The Game of Chess In how many ways can the first four moves (two from each side) be made in a game of chess?
I've seen and solved one on the first two moves. Now I wonder what the answer will look like for the first four moves

I bielive it's better to ask computer (implement brute force algorithm) than to try to calculate this number by hand.
For any possible (or imaginary) position there are no more than 121 moves for each side.
Let count roughly: 2 for each pawn (en passant is possible to be the third or the fourth move of the pawn only and tranformation to figure needs 5 or 6 moves of the same pawn), 8 for each knight, 13 for each bishop, 14 for each rook, 27 for queen and 8 for king. Totally 121 is upper bound (definitely unreachable). Counting $121^4$ moves will take at most one second for computer.
But is you really want to do it by hand... Try to count number of two moves made by the same side, square it and then count number of impossible combinations (when there is a check or when pawn is supposed to move, but it can't due to enemy pawn in front of it or due to covering king from check). But there are really many cases to consider.
