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Variant of this lovely puzzle

The playable version of the game

You are given an empty chessboard as in the link above as 6x6.

  1. Place one knight anywhere you like.
  2. Then place as many pawns as possible on the board.

6 by 6 checkerboard

The goal is:

  • The knight must be able to capture all the pawns.
  • The knight must capture them one by one, making exactly one move per pawn.
  • The knight must have only one possible path to do this.

Rules:

  • Pawns don’t move.
  • Pawns don’t protect each other.
  • You can’t capture more than one pawn in a single move.

Challenge: At most how many pawns can you place?

The playable version of the game

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  • $\begingroup$ The playable version immediately jumps away after getting the max answer. It didn't even let me copy my solution out. $\endgroup$ Commented Aug 3 at 20:51
  • $\begingroup$ @ralphmerridew wow you are fast :) I need to fix it, you are right. $\endgroup$ Commented Aug 3 at 20:53
  • $\begingroup$ @ralphmerridew: You can hit the back arrow. It'll save your solution. $\endgroup$ Commented Aug 3 at 20:53
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    $\begingroup$ How do you mean the 'only one possible path' condition? It doesn't mean there is only one possible move at each step. So the solution path should be globally unique given the pawn configuration? What does that give you? It also seems highly non-trivial to check. $\endgroup$ Commented Aug 4 at 7:21
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    $\begingroup$ @quarague It is trivial to check with a computer program, which Oray has conveniently provided. $\endgroup$ Commented Aug 4 at 11:09

3 Answers 3

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9
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With some brute force searching:

32 pawns:
32 pawns

Previous solution, found using the playable version:

25 pawns:
25 pawns

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  • $\begingroup$ nice! I didn't know there were better answer, I changed the limits accordingly, maybe there is even better one, I will wait a bit more for 6x6. $\endgroup$ Commented Aug 3 at 21:16
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    $\begingroup$ great idea! i will do that tomorrow morning as i am in bed now :) $\endgroup$ Commented Aug 3 at 21:27
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    $\begingroup$ @msh210 Oray's program confirms it. $\endgroup$ Commented Aug 4 at 11:09
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    $\begingroup$ I confirmed optimality via integer linear programming. $\endgroup$ Commented Aug 5 at 15:58
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    $\begingroup$ @Oray I also confirmed via exhaustive search that there are no better solutions. Another exhaustive search targeting this maximal number of pawns identified a total of 36 distinct solutions. $\endgroup$ Commented Aug 5 at 23:08
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I managed to get 28, although I don't know if it is the max:

1

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  • $\begingroup$ wow! only 7 locations are empty! pretty impressive. $\endgroup$ Commented Aug 4 at 7:19
  • $\begingroup$ I also got another solution with 28, cant find 29 myself. this could be maximum. $\endgroup$ Commented Aug 4 at 7:28
  • $\begingroup$ Can you demonstrate that the knight has only one possible path to capture these 28 pawns? $\endgroup$ Commented Aug 4 at 8:57
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    $\begingroup$ @msh210 The playable game checks that for you. $\endgroup$ Commented Aug 4 at 9:49
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    $\begingroup$ Can you put one more pawn in the spot just north of where the horse starts? $\endgroup$ Commented Aug 5 at 13:45
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On a 6×6 board, you can capture

21 pawns (outdated—see JS1's and Daniel Mathias' answers): 6×6 solution Algebraic notation: a6 b4 c6 a5 b3 c5 a4 b6 c4 a3 b1 d2 f1 e3 f5 d4 b2 f4 d5 f6 e4 f2.

On an 8×8 board, you can capture

30 pawns (outdated—Daniel Mathias has a better one for 6×6): 8×8 solution Algebraic notation: a8 b6 c8 a7 b5 c7 a6 b8 c6 a5 b7 c5 a4 b2 d1 c3 b1 a3 c2 e3 f1 d2 e4 f6 h5 g7 e6 f8 g6 h8 f7.

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    $\begingroup$ The 8x8 is also outdated, since there is already a 32 solution for 6x6. $\endgroup$ Commented Aug 5 at 3:53

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