# Chess - Mathematics and Computers

Mathematics and Computers

See also: Computer chess, List of mathematicians who studied chess, Human-computer chess matches, Deep Blue versus Garry Kasparov, and Chess engine

The game structure and nature of chess is related to several branches of mathematics. Many combinatorical and topological problems connected to chess were known of for hundreds of years. In 1913, Ernst Zermelo used chess as a basis for his theory of game strategies, which is considered as one of the predecessors of game theory.

The number of legal positions in chess is estimated to be between 1043 and 1047 (a provable upper bound), with a game-tree complexity of approximately 10123. The game-tree complexity of chess was first calculated by Claude Shannon as 10120, a number known as the Shannon number. Typically an average position has thirty to forty possible moves, but there may be as few as zero (in the case of checkmate or stalemate) or as many as 218.

One of the most important mathematical challenges of chess is the development of algorithms that can play chess. The idea of creating a chess-playing machine dates to the 18th century; around 1769, the chess-playing automaton called The Turk became famous before being exposed as a hoax. Serious trials based on automatons, such as El Ajedrecista, were too complex and limited to be useful.

Since the advent of the digital computer in the 1950s, chess enthusiasts, computer engineers and computer scientists have built, with increasing degrees of seriousness and success, chess-playing machines and computer programs. The groundbreaking paper on computer chess, "Programming a Computer for Playing Chess", was published in 1950 by Shannon. He wrote:

The chess machine is an ideal one to start with, since: (1) the problem is sharply defined both in allowed operations (the moves) and in the ultimate goal (checkmate); (2) it is neither so simple as to be trivial nor too difficult for satisfactory solution; (3) chess is generally considered to require "thinking" for skillful play; a solution of this problem will force us either to admit the possibility of a mechanized thinking or to further restrict our concept of "thinking"; (4) the discrete structure of chess fits well into the digital nature of modern computers.

The Association for Computing Machinery (ACM) held the first major chess tournament for computers, the North American Computer Chess Championship, in September 1970. CHESS 3.0, a chess program from Northwestern University, won the championship. Nowadays, chess programs compete in the World Computer Chess Championship, held annually since 1974. At first considered only a curiosity, the best chess playing programs, for example Rybka, have become extremely strong. In 1997, a computer won a chess match against a reigning World Champion for the first time: IBM's Deep Blue beat Garry Kasparov 3½–2½ (it scored two wins, one loss, and three draws). In 2009, a mobile phone won a category 6 tournament with a performance rating 2898: chess engine Hiarcs 13 running on the mobile phone HTC Touch HD won the Copa Mercosur tournament with nine wins and one draw. The best chess programs are now able to beat the strongest human players.

With huge databases of past games and high analytical ability, computers can help players to learn chess and prepare for matches. Internet Chess Servers allow people to find and play opponents all over the world. The presence of computers and modern communication tools have raised concerns regarding cheating during games, most notably the "bathroom controversy" during the 2006 World Championship.

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