The Shannon number, named after the American mathematician

Claude Shannon Claude Elwood Shannon (April 30, 1916 – February 24, 2001) was an American mathematician, electrical engineer, and cryptographer known as a "father of information theory". As a 21-year-old master's degree student at the Massachusetts In ...

, is a conservative lower bound of the game-tree complexity of

chess Chess is a board game for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to dist ...

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 Brute Force or brute force may refer to: Techniques * Brute force method or proof by exhaustion, a method of mathematical proof * Brute-force attack, a cryptanalytic attack * Brute-force search, a computer problem-solving technique People * Brut ...

, in his 1950 paper "Programming a Computer for Playing Chess". (This influential paper introduced the field of computer chess.) Shannon also estimated the number of possible positions, "of the general order of

\scriptstyle \frac

, 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. 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 In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an elem ...

of 5×10⁵² for the number of positions, and estimated the true number to be about 10⁵⁰. Recent results improve that estimate, by proving an upper bound of 8.7x10⁴⁵, and showing an upper bound 4×10³⁷ in the absence of promotions.

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^

, based on an efficiently computable bijection between integers and chess positions.

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)."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.

Notes and references

External links

Mathematics and chess
{{Large numbers Mathematical chess problems Large integers Combinatorial game theory Computer chess Claude Shannon