user16483: It's White to move-can they force a win?. The left bottom is a1 as usual.
[FEN "4k2r/6p1/p4pKb/p2N3p/p7/p5Q1/p1n5/qrn5 w - - 9 9"]

Note for people who saw an earlier version: the position was edited (including a swap of the colors) for a more beautiful puzzle by Laska's suggestions. The answer has now been fixed accordingly.
History: I came up with this puzzle when I was a student, almost 20 years ago. It was published in the magazine of my local chess club, but nobody found the answer. A friend popularized it with his new chess-friends after he emigrated to Canada.

user16524: As the tags suggests, the problem is a retrograde problem. If Black may no longer castle, there is an easy solution-1. Qc7 and mate on the second move.
The Black pawns on a6-a2 must have taken 10 white pieces or pawns. White has 3 pieces left, so the pawns could have taken 5 pieces and the pawns from a2 to d2. So one of the other pawns (e2-h2) must have been taken too. That only could have happened if one of those pawns promoted. But with the Black king on e8 and the black rook on h8, this is impossible, taking in account that the Black pawns are on f6, g7 and h5.
Conclusion: the Black king and/or rook must have moved, so castling is illegal and 1. Qc7 wins.

user7297: The solution is

 1. Qc7 followed by Qe7, Qf7 or Qc8 mate (depending on Black's second move).

This works because

 Black can't castle because either his king or rook on h8 have moved already. As Brian Towers notes, the black a-pawns need to have captured 10 white pieces and/or pawns; there are enough pieces missing, but capturing the white e-pawn (or to the right) doesn't bring you any closer to the a-file. So, one white pawn on the queenside must have been captured on another file; this is possible via promotion, but since there's just one black piece missing (a white-squared bishop) the promotion could have happened on the d-, e-, f-, or h-file. The pawn has either attacked the black king at f7 or d7, or it promoted on e8, forcing the black King to move in all cases. Or it promoted on h8, forcing the black rook to move. Another possibility, as @Laska noted, is that the e-pawn captures the black bishop on the d-file and is captured itself before promotion; however, we only need a single capture on the d-file, and the original white d-pawn must be captured too.


16483: <p>It's White to move-can they force a win?. The left bottom is a1 as usual.</p>
<pre><code>[FEN &quot;4k2r/6p1/p4pKb/p2N3p/p7/p5Q1/p1n5/qrn5 w - - 9 9&quot;]
</code></pre>
<p>Note for people who saw an earlier version: the position was edited (including a swap of the colors) for a more beautiful puzzle by Laska's suggestions. The answer has now been fixed accordingly.</p>
<p>History: I came up with this puzzle when I was a student, almost 20 years ago. It was published in the magazine of my local chess club, but nobody found the answer. A friend popularized it with his new chess-friends after he emigrated to Canada.</p>
 16524: <p>As the tags suggests, the problem is a retrograde problem. If Black may no longer castle, there is an easy solution-1. Qc7 and mate on the second move.</p>
<p>The Black pawns on a6-a2 must have taken 10 white pieces or pawns. White has 3 pieces left, so the pawns could have taken 5 pieces and the pawns from a2 to d2. So one of the other pawns (e2-h2) must have been taken too. That only could have happened if one of those pawns promoted. But with the Black king on e8 and the black rook on h8, this is impossible, taking in account that the Black pawns are on f6, g7 and h5.</p>
<p>Conclusion: the Black king and/or rook must have moved, so castling is illegal and 1. Qc7 wins.</p>
 7297: <p>The solution is</p>

<blockquote class="spoiler">
  <p> 1. Qc7 followed by Qe7, Qf7 or Qc8 mate (depending on Black's second move).</p>
</blockquote>

<p>This works because</p>

<blockquote class="spoiler">
  <p> Black can't castle because either his king or rook on h8 have moved already. As Brian Towers notes, the black a-pawns need to have captured 10 white pieces and/or pawns; there are enough pieces missing, <em>but</em> capturing the white e-pawn (or to the right) doesn't bring you any closer to the a-file. So, one white pawn on the queenside must have been captured on another file; this is possible via promotion, but since there's just one black piece missing (a white-squared bishop) the promotion could have happened on the d-, e-, f-, or h-file. The pawn has either attacked the black king at f7 or d7, or it promoted on e8, forcing the black King to move in all cases. Or it promoted on h8, forcing the black rook to move. Another possibility, as @Laska noted, is that the e-pawn captures the black bishop on the d-file and is captured itself before promotion; however, we only need a single capture on the d-file, and the original white d-pawn must be captured too.</p>
</blockquote>
