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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:

  1. I have found a game that completes all captures in 97 turns. Appendix I contains the complete game.
  2. 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
  3. 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: Ranks 1, 2, 7, and 8 are highlighted - these are the four rows that all of the pieces are on at the start of a chess game.. 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: PGN notation: 1. Ke3 e6 2. Ke5
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  • 1
    $\begingroup$ I wonder if this question might be appropriate for the chess.stackexchange.com site? $\endgroup$ Sep 26, 2022 at 4:59
  • 4
    $\begingroup$ @WillOctagonGibson : I think the question is on-topic both on Chess.SE and here. $\endgroup$
    – Evargalo
    Sep 26, 2022 at 6:21
  • 2
    $\begingroup$ @stewartIM yes, SE discourage cross posting, although from my experience sometimes it's ok if there are complete enough attempts here and the problem is still unsolved. Then we can post it again on another stack, referencing the earlier one. So in this case, you should at least wait for good answers here first, since you already posted it here. $\endgroup$
    – justhalf
    Sep 26, 2022 at 11:28
  • 3
    $\begingroup$ I am a bit perturbed by the notation a,b,...,h for ranks, when traditionally (including in the first solution you give just above) files are referred to by letters and ranks by numbers... $\endgroup$
    – Evargalo
    Sep 26, 2022 at 12:30
  • 1
    $\begingroup$ I'm perturbed too... I had no idea I had those backwards until you mentioned it. Fixed! $\endgroup$
    – stewartIM
    Sep 27, 2022 at 0:20

1 Answer 1

7
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Score :

I got it down to 87 moves (if I did not do something stupid/ it's a chaotic challenge)

Game :

1. h4 g5 2. h5 g4 3. Rh3 gxh3 4. g4 h2 5. g5 h1=Q 6. g6 Qxg1 7. g7 Qg6 8. hxg6 h5 9. gxf8=B h4 10. Bxe7 Nxe7 11. b4 h3 12. b5 h2 13. b6 d5 14. bxc7 Be6 15. g7 b5 16. g8=N b4 17. c8=B b3 18. Nxe7 Kxe7 19. Bxe6 Kxe6 20. c4 d4 21. Nc3 dxc3 22. c5 b2 23. c6 h1=N 24. c7 Nxf2 25. c8=R Nxd1 26. Rxb8 Qxd2+ 27. Bxd2 c2 28. Rxa8 c1=R 29. Bxc1 b1=Q 30. Rxb1 f5 31. Rb2 Nxb2 32. Bxb2 Rh1 33. Rxa7 f4 34. a4 Rxf1+ 35. Kd2 f3 36. a5 f2 37. a6 Rc1 38. Rf7 f1=Q 39. a7 Qxf7 40. a8=Q Kf5 41. Qa2 Qxa2 42. e4+ Kxe4 43. Kxc1 Qxb2+ 44. Kxb2

Animated diagram :

Game with all types of captures

Strategic considerations:

the opening:

move 1-3: sacrificing a rook makes sense for several reasons
- the rook is hindered in the early game by its own pawns, while quirte manuevrable at the end.
- cleaning a side keeps the game more compact making it easier to exchange pieces later.
- clearing a king early may lead to more check related problems.
move 4 to 8: promoting to queen makes sense because:
- a rook would be more useful later (and is already sacrificed to a pawn)
- a bishop would be out of position.
- a knight makes some sence but freeing the f pawn early does not help much, since the white king and bischop are both in the way.
- a queen promotion is needed 4 times, and she can be effiently used to give black playing room (with its h-pawn).

the strategic part:

The plan is to work a lot with the knights and bishops early , since they are least maneuvrable.
Since they cannot hit the end rows (without extra moves), it should be done as much as possible at row 2 or 7.
The white g pawns are ideal, and using white as much as possible can also move the black king towards the white pawns for a pawn capture.
move 9-17
A rook has taken a bishop, and more 'lesser' capture have been prepared. Moving the king forward immediately will cost flexibility.
move 18,19
I still did it immediately for my own overview, and since that would be moves that I probably would not want to take back while finding a solution.

the mid game:

move 20,21: unblocking some pawns
The pawns still have to take 1 piece: a knight. Doing c*b8 later seems obvious but Nc3 will unblock the black b and d pawns for only 1 move and generate move room to maneuver. move 22-25: some knight captures
the knight is the least maneuverable piece, and only a promoted h pawn can perform several captures; so, let us get that out of the way.
move 26-32: maneuvering around b2
We have to handle the promotions here wisely: the bishop wants a queen, a rook and a knight, the rook wants a queen, and the knight wants a horse.
In addition we have to consider if we want to promote the white or black a-pawn. Counting still mandatory turns makes 7 captures+8 pawn moves to prmotion for black v.s. 8 captures + 5 pawn to promotion moves for white. Less for white, so let us in addition promote the white a pawn.

the end game:

Be efficient with white; get everything aligned
move 33-38
The white king should not take the rook and block the black pawn, but move towards his bishop. In addition the rook should be moved into position for the black queens final tour.
move 42+
everything works out perfectly at the end as long as we move the pawn first.

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1
  • $\begingroup$ It looks valid to me! That's much closer to optimal than the game I had come up with. Question: what methods did you use to end up with the final sequence of moves? For mine for example, I planned a few specific combinations before the game started, but mostly did trial and error for the rest. $\endgroup$
    – stewartIM
    Sep 30, 2022 at 0:17

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