Search for permutations of objects in a circle for which any pair of objects is at a different minimal distance along the circle than originally; see http://math.stackexchange.com/questions/281094.

circle.pl

#!/usr/bin/perl

my $M = shift() || 15;

# A node has [ N, [1, ... ] ]
# where N is the max of the [ 1, ... ]
sub children {
    my ($node) = @_;
    my ($N, $circ) = @$node;
    my @children;
    return if $N == $M;;
    for my $i (0 .. $M-1) {
  next if defined $circ->[$i];
	my @new_circ = @$circ;
	$new_circ[$i] = $N+1;
	my $newnode = [$N+1, \@new_circ];
	push @children, $newnode
	    if okay($i, $newnode);
    }
    return grep okay($_), @children;
}


sub okay {
    my ($j, $node) = @_;
    my ($N, $circ) = @$node;
    my $v = $circ->[$j];
    for my $i (0 .. $M-1) {
	next unless defined $circ->[$i];
	my $d = minus($j, $i);
	my $dd = minus($v, $circ->[$i]);
	return if $d != 0 && $dd == $d;
    }
    return 1;
}

sub root {
    return [1, [1]];
}

sub minus {
    my ($p, $q) = @_;
    my $d1 = abs($p-$q);
    my $d2 = $M - $d1;
    return $d1 < $d2 ? $d1 : $d2;
}

my @queue = root();

$| = 1;
while (my $node = shift @queue) {
    my ($N, $circ) = @$node;
    print STDERR "| " x $N, "@$circ\n" if $N % 5 == 0;
    if ($N == $M) {
	my $circ = $node->[1];
	print "@$circ\n";
    }
    unshift @queue, children($node);
}