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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.

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.

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

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.

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Shannon also estimated the number of possible positions, of the general order of 63 ! 32 ! 8 ! 2 {\displaystyle {\frac {63!}{32!{8!}^{2}}}} , or roughly 3.7 ∗ 10 43 {\displaystyle 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.

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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.)

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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]

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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]

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