Can this be worked out mathematically?

Hardy (1999, p. 17) estimated the number of possible games of chess as . In a game of 40 moves, the number of possible board positions is at least according to Peterson (1996). However, this value does not agree with the possible positions given by Beeler et al. (1972), which was obtained by estimating the number of pawn positions (in the no-captures situation, this is ), multiplying by the possible positions for all pieces, then dividing by two for each of the (rook, knight) pairs that are interchangeable, and dividing by two for each pair of bishops (since half the positions will have the bishops on the same color squares). (However, note that there are more positions with one or two captures, since the pawns can then switch columns; Schroeppel 1996.) Shannon (1950) gave the estimate

Rex Stout's fictional detective Nero Wolfe quotes the number of possible games after ten moves as follows: "Wolfe grunted. One hundred and sixty-nine million, five hundred and eighteen thousand, eight hundred and twenty-nine followed by twenty-one ciphers. The number of ways the first ten moves, both sides, may be played" (Stout 1983). To be precise, the number of distinct chess positions after moves for , 2, ... are 20, 400, 5362, 71852, 809896?, 9132484?, ... (Schwarzkopf 1994, Sloane's A019319). The number of chess games that end in exactly moves (including games that mate in fewer than plies) for , 2, 3, ... are 20, 400, 8902, 197742, 4897256, 120921506, 3284294545, ... (K. Thompson, Sloane's A006494).