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 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". [1] (This influential paper introduced the field of computer chess.)
Shannon also estimated the number of possible positions, of the general order of , or roughly 3.7*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.
| Number of plies (half-moves) | Number of possible positions [2] | Number of checkmates [3] |
|---|---|---|
| 1 | 20 | 0 |
| 2 | 400 | 0 |
| 3 | 8,902 | 0 |
| 4 | 197,281 | 8 |
| 5 | 4,865,609 | 347 |
| 6 | 119,060,324 | 10,828 |
| 7 | 3,195,901,860 | 435,767 |
| 8 | 84,998,978,956 | 9,852,036 |
| 9 | 2,439,530,234,167 | 400,191,963 |
| 10 | 69,352,859,712,417 | 8,790,619,155 |
| 11 | 2,097,651,003,696,806 | 362,290,010,907 |
| 12 | 62,854,969,236,701,747 | 8,361,091,858,959 |
| 13 | 1,981,066,775,000,396,239 | 346,742,245,764,219 |
| 14 | 61,885,021,521,585,529,237 | |
| 15 | 2,015,099,950,053,364,471,960 |
After each player has moved a piece 5 times each (10 ply) there are 69,352,859,712,417 possible games that could have been played.
Taking Shannon's numbers into account, Victor Allis calculated an upper bound of 5×1052 for the number of positions, and estimated the true number to be about 1050. [4] Recent results [5] improve that estimate, by proving an upper bound of 8.7x1045, and showing [6] [7] an upper bound 4×1037 in the absence of promotions.
Allis also estimated the game-tree complexity to be at least 10123, "based on an average branching factor of 35 and an average game length of 80". As a comparison, the number of atoms in the observable universe, to which it is often compared, is roughly estimated to be 1080.
John Tromp and Peter Österlund estimated the number of legal chess positions with a 95% confidence level at , based on an efficiently computable bijection between integers and chess positions. [5]
As a comparison to the Shannon number, if chess is analyzed for the number of "sensible" games that can be played (not counting ridiculous or obvious game-losing moves such as moving a queen to be immediately captured by a pawn without compensation), then the result is closer to around 1040 games. This is based on having a choice of about three sensible moves at each ply (half-move), and a game length of 80 plies (or, equivalently, 40 moves). [8]
Chess is a board game for two players, called White and Black, each controlling an army of chess pieces, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to distinguish it from related games such as xiangqi and shogi. The recorded history of chess goes back at least to the emergence of a similar game, chaturanga, in seventh century India. The rules of chess as they are known today emerged in Europe at the end of the 15th century, with standardization and universal acceptance by the end of the 19th century. Today, chess is one of the world's most popular games, and is played by millions of people worldwide.
Claude Elwood Shannon was an American mathematician, electrical engineer, computer scientist and cryptographer known as the "father of information theory". He is credited alongside George Boole for laying the foundations of the Information Age.
Shogi, also known as Japanese chess, is a strategy board game for two players. It is one of the most popular board games in Japan and is in the same family of games as Western chess, chaturanga, xiangqi, Indian chess, and janggi. Shōgi means general's board game.
Hex is a two player abstract strategy board game in which players attempt to connect opposite sides of a rhombus-shaped board made of hexagonal cells. Hex was invented by mathematician and poet Piet Hein in 1942 and later rediscovered and popularized by John Nash.
A solved game is a game whose outcome 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.
Computer chess includes both hardware and software capable of playing chess. Computer chess provides opportunities for players to practice even in the absence of human opponents, and also provides opportunities for analysis, entertainment and training. Computer chess applications that play at the level of a chess grandmaster or higher are available on hardware from supercomputers to smart phones. Standalone chess-playing machines are also available. Stockfish, Leela Chess Zero, GNU Chess, Fruit, and other free open source applications are available for various platforms.
This list contains selected positive numbers in increasing order, including counts of things, dimensionless quantities and probabilities. Each number is given a name in the short scale, which is used in English-speaking countries, as well as a name in the long scale, which is used in some of the countries that do not have English as their national language.
In the context of combinatorial game theory, which typically studies sequential games with perfect information, a game tree is a graph representing all possible game states within such a game. Such games include well-known ones such as chess, checkers, Go, and tic-tac-toe. This can be used to measure the complexity of a game, as it represents all the possible ways a game can pan out. Due to the large game trees of complex games such as chess, algorithms that are designed to play this class of games will use partial game trees, which makes computation feasible on modern computers. Various methods exist to solve game trees. If a complete game tree can be generated, a deterministic algorithm, such as backward induction or retrograde analysis can be used. Randomized algorithms and minmax algorithms such as MCTS can be used in cases where a complete game tree is not feasible.
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.
Connect Four is a game in which the players choose a color and then take turns dropping colored tokens into a six-row, seven-column vertically suspended grid. The pieces fall straight down, occupying the lowest available space within the column. The objective of the game is to be the first to form a horizontal, vertical, or diagonal line of four of one's own tokens. It is therefore a type of M,n,k-game with restricted piece placement. Connect Four is a solved game. The first player can always win by playing the right moves.
Combinatorial game theory measures game complexity in several ways:
In chess, fool's mate is the checkmate delivered after the fewest possible moves from the game's starting position. It arises from the following moves, or similar:
Hexapawn is a deterministic two-player game invented by Martin Gardner. It is played on a rectangular board of variable size, for example on a 3×3 board or on a regular chessboard. On a board of size n×m, each player begins with m pawns, one for each square in the row closest to them. The goal of each player is to either advance a pawn to the opposite end of the board or leave the other player with no legal moves, either by stalemate or by having all of their pieces captured.
In chess, an endgame tablebase, or simply tablebase, is a computerised database containing precalculated evaluations of endgame positions. Tablebases are used to analyse finished games, as well as by chess engines to evaluate positions during play. Tablebases are typically exhaustive, covering every legal arrangement of a specific selection of pieces on the board, with both White and Black to move. For each position, the tablebase records the ultimate result of the game and the number of moves required to achieve that result, both assuming perfect play. Because every legal move in a covered position results in another covered position, the tablebase acts as an oracle that always provides the optimal move.
Connect6 introduced in 2003 by Professor I-Chen Wu at Department of Computer Science and Information Engineering, National Chiao Tung University in Taiwan, is a two-player strategy game similar to Gomoku.
Minichess is a family of chess variants played with regular chess pieces and standard rules, but on a smaller board. The motivation for these variants is to make the game simpler and shorter than standard chess. The first chess-like game implemented on a computer was the 6×6 chess variant Los Alamos chess. The low memory capacity of early computers meant that a reduced board size and a smaller number of pieces were required for the game to be implementable on a computer.
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 development of the surreal numbers and contributed to development of combinatorial game theory (with Go Infinitesimals being a specific example of its use in Go).
Solving chess consists of finding an optimal strategy for the game of chess; that is, one by which one of the players can always force a victory, or either can force a draw. It is also related to more generally solving chess-like games such as Capablanca chess and infinite chess. In a weaker sense, solving chess may refer to proving which one of the three possible outcomes is the result of two perfect players, without necessarily revealing the optimal strategy itself.
The following outline is provided as an overview of and topical guide to chess:
