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 6331 (8!)-2 (where the ! represents the factorial and the underlined superscript represents a falling factorial), or roughly 3.7×1034. 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 games [2] | Number of possible positions [3] | Number of checkmates [4] |
|---|---|---|---|
| 1 | 20 | 20 | 0 |
| 2 | 400 | 400 | 0 |
| 3 | 8,902 | 5362 | 0 |
| 4 | 197,281 | 72,078 | 8 |
| 5 | 4,865,609 | 822,518 | 347 |
| 6 | 119,060,324 | 9,417,681 | 10,828 |
| 7 | 3,195,901,860 | 96,400,068 | 435,767 |
| 8 | 84,998,978,956 | 988,187,354 | 9,852,036 |
| 9 | 2,439,530,234,167 | 9,183,421,888 | 400,191,963 |
| 10 | 69,352,859,712,417 | 85,375,278,064 | 8,790,619,155 |
| 11 | 2,097,651,003,696,806 | 726,155,461,002 | 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. [5] Later work proved an upper bound of 8.7×1045, [6] and showed an upper bound 4×1037 in the absence of promotions. [7] [8]
John Tromp and Peter Österlund estimated the number of legal chess positions with a 95% confidence level at (4.822±0.028)×1044, based on an efficiently computable bijection between integers and chess positions. [6]
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.
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). [9]
{{cite journal}}: ISBN / Date incompatibility (help)