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All,
YuckFoo once asked if there was a more efficient way to verify that any non-negative whole integer could be represented as the sum of 3 triangular numbers. particle came up with a pretty cool solution using bit strings. While fast, it blew up when I asked it for 987654321.

My usual approach to these kind of problems is to translate how I would solve the problem with a pencil and paper in code, make any obvious performance changes, and see if it is fast enough. I usually don't start thinking about more efficient algorithms unless it is a "challenge" or my initial approach was just plain abysmal. Here was my go at it:

#!/usr/bin/perl
use strict;
use warnings;

my $target = defined $ARGV[0] ? $ARGV[0] : 987654321;
print join ', ' , get_three( $target );

sub get_three {
    my $num = shift;
    my $prev = p_tri( $num );

    return ($num, 0, 0) if ! $prev;

    while ( $prev ) {
        my $p_tri = ($prev * $prev + $prev) / 2;
        my $i = 0;
        my $j = $num - $p_tri;

        while ( $j >= $i ) {
            return $p_tri, $i, $j if ! p_tri( $i ) && ! p_tri( $j );
            ++$i;
            --$j;
        }
        --$prev;
    }
    die "Something went horribly wrong : $num\n";
}

sub p_tri {

    my $num = shift;
    my $x = ( sqrt( 8 * $num + 1 ) + 1 )/ 2;
    my $t = int $x;
    return $t == $x ? 0 : --$t;
}
[download]

Now without explaining how this works^*, I will ask the same question as YuckFoo - Any suggestions to optimize the search method?

Cheers - L~R

* If anyone wants to know, feel free to ask. If anyone wants to offer their own explanation, feel free to post.

In reply to Triangle Numbers Revisited by Limbic~Region

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