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ChessPositionRanking
Software suite for ranking chess positions and accurately estimating the number of legal chess positions
Taking "a sample of random legal positions" is nowhere near as trivial as you make it sound. You can have a position that "looks" legal (as in: at most one King is in check, there are no impossible multiple checks, the Kings are not adjacent, the "correct" side is to move, etc., etc.) but are not legal. As in, they couldn't have been reached through any legal sequence of moves from the starting position. Then there's also the question of: does that actually matter? What if the position is reachable only in 960? How would discarding such positions contribute to solving "chess"? And so on...
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