Progress So Far
So far, the shortest known solution takes 87 turns (9 turns from optimal), and was found by Retudin.
Progress over time:
Date | Moves | Found by |
---|---|---|
2022-09-29 | 87 | Retudin |
(initial) | 97 | stewartIM |
In chess, there are 6 types of pieces that can capture other pieces, and there are 5 types of pieces that can be captured. Ignoring piece color, this means that there are thirty distinct possible types of captures:
PxP | PxR | PxN | PxB | PxQ |
RxP | RxR | RxN | RxB | RxQ |
NxP | NxR | NxN | NxB | NxQ |
BxP | BxR | BxN | BxB | BxQ |
QxP | QxR | QxN | QxB | QxQ |
KxP | KxR | KxN | KxB | KxQ |
What is the shortest possible game in which each of these captures takes place exactly once?
Constraints:
- There are thirty capturable pieces at the start of a typical game of chess. The game ends when all of these has been captured, at which point the two kings are the only pieces left on the board.
- Ten pawns must promote into other pieces: two knights, two bishops, two rooks, and four queens.
- All standard chess rules apply, including conditions for checks and draws.
- Both players are cooperating with each other to achieve the goal.
- Length of the game is measured in turns. A turn is when one piece is moved by one player. A move is when both players move one piece.
I have been investigating this problem with a friend of mine over the summer. Below are our findings so far:
- I have found a game that completes all captures in 97 turns. Appendix I contains the complete game.
- The shortest game must take a minimum of 75 turns. Explanation:
- Each of the 30 captures takes one turn.
- 10 pawns have to be promoted in 5 turns each for a total of 50 turns.
- 5 captures involve a pawn capturing another piece. These also move the pawn one step to promotion, so these turns can count towards both totals above.
- Total: 50 + 30 - 5 = 75
- In practice, the shortest game will likely take at least 78 turns, with three additional turns being required for the KxP capture. Appendix II is our line of reasoning for this - feel free to skip it.
Appendix I
Below is the PGN notation for my 97-turn game, which is compatible with LiChess.
1. f4 h5
2. f5 h4
3. f6 h3
4. fxe7 g5
5. exd8=N g4
6. Nxb7 Bxb7
7. b4 Bxg2
8. Bxg2 hxg2
9. Nh3 Ke7
10. e4 g3
11. Ke2 g1=R
12. b5 Rxd1
13. b6 Rxc1
14. b7 Rxb1
15. bxa8=Q Rxa1
16. Qxb8 Rxa2
17. Qxc7 Ke6
18. e5 Kd5
19. d4 Kxd4
20. Ng5 Rh7
21. Nxh7 g2
22. Nxf8 g1=Q
23. h4 Qxh1
24. h5 a5
25. h6 a4
26. h7 a3
27. e6 d5
28. e7 f5
29. e8=R f4
30. Re4+ Kxe4
31. hxg8=B d4
32. Bxa2 d3+
33. Kd2 f3
34. Bd5+ Kxd5
35. Qh7 a2
36. c4+ Ke5
37. Ke3 f2
38. c5 d2
39. c6 a1=N
40. c7 d1=B
41. c8=Q f1=Q
42. Qcc2 Bxc2
43. Qxc2 Qc4
44. Qxc4 Nb3
45. Qc5+ Nxc5
46. Ne6 Kd5
47. Nxc5 Kxc5
48. Kf2 Qg2+
49. Kxg2
Note: One condition in chess is that if there is "insufficient material" for one side to checkmate the other then the game is declared a draw. If this condition is ignored then the game would take only 96 turns:
46. ... Qf3+
47. Kxf3 Kd5
48. Nxc5 Kxc5
Appendix II
Below is our line of reasoning for why at least 78 moves are required:
- Suppose that we could find a "perfect" chess game that takes exactly 75 moves. In this game, every move has to be either a capture or a pawn movement towards promotion. Any other type of move would be considered "waste."
- Lemma: Every capture in this "perfect" game must take place on rank 1, rank 2, rank 7, or rank 8:
. Explanation:
- Every piece in the game starts on one of ranks 1, 2, 7, or 8
- Ignoring pawn promotions, the only way that pieces can move is by capturing other pieces. Even if the capture is an "en passant" capture, every attacking piece ends up on the same rank as the piece that it captures. This means that every capture that doesn't involve pawns must take place on one of those four ranks as well.
- Once a pawn has moved, it must be promoted. Even though it will pass through ranks 3, 4, 5, and/or 6, it can't be captured. If it were, then the moves that it used to end up in the middle were wasted.
- Once a pawn has finished promoting, the promoted piece is on rank 1 or 8, and the only way that it can move is by capturing. At this point, it works the same as a regular non-pawn piece.
- Does that make sense?
- Now consider the KxP capture. Neither piece can move more than one square at a time in normal play, which means that several moves must be wasted in order for it to occur. We have found that at best this capture must waste at least 3 moves, as shown below: