeroes until you get tired". Kasner decided to adopt a more formal definition because "different people get tired at different times and it would never do to have Carnera ea better mathematician than Dr. Einstein, simply because he had more endurance and could write for longer". It thus became standardized to 10(10100) = 1010100, due to the right-associativity of exponentiation. Size A typical book can be printed with 10 zeros (around 400 pages with 50 lines per page and 50 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Game-tree Complexity
Combinatorial game theory has several ways of measuring game complexity. This article describes five of them: state-space complexity, game tree size, decision complexity, game-tree complexity, and computational complexity. Measures of game complexity State-space complexity The state-space complexity of a game is the number of legal game positions reachable from the initial position of the game. When this is too hard to calculate, an upper bound can often be computed by also counting (some) illegal positions, meaning positions that can never arise in the course of a game. Game tree size The game tree size is the total number of possible games that can be played: the number of leaf nodes in the game tree rooted at the game's initial position. The game tree is typically vastly larger than the state space because the same positions can occur in many games by making moves in a different order (for example, in a tic-tac-toe game with two X and one O on the board, this position co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Victor Allis
Louis Victor Allis (born 19 May 1965) is a Dutch computer scientist working in the artificial intelligence (AI) field. In his graduate work, he revealed AI solutions for Connect Four, Qubic, and Gomoku. His dissertation introduced two new game search techniques: proof-number search and dependency-based search. Proof-number search has seen further successful application in computer Go tactical search and many other games. Career Allis holds a Ph.D. in Artificial Intelligence from Maastricht University, The Netherlands, and graduated ''cum laude'' with a M. Sc. in Computer Science from the Vrije Universiteit, The Netherlands. He has more than 30 publications to his name; the majority of his published work reports on research in search technologies. He started his career in 1987 as a freelance teacher, course developer and mentor of various AMBI courses for NOVI. Allis has lectured at the Vrije Universiteit in Amsterdam as an assistant professor in artificial intelligence. In 19 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Scientist
A computer scientist is a person who is trained in the academic study of computer science. Computer scientists typically work on the theoretical side of computation, as opposed to the hardware side on which computer engineers mainly focus (although there is overlap). Although computer scientists can also focus their work and research on specific areas (such as algorithm and data structure development and design, software engineering, information theory, database theory, computational complexity theory, numerical analysis, programming language theory, computer graphics, and computer vision), their foundation is the theoretical study of computing from which these other fields derive. A primary goal of computer scientists is to develop or validate models, often mathematical, to describe the properties of computational systems (processors, programs, computers interacting with people, computers interacting with other computers, etc.) with an overall objective of discovering des ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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