At first I had ignored this, then decided to do it. It was a more fun challenge than I thought. There are, not counting the order of the moves, actually 4 solutions in 15 moves for a 5x5 board. What follows is the throw-away script I wrote to find this. By default it solves a 5x5 board. Pass it an argument and it will solve an nxn board. (I tried it in the 1..10 range and found that there is 1 solution for 1, 2, 3, 6, 7, 8 and 10. As I mentioned, there are 4 for 5, plus 16 for 4 and 256 for 9. Don't ask me why, I merely report what I found...)

It would not be hard to extend this to handle arbitrary rectangular boards. I also didn't need the globals but this is throw-away code and it was easier that way. I make no apologies for the huge numbers of anonymous functions. The fact that I can feasibly find all 64 solutions for an 11x11 board by brute-force search on my old laptop speaks loudly enough for the efficiency of the method...

use strict;
use Carp;
use vars qw($min $max @board @soln @toggles);
$min = 1;
$max = shift(@ARGV) || 5;
@board = map [map 0, $min..$max], $min..$max;
foreach my $x ($min..$max) {
  foreach my $y ($min..$max) {
    push @toggles, ["$x-$y", ret_toggle_square($x, $y)];
  }
}

find_soln();

sub find_soln {
  if (! @toggles) {
    # Solved!
    print join " ", "Solution:", map $_->[0], @soln;
    print "\n";
  }
  else {
    my $toggle = shift(@toggles);
    # Try with, then without
    if ($toggle->[1]->()) {
      push @soln, $toggle;
      find_soln();
      pop @soln;
    }
    if ($toggle->[1]->()) {
      find_soln();
    }
    unshift @toggles, $toggle;
  }
}

# Returns a function that switches one square and returns
# true iff the new color is black
sub ret_swap_square {
  my ($x, $y) = @_;
  #print "Generated with $x, $y\n";
  my $s_ref = \($board[$x-1][$y-1]);
  return sub {$$s_ref = ($$s_ref + 1) %2;};
}

# Returns a function that toggles one square and its
# neighbours, and returns whether or not any neighbour
# has turned to white and cannot return to black without
# swapping again with $x lower or $x the same and $y lower.
sub ret_toggle_square {
  my ($x, $y) = @_;
  my @fin_swaps;
  my @other_swaps;
  unless ($x == $min) {
    push @fin_swaps, ret_swap_square($x - 1, $y);
  }
  if ($x == $max) {
    unless ($y == $min) {
      push @fin_swaps, ret_swap_square($x, $y - 1);
    }
    if ($y == $max) {
      push @fin_swaps, ret_swap_square($x, $y);
    }
    else {
      push @other_swaps, ret_swap_square($x, $y);
      unless ($y == $max) {
        push @other_swaps, ret_swap_square($x, $y+1);
      }
    }
  }
  else {
    unless ($y == $min) {
      push @other_swaps, ret_swap_square($x, $y - 1);
    }
    push @other_swaps, ret_swap_square($x, $y);
    push @other_swaps, ret_swap_square($x + 1, $y);
    unless ($y == $max) {
      push @other_swaps, ret_swap_square($x, $y + 1);
    }
  }
  return sub {
    $_->() foreach @other_swaps;
    my $ret = 1;
    $ret *= $_->() foreach @fin_swaps;
    return $ret;
  }
}
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