Me and Peter Österlund (but mostly Peter and his program Texel) have completed analyzing all 1 million positions in the random 1 million sample and determined that with 56011 legal positions with average inverse multiplicity of ~0.98013, the most accurate estimate of the number of chess positions is (4.79 +- 0.04) * 10^44 (at 95% confidence level).
So approximately 4.8 x 10^44, with 2 digits of accuracy.
https://github.com/tromp/ChessPositionRanking
The original 1 million sample was generated with version 1.1 of the Haskell package System.Random
Using the newer 1.2 revision instead causes `make testRnd1mFENs` to generate a completely different 1m sample. We have analyzed this new sample, yielding a slightly different estimate of (4.853 +- 0.039) * 10^44.
Combining the two estimates yields the sqrt(2) more accurate combined estimate of
(4.822 +- 0.028) * 10^44.
#124
OK, so there are (4.822 +- 0.028) * 10^44 legal chess positions.
A legal position is a position that can result from the initial position by a proof game of legal moves.
Most of these positions are not sensible.
A sensible position is a legal position for which the proof game has > 50% accuracy.
This paper
https://arxiv.org/pdf/2112.09386.pdf
arrives at an upper bound of 3.8521 . . . × 10^37
for the number of chess diagrams without promotions to pieces not previously captured.
#114
I did look at the sample of 30 you provided, though most are illegal. I am willing to look at your 200 sample, though most are illegal, I do it manually and I also have other things to do.
Any progress on analyzing the 200 potentially legal in the 10k noproms sample?
Any progress on pinning down your definition of sensible?