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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