The Shannon number, named after the American mathematician Claude Shannon, is a conservative lower bound of the game-tree complexity of chess of 10<sup>120</sup>, based on an average of about 10<sup>3</sup> 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 showed a calculation for the lower bound of the game-tree complexity of chess, resulting in about 10<sup>120</sup> possible games, to demonstrate the impracticality of solving chess by brute force, 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 63<sup style="text-decoration:underline;">31</sup> (8!)<sup>-2</sup> (where the ! represents the factorial and the underlined superscript represents a falling factorial), or roughly . 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.
Taking Shannon's numbers into account, Victor Allis calculated an upper bound of 5ÃÂ10<sup>52</sup> for the number of positions, and estimated the true number to be about 10<sup>50</sup>. Later work proved an upper bound of 8.7ÃÂ10<sup>45</sup>, and showed an upper bound 4ÃÂ10<sup>37</sup> in the absence of promotions.
John Tromp and Peter ÃÂsterlund estimated the number of legal chess positions with a 95% confidence level at , based on an efficiently computable bijection between integers and chess positions.
Allis also estimated the game-tree complexity to be at least 10<sup>123</sup>, "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 10<sup>80</sup>.
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 10<sup>40</sup> 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).