=begin output


##</code><code>##

50 = 28 + 1 + 21
51 = 0 + 15 + 36
52 = 21 + 3 + 28
53 = 10 + 28 + 15
54 = 6 + 3 + 45

##</code><code>##


=cut 

use warnings; use 5.010; use strict;
use POSIX qw"floor";


##</code><code>##

sub triang_decompose_1 {
	my($goal, $found, $n0, $n1, $n2);
	for $goal (0 .. 80) {
		$found = 0;
		for $n0 (0 .. 12) {
			for $n1 (0 .. 12) {
				for $n2 (0 .. 12) {
					if ($goal == $n0*($n0+1)/2 + $n1*($n1+1)/2 + $n2*($n2+1)/2 && !$found) {
						say $goal, " = ", $n0*($n0+1)/2, " + ", $n1*($n1+1)/2, " + ", $n2*($n2+1)/2;
						$found = 1;
					}
				}
			}
		}
	}
}


##</code><code>##

sub triang_decompose_2 {
	my($goal, $found, $n0, $n1, $n2);
	for $goal (0 .. 80) {
		$found = 0;
		$n0 = 0;
		while (!$found && $n0*($n0+1)/2 <= $goal) {
			$n1 = 0;
			while (!$found && $n1 <= $n0) {
				$n2 = 0;
				while (!$found && $n2 <= $n1) {
					if ($goal == $n0*($n0+1)/2 + $n1*($n1+1)/2 + $n2*($n2+1)/2 && !$found) {
						say $goal, " = ", $n0*($n0+1)/2, " + ", $n1*($n1+1)/2, " + ", $n2*($n2+1)/2;
						$found = 1;
					}
				$n2++;
				}
			$n1++;
			}
		$n0++;
		}
		if (!$found) {
			say $goal, " has no decomposition.";
		}
	}
}


##</code><code>##

sub triang_decompose_3 {
	my($goal, $found, $n0, $n1, $n2, $rem2, $rem1);
	for $goal (0 .. 80) {
		$found = 0;
		$n0 = int(sqrt(2*$goal+1) - 1/2);
		while (!$found && 0 <= $n0) {
			$rem2 = $goal - $n0*($n0+1)/2;
			$n1 = floor(sqrt(2*$rem2+1) - 1/2);
			while (!$found && 0 <= $n1) {
				$rem1 = $rem2 - $n1*($n1+1)/2;
				$n2 = floor(sqrt(2*$rem1+1) - 1/2);
				if ($rem1 == $n2*($n2+1)/2) {
					say $goal, " = ", $n0*($n0+1)/2, " + ", $n1*($n1+1)/2, " + ", $n2*($n2+1)/2;
					$found = 1;
				}
				$n1--;
			}
			$n0--;
		}
		if (!$found) {
			say $goal, " has no decomposition.";
		}
	}
}

##</code><code>##


say "Solution 1:";
triang_decompose_1();
say "Solution 2:";
triang_decompose_2();
say "Solution 3:";
triang_decompose_3();

__END__