Shannon number

Claude Shannon

The Shannon number, named after Claude Shannon, is a conservative lower bound (not an estimate) 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 one for Black, and a typical game lasting about 40 such pairs of moves. Shannon calculated it 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 $\scriptstyle \frac{64!}{32!{8!}^2{2!}^6}$, or roughly 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. Taking these 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.^[2] Recent results^[3] improve that estimate, by proving an upper bound of only 2¹⁵⁵, which is less than 10^46.7 and showing^[4] an upper bound 2×10⁴⁰ 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 estimated to be between 4×10⁷⁹ and 4×10⁸¹.

See also

• Go and mathematics

Notes and references

1. Claude Shannon (1950). "Programming a Computer for Playing Chess" (PDF). Philosophical Magazine 41 (314).
2. Victor Allis (1994). Searching for Solutions in Games and Artificial Intelligence (PDF). Ph.D. Thesis, University of Limburg, Maastricht, The Netherlands. ISBN 90-900748-8-0.
3. John Tromp (2010). "John's Chess Playground".
4. S. Steinerberger (2014). International Journal of Game Theory.

External links

• Mathematics and chess