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There have been several questions regarding possible number of chess games. for example - Database of every possible move in chess

However Can there be an estimate on the number of "allowable" chess positions.

For example White Pawns cannot occupy first row and Black Pawns cannot occupy eighth row. If a given chess piece is occupying a square, then another chess piece cannot occupy that. Using simple elimination rules coded, can we have an estimate on the number of unique positions that chess pieces can have. Since they would be reducing in number. An upper bound on positions before applying any elimination rules would be sum of (64 C 32 + 64 C 32 ... + 64 C 2). This is less than 10^20. With elimination rules, they should be significantly lower.

Any ideas on how elimination rules be written for chess pieces?


  • Your estimate is too low. You shouldn't just compute combinations, it matters which pieces are on which squares. – Dag Oskar Madsen Jun 16 '14 at 10:36
  • The maths goes like you choose the 32 squares possible, and find number of pieces you can place on each square. Then multiply 64 C 32 – QuIcKmAtHs Dec 27 '17 at 12:25
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Upper bounds on the number of allowable positions are discussed on the Shannon number Wikipedia page.

There is an accepted upper bound of 5×10^52 by Vicor Allis, and a less verified upper bound of 2^155 (approx 10^46.7) by John Trump.

By comparison, the number of atoms on earth is about 10^50.

  • The article fairly estimates the positions to be close to 10^50. Link is nwchess.com/articles/misc/Chess_Board_Positions_article.pdf – shoonya Jun 16 '14 at 9:34
  • However this can be computed to a far better estimate, especially the cases when Kings are already in checks, Kings are simultaneously in check. The key thing being estimating this can allow us to typically find the optimal moves for a given position. – shoonya Jun 16 '14 at 10:11

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