Project Euler – Problem 9

Problem description:

http://projecteuler.net/index.php?section=problems&id=9

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My solution:

The logic behind:

My initial approach was to simplify the equation by using the given information and some algebra, resulting in the following equation:

b^2 = c² – a²
b = 1000 – a – c

(1000 – a – c)² = c² – a²
1000² – 1000a – 1000c – 1000a – a² – ac – 1000c – ac – c² = c² – a²
2000a + 2000c + 2ac = 1000²
1000a + 1000c + ac = 500000

Eventually i reconsidered my approach.
As i later was to discover, this pitch wasn’t a bad idea after all, but i eventually went for a good ol’ brute force.
The equation could have been used to shorten the time required to solve the problem by brute force a lot, but my solution spits out the correct answer within 5 secs so what the heck.

Solution (PHP):

 <?php

for ($a = 1; $a < 800; $a++)
{

for ($b = $a; $b < 1000; $b++)
{
$c = 1000-$a-$b;

if (bcpow($a,2)+bcpow($b,2))==bcpow($c,2) //GREAT SUCCESS!
{
echo $a*$b*$c;
}
}
}
?>