In chess, there are 6 varieties of items that may seize different items, and there are 5 varieties of items that may be captured. Ignoring piece shade, because of this there are thirty distinct attainable varieties of captures:

What is the shortest attainable recreation through which every of those captures takes place precisely as soon as?

Constraints:

• There are thirty capturable items at first of a typical recreation of chess. The recreation ends when all of those has been captured, at which level the 2 kings are the one items left on the board.
• Ten pawns should promote into different items: two knights, two bishops, two rooks, and 4 queens.
• All customary chess guidelines apply, together with situations for checks and attracts.
• Both gamers are cooperating with one another to realize the objective.
• Length of the sport is measured in turns. A flip is when one piece is moved by one participant. A transfer is when each gamers transfer one piece.

I’ve been investigating this drawback with a buddy of mine over the summer season. Below are our findings thus far:

1. I’ve discovered a recreation that completes all captures in 97 turns. Appendix I comprises the whole recreation.
2. The shortest recreation should take a minimal of 75 turns. Explanation:
   • Each of the 30 captures takes one flip.
   • 10 pawns should be promoted in 5 turns every for a complete of fifty turns.
   • 5 captures contain a pawn capturing one other piece. These additionally transfer the pawn one step to promotion, so these turns can depend in the direction of each totals above.
   • Total: 50 + 30 – 5 = 75
3. In observe, the shortest recreation will possible take at the very least 78 turns, with three further turns being required for the KxP seize. Appendix II is our line of reasoning for this – be happy to skip it.

Progress So Far

So far, the shortest identified answer takes 97 turns.

Appendix I

Below is the PGN notation for my 97-turn recreation, which is appropriate 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 situation in chess is that if there’s “inadequate materials” for one facet to checkmate the opposite then the sport is asserted a draw. If this situation is ignored then the sport would take solely 96 turns:}

46.    ...    Qf3+
47.   Kxf3     Kd5
48.   Nxc5    Kxc5

Appendix II

Below is our line of reasoning for why at the very least 78 strikes are required:

• Suppose that we might discover a “good” chess recreation that takes precisely 75 strikes. In this recreation, each transfer needs to be both a seize or a pawn motion in the direction of promotion. Any different kind of transfer could be thought-about “waste.”
• Lemma: Every seize on this “good” recreation should happen on ranks a, b, g, or h:
  
  Explanation:
  • Every piece within the recreation begins on ranks a, b, g, or h.
  • Ignoring pawn promotions, the one approach that items can transfer is by capturing different items. Even if the seize is an “en passant” seize, each attacking piece finally ends up on the identical rank because the piece that it captures. This implies that each seize that does not contain pawns should happen on a type of 4 ranks as properly.
  • Once a pawn has moved, it should be promoted. Even although it should go by means of the c, d, e, and/or f ranks, it may’t be captured. If it had been, then the strikes that it used to finish up within the center had been wasted.
  • Once a pawn has completed selling, the promoted piece is on the a or h rank, and the one approach that it may transfer is by capturing. At this level, it really works the identical as an everyday non-pawn piece.
  • Does that make sense?
• Now think about the KxP seize. Neither piece can transfer a couple of sq. at a time in regular play, which implies that a number of strikes should be wasted to ensure that it to happen. We have discovered that at finest this seize should waste at the very least 3 strikes, as proven beneath: