# A Plethora of Puzzled Pawns

Inspired by this question

According to FIDE rules,

the pawn may move to a square occupied by an opponent’s piece diagonally in front of it on an adjacent file, capturing that piece.

So, what's in front of a pawn? I'll let you choose.

Find a minimal sequence of moves for the following board, such that one white and one black pawn remain after the sequence of moves is executed.

Rules:

• Each pawn may only capture pieces of its own color
• Each pawn may face up, down, left, or right at the beginning
• Each pawn may only have one orientation for the entire sequence
• Each pawn may move to a square diagonally in front of it on an adjacent file/rank, capturing that piece.

Hint:

A minimal sequence requires 62 moves.

Since each pawn can face whichever way we like at the start, have all the pawns capture to the northeast or southwest toward one long diagonal, and then along it. For example, for white: h8xg7, g7xf6, f6xe5, a1xb2, b2xc3, c3xd4, d4xe5, and now the e5 pawn can only move up or to the right. f8xe7, e7xd6, a3xb4, b4xc5, c5xd6, and now the d6 pawn can only move up or to the right. Do the same for the other diagonals and then clean up the b8-h2 diagonal the same way.

• Give the sequence for black, not white - It's a little bit trickier to find. Of course, if you can give both, that's even better :)
– Avi
Commented Dec 12, 2019 at 22:45
• @Avi, black and white are exactly the same: they're mirror images of one another. Commented Dec 12, 2019 at 22:51
• Darn, sniped by a few minutes, and yours is a much more elegant solution that mine. +1!
– HTM
Commented Dec 12, 2019 at 22:51
• Oh, you're right - that's clever. I found a different way to do it where both pawns end up at g8 and h8, but I guess that's irrelevant.
– Avi
Commented Dec 12, 2019 at 22:56

Here's one sequence for white - the sequence for black is analogous:

1. d8xc7 2. c7xb6 3. f8xe7 4. e7xd6 5. d6xc5 6. c5xb4 7. h8xg7 8. g7xf6 9. f6xe5 10. e5xd4 11. d4xc3 12. c3xb2 13. h6xg5 14. g5xf4 15. f4xe3 16. e3xd2 17. h4xg3 18. g3xf2 19. b8xa7 20. a7xb6 21. b6xa5 22. a5xb4 23. b4xa3 24. a3xb2 25. h2xg1 26. g1xf2 27. f2xe1 28. e1xd2 29. d2xc1 30. c1xb2 31. b2xa1

Basically what we are doing here is

setting up the b8 and h2 pawns to zigzag capture their way to a1. For black, the set up is similar but with the b1 and h7 pawns going to a8.

Start with the shortest diagonal on one side of the board and capture the next pawn on the diagonal if there is one, otherwise capture the pawn on the next diagonal.

Then start at one end of the next diagonal and continue in the same manner until you capture a pawn on the longest diagonal.

Then do the same from the other side.

Finally, start capturing on one end of the longest diagonal until the end and you’ll be left with one pawn in one of the corners.

Do this for both white and black. Altogether it takes 62 moves.