As a demonstration of how finding the pattern (perfect squares) is so much better than brute force, a benchmark: 

use strict;
use warnings;
$|=1;

sub sol1 {
    #init;
    my $lights = 20_000;
    my @lit = map {1} (1..$lights);
    
    my @sol;
    
    #flip;
    for (2..$lights) { 
        my $cnt = $_-1;
        while ($cnt < $lights) { 
            $lit[$cnt] = !$lit[$cnt];
            $cnt+=$_;
        }
    }
    
    #answer;
    for (0..$#lit) { push @sol, $_+1 if $lit[$_] }
}

sub sol2 {
    my $lights = 20_000;
    
    my @sol;
    
    for (1..$lights) {
        my $sqr = sqrt($_);
        push @sol, $_ if int($sqr) == $sqr;
    }
}

use Benchmark ':all';
cmpthese( 100, {
    sol1 => \&sol1,
    sol2 => \&sol2,
});

__END__
       Rate  sol1  sol2
sol1 6.58/s    --  -93%
sol2 98.5/s 1398%    --

[download]

These two solutions are abstracted to work for any number of lights, and the first is meant to be something someone of average coding skill might come up with (i.e. my crack at the solution ;-) ). Notice how sol2, where we just find all the perfect squares in range, is tremendously faster.

Just goes to show that the best optimization is done by redefining the problem. In this case, redefining from "toggle every nth light for n=2..max" to "find all lights that are perfect squares" resulted in a performance gain of nearly 1700%1400%!

Updates:

<-radiant.matrix->
A collection of thoughts and links from the minds of geeks
The Code that can be seen is not the true Code
"In any sufficiently large group of people, most are idiots" - Kaa's Law

In reply to Re: CarTalk Puzzler by radiantmatrix
in thread CarTalk Puzzler by freddo411

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