What is the Shannon number and how is it calculated?

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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, estimated to be around 10¹²⁰. This number represents the total number of possible chess games and was calculated based on an average of about 10³ possibilities for a pair of moves (one for White and one for Black), with a typical game lasting about 40 such pairs of moves13. To understand how the Shannon number is calculated, it's important to understand the term "ply". In chess, a ply refers to a single move by one player. Therefore, a pair of moves (one by White and one by Black) is considered two plies. Shannon's calculation was based on the assumption that there are about 30 legal moves from any position in the game of chess, and a typical game lasts about 40 pairs of moves (or 80 plies)36. So, Shannon's calculation can be represented as follows:
  • For each ply, there are about 30 possible moves.
  • For a pair of moves (two plies), there are about 30 * 30 = 900 possibilities.
  • For a typical game of 40 pairs of moves (80 plies), the total number of possible games is roughly 900^40, which is approximately 10¹²⁰6.
This number is so large that it exceeds the estimated number of atoms in the observable universe, which is about 10⁸¹35. It's important to note that the Shannon number is a conservative estimate and other calculations have proposed even larger numbers3. However, the Shannon number is widely accepted as a demonstration of the impracticality of solving chess by brute force, i.e., by calculating all possible games1.
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