first, we expand the rules of chess to define the movement of a pawn on the first rank. this is immediately useful for compositions and for chess variants with non-standard initial positions.
let a pawn on the 1st rank be permitted to move forward 2 squares, ending on the 3rd rank. of course, it also retains its regular pawn moves and captures from the 1st to the 2nd rank.
a pawn on the 1st rank may move to the 2nd rank in one move (regular pawn movement), then move 2 squares to the 4th rank on a subsequent move. the ability to move 2 squares depends only on its position. any pawn on the 2nd rank may move forward 2 squares: it does not matter whether it started on the 2nd rank or got there from the 1st rank.
a pawn on the 1st rank may not move 3 squares directly to the 4th rank (proposed in other chess variants). 2 squares is the fastest a pawn may ever move. not permitting 3 or more squares on larger boards keeps the pawn a "small" piece, its power not growing as board size increases. (knight and king also remain small pieces.)
next, generalize to movement on boards of arbitrary size: a pawn may move 2 squares if its destination is not on or beyond the middle of the board.
for example, in 8x8 chess, the middle of the board is the line between the 4th and 5th ranks, so Pe2e4 and Pe1e3 are permitted. in 7x7 chess, the midline runs through the 4th rank, so Pe1e3 is permitted but Pe2e4 is not. in 5x5 chess, no 2-square-advance moves are permitted because the midline runs through the 3rd rank. in 9x9 chess, the midline runs through the 5th rank, so Pe1e3 and Pe2e4 are permitted but Pe3e5 is not.
on larger boards (10 squares or taller), a pawn might move 2 squares multiple times, for example Pe1e3 followed by Pe3e5. en passant capture continues to check the power of this ability.
on odd-sized boards, we prohibit the 2-square move exactly to the middle rank because that would allow opposing pawns in adjacent files to pass each other. the foot soldiers may not avoid fighting one another.
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