Alternative way to estimate the number of possible chess positions:

Chesserroo2 · 2011-07-28T07:40:55-07:00

Using permutation math and 32 pieces and 64 squares, it may sound simple to estimate the number of possible chess positions. But not only are most not legal, many are impossible to arrive at. The math problem may then sound more difficult. But I have a plan: Over 2 million or so games have been play

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HGMuller

Jul 29, 2011

#21

The method you propose seems very unreliable, and frought with systematical errors. For one, you should not use games that are actually played by people with any skill level, as those will not offer a representative selection of positions. Because they only play moves that make sense, and the number of positions that do not make sense or can only be reached through moves that do not make sense is astronomically larger.

I think you also over-estimate the reachability problem. Virtually all positions that have a reachable Pawn structure (and a number of pieces consistent with the minimum number of captures it takes to reach it) will in practice be reachable. So it would just be a matter of having a computer calculate the reachable Pawn structures, using combinatorics to calculate in how many ways the other pieces can be put on the board, and doing some random sampling of those to figure out which fraction of these is illegal.

Further note that the set of positions will be very much dominated by positions that do not have any pieces captured. Each extra piece causes the number of positions to increase by a factor 16 to 64. And for positions without any pieces captured, it is comparatively easy to calculate the number of reachable Pawn structures.

julianosam

Mar 21, 2023

#22

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