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 . 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 | |
| 16 | (64!/(32!(8!)^2 (2!^6)) = 4634726695587809641192045982323285670400000. |
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] Later work proved an upper bound of 8.7×1045,[5] and showed an upper bound 4×1037 in the absence of promotions.[6][7]
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]- ^ Shannon, Claude E. (March 1950). Levy, David (ed.). "XXII. Programming a computer for playing chess" (PDF). Philosophical Magazine. 7. 41 (314). New York, NY: Springer: 256–275. doi:10.1080/14786445008521796. ISBN 978-1-4757-1970-3. ISSN 1941-5982. Archived from the original (PDF) on 2020-05-23.
- ^ "A048987". OEIS.
- ^ "A079485". OEIS.
- ^ Allis, Victor (1994). Searching for Solutions in Games and Artificial Intelligence (PDF). Maastricht, The Netherlands: Ph.D. Thesis, University of Limburg. ISBN 978-90-900748-8-7.
- ^ a b Tromp, John (2022). "Chess Position Ranking". GitHub.
- ^ Steinerberger, Stefan (August 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. ISSN 0020-7276. S2CID 31972497.
- ^ Gourion, Daniel (12 October 2022). "An upper bound for the number of chess diagrams without promotion". ICGA Journal. 44 (2) (version 2 ed.): 44–55. arXiv:2112.09386. doi:10.3233/ICG-220210. Retrieved 2021-12-18.
- ^ Grime, James (24 July 2015). How many chess games are possible? (Video). Numberphile – via YouTube. Dr. James Grime talking about the Shannon Number and other chess stuff (films by Brady Haran). MSRI, Mathematical Sciences.