The Shannon number, named after the American mathematician

Claude Shannon Claude Elwood Shannon (April 30, 1916 – February 24, 2001) was an American people, American mathematician, electrical engineering, electrical engineer, and cryptography, cryptographer known as a "father of information theory". As a 21-year-o ...

, is a conservative lower bound of the

game-tree complexity Combinatorial game theory has several ways of measuring game complexity. This article describes five of them: state-space complexity, game tree size, decision complexity, game-tree complexity, and computational complexity. Measures of game comple ...

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 disti ...

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 Solving chess means finding an optimal strategy for the game of chess, that is, one by which one of the players ( White or Black) can always force a victory, or either can force a draw (see solved game). It also means more generally solving ''chess ...

by brute force, in his 1950 paper "Programming a Computer for Playing Chess". (This influential paper introduced the field of

computer chess Computer chess includes both hardware (dedicated computers) and software capable of playing chess. Computer chess provides opportunities for players to practice even in the absence of human opponents, and also provides opportunities for analysi ...

.) 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 Ply, Pli, Plies or Plying may refer to: Common uses * Ply (layer), typically of paper or wood ** Plywood, made of layers of wood ** Tire ply, a layer of cords embedded in the rubber of a tire Places * Plymouth railway station, England, station ...

) 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 Louis Victor Allis (born 19 May 1965) is a Dutch computer scientist working in the artificial intelligence (AI) field. In his graduate work, he revealed AI solutions for Connect Four, Qubic, and Gomoku. His dissertation introduced two new game s ...

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 element ...

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 John Tromp is a Dutch computer scientist. He formerly worked for Dutch Centre for Mathematics and Computer Science. Tromp discovered the number of legal states of the board game Go (game), Go, and co-authored with Bill Taylor the Rules of Go#Basic ...

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