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[edit]
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.
Tighter bounds[edit]
Upper[edit]
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.
Lower[edit]
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.
Accurate estimates[edit]
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]
Number of sensible chess games[edit]
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]
See also[edit]
Notes and references[edit]
- ^ Claude Shannon (1950). "Programming a Computer for Playing Chess" (PDF). Philosophical Magazine. 41 (314). Archived from the original (PDF) on 2020-05-23.
- ^ "A048987 - Oeis".
- ^ "A079485 - Oeis".
- ^ Victor Allis (1994). Searching for Solutions in Games and Artificial Intelligence (PDF). Ph.D. Thesis, University of Limburg, Maastricht, The Netherlands. ISBN 978-90-900748-8-7.
- ^ a b John Tromp (2022). "Chess Position Ranking". GitHub.
- ^ S. Steinerberger (2015). "On the Number of Positions in Chess Without Promotion". International Journal of Game Theory. 44 (3): 761–767. doi:10.1007/s00182-014-0453-7. S2CID 31972497.
- ^ Gourion, Daniel (2021-12-16), An upper bound for the number of chess diagrams without promotion, arXiv:2112.09386, retrieved 2021-12-18
- ^ "How many chess games are possible?" Dr. James Grime talking about the Shannon Number and other chess stuff (films by Brady Haran). MSRI, Mathematical Sciences.