sub check_scheme {
    my @scheme = @_;

    ## check that each round measures the same amount of balls
    for my $round (0 .. 2) {
        my $lhs = grep { $scheme[$round][$_] eq "L" } 0 .. 11;
        my $rhs = grep { $scheme[$round][$_] eq "R" } 0 .. 11;
        return 0 if $lhs != $rhs
    }

    ## simulate the scheme with each of the balls being the odd one
    for my $item (0 .. 11) {
        my @candidates = (0 .. 11);

        for my $round (0 .. 2) {
            if ($scheme[$round][$item] eq "-") {
                @candidates = grep { $scheme[$round][$_] ne "-" } @candidates;
            } else {
                @candidates = grep { $scheme[$round][$_] eq "-" } @candidates;
            }
        }

        ## by this point we should have eliminated all but one ball!
        return 0 if @candidates != 1;
    }
    return 1
}