The **Shannon number**, named after the American mathematician Claude Shannon, is a conservative lower bound of the game-tree complexity of chess of 10^{120}, based on an average of about 10^{3} 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 10^{120} 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.)

Shannon also estimated the number of possible positions, "of the general order of , or roughly 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.

Number of plies (half-moves) |
Number of possible games |
---|---|

1 | 20 |

2 | 400 |

3 | 8,902 |

4 | 197,281 |

5 | 4,865,609 |

6 | 119,060,324 |

7 | 3,195,901,860 |

8 | 84,998,978,956 |

9 | 2,439,530,234,167 |

10 | 69,352,859,712,417 |

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 of 5×10^{52} for the number of positions, and estimated the true number to be about 10^{50}.^{[2]} Recent results^{[3]} improve that estimate, by proving an upper bound below 2^{155}, which is less than 10^{46.7} and showing^{[4]} an upper bound 2×10^{40} in the absence of promotions.

### Lower

Allis also estimated the game-tree complexity to be at least 10^{123}, "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^{80} hydrogen atoms.

## 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 10^{40} games. This is based on having a choice of about three sensible moves at each ply (half-move), and a game length of 80 moves.^{[5]}

## See also

## Notes and references

**^**Claude Shannon (1950). "Programming a Computer for Playing Chess" (PDF).*Philosophical Magazine*.**41**(314).**^**Victor Allis (1994).*Searching for Solutions in Games and Artificial Intelligence*(PDF). Ph.D. Thesis, University of Limburg, Maastricht, The Netherlands. ISBN 978-90-900748-8-7.**^**John Tromp (2010). "John's Chess Playground".**^**S. Steinerberger (2014). "International Journal of Game Theory".*International Journal of Game Theory*.**44**(3): 761–767. doi:10.1007/s00182-014-0453-7.**^**"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.