user24675: Here is a chess problem that uses retrograde analysis. I’ve also posted it on the “France-Echecs” chess forum, which is in French of
course, and on my blog.
[FEN "3qk3/3pp3/8/8/8/3PP3/1PP2PP1/RNBQKBNR w - - 0 1"]

Each of the eight White pieces on the first rank has made exactly three movements, except one of them who made only one. Which piece is it?

user4572: According to the pawn structure, one of the white bishop must be a promoted one. The knights have switched place, the other pieces can easily make three movements.
The only ones that can make a single one are the rooks, either by

Castling
Taking the other rook's square
By coming from a promoted pawn on a8 or h8

All possibilities seems doable as white can promote two pawns, and can use a captured piece to capture all black pieces.
Here is an example:
[fen "rnbqrnbq/ppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w - -"]

1. d3 a5 2. e3 h5 3. Bd2 b6 4. Be2 g6 5. Bxa5 Bb7 6. Bxh5 Bg7 7. Bxb6 Qc8 8. Bxg6 Kf8 9. Bxc7 Nc6 10. Bxf7 Nf6 11. a4 Nd8 12. h4 Ne8 13. Bxd8 Qxd8 14. Bxe8 Kxe8 15. a5 Qc7 16. h5 Kf7 17. a6 Qd8 18. h6 Ke8 19. axb7 Qc7 20. hxg7 Kf7 21. bxa8=B Qd8 22. gxh8=B Ke8 23. Nc3 Qc7 24. Nf3 Kf7 25. Qd2 Qd8 26. Ke2 Ke8 27. Ra5 Qc7 28. Rha1 Kf7 29. Rh5 Qd8 30. Rhh1 Ke8 31. Qe1 Qc7 32. Kd2 Kf7 33. Qd1 Qd8 34. Ke1 Ke8 35. Ne2 Qc7 36. Nd2 Kf7 37. Nb1 Qd8 38. Ng1 Ke8 39. Bf3 Qc7 40. Bc3 Kf7 41. Bd2 Qc8 42. Be2 Kf8 43. Bc1 Qd8 44. Bf1 Ke8

In this example rook h1 makes a single move while other white pieces on the first row make three moves.

24675: <p>Here is a chess problem that uses retrograde analysis. I’ve also posted it on the <a href="https://www.france-echecs.com/article.php?art=20200909190151964" rel="nofollow noreferrer">“France-Echecs”</a> chess forum, which is in French of
course, and on my <a href="http://cinquantesignes.blogspot.com/2020/09/symmetry.html" rel="nofollow noreferrer">blog.</a></p>
<pre><code>[FEN &quot;3qk3/3pp3/8/8/8/3PP3/1PP2PP1/RNBQKBNR w - - 0 1&quot;]
</code></pre>
<p>Each of the eight White pieces on the first rank has made exactly three movements, except one of them who made only one. Which piece is it?</p>
 4572: <p>According to the pawn structure, one of the white bishop must be a promoted one. The knights have switched place, the other pieces can easily make three movements.</p>
<p>The only ones that can make a single one are the rooks, either by</p>
<ol>
<li>Castling</li>
<li>Taking the other rook's square</li>
<li>By coming from a promoted pawn on a8 or h8</li>
</ol>
<p>All possibilities seems doable as white can promote two pawns, and can use a captured piece to capture all black pieces.</p>
<p>Here is an example:</p>
<pre><code>[fen &quot;rnbqrnbq/ppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w - -&quot;]

1. d3 a5 2. e3 h5 3. Bd2 b6 4. Be2 g6 5. Bxa5 Bb7 6. Bxh5 Bg7 7. Bxb6 Qc8 8. Bxg6 Kf8 9. Bxc7 Nc6 10. Bxf7 Nf6 11. a4 Nd8 12. h4 Ne8 13. Bxd8 Qxd8 14. Bxe8 Kxe8 15. a5 Qc7 16. h5 Kf7 17. a6 Qd8 18. h6 Ke8 19. axb7 Qc7 20. hxg7 Kf7 21. bxa8=B Qd8 22. gxh8=B Ke8 23. Nc3 Qc7 24. Nf3 Kf7 25. Qd2 Qd8 26. Ke2 Ke8 27. Ra5 Qc7 28. Rha1 Kf7 29. Rh5 Qd8 30. Rhh1 Ke8 31. Qe1 Qc7 32. Kd2 Kf7 33. Qd1 Qd8 34. Ke1 Ke8 35. Ne2 Qc7 36. Nd2 Kf7 37. Nb1 Qd8 38. Ng1 Ke8 39. Bf3 Qc7 40. Bc3 Kf7 41. Bd2 Qc8 42. Be2 Kf8 43. Bc1 Qd8 44. Bf1 Ke8
</code></pre>
<p>In this example rook h1 makes a single move while other white pieces on the first row make three moves.</p>
