Grind Report 10

Problem 9

A Pythagorean triplet is a set of three natural numbers, a < b < c, for which,

a² + b² = c²

For example, 3² + 4² = 9 + 16 = 25 = 5².

There exists exactly one Pythagorean triplet for which a + b + c = 1000.
Find the product abc.

*Warning: This contains the solution!*

As I began working on this problem I decided I should learn a little bit more about Pythagorean triplets. So I went to the Wikipedia page for them. Here I found the useful Euclid's formula.

I started playing around with this...

a + b + c = 1000
2mn + (m² - n²) + (m² + n²) = 1000
2mn + 2m² = 1000
mn + m² = 500

Unfortunately I only have one equation and 2 unknowns so I wasn't quite sure what to do with this by hand. There are two things I do know though.

1. m > n > 0
   
2. both m and n are integral
   
3. m must be less than sqrt(500) = 22.36

In order to solve this I quickly threw together a simple program to figure out m and n.

 for (int m = 1; m <= 22; ++m)
{
for (int n = 1; n < m; ++n)
{
if (m * n + m * m == 500)
 cout << m << ", " << n;
}
}

This could have been much more efficient like starting from 22 and working down.

20, 5 was the output to this program. I simply had to plug them back in and then solve for abc. Very simple and easy. I may come back to this and write something to be able to solve for any number - not just 1000. We'll see.

*saved game*