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

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 ${\frac {63!}{32!{8!}^{2}}}$ , or roughly $3.7*10^{43}$ . 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

Upper

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

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

John Tromp and Peter Österlund estimated the number of legal chess positions with a 95% confidence level at $(4.822\pm 0.028)\times 10^{44}$ , based on an efficiently computable bijection between integers and chess positions.^[5]

Number of sensible chess games

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]

Notes and references

^ 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.

External links

Mathematics and chess

[1] 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.

[2] "A048987". OEIS.

[3] "A079485". OEIS.

[4] 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.

[TrompCPR-5] Tromp, John (2022). "Chess Position Ranking". GitHub.

[6] 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.

[7] 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.

[8] 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.

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