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Say that a solution to a Sokoban puzzle with only 2 boxes is box-isolating if and only if there are step numbers $m$ and $n$ such that for one of the boxes:

  • All of its pushes were between step $m$ and step $n$, and:
  • The other box was not pushed between step $m$ and step $n$

Is there a Sokoban puzzle with only 2 boxes such that there is a solution but there is no box-isolating solution?

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Yes, here is such a board
#######
###F###
#...###
#.#.AS#
#.#.B##
#.##.##
#....F#
#######
S is the starting position, A and B are the initial positions of the boxes, and F are the final positions. The A-box has to be touched in the first step and the B-box in the second step. But there is no way to take B to a final position without touching A again.

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