Problem #142 say:

Find the smallest x + y + z with integers x > y > z > 0 such that x + y, x − y, x + z, x − z, y + z, y − z are all perfect squares.

Now one can brute force this by looping over x, y and z. That takes around 4 hours on a modern computer. However there are a few relationships we can take advantage of:

x+y=a
x-y=b
x+z=c
x-z=d
y+z=e
y-z=f

e=a-d
f=a-c
b=c-e

(x+z) = (x+y)-(x-z)
(y-z) = (x+y)-(x+z)
(x-y) = (x+z)-(y+z)

x=(a+b)/2
y=(e+f)/2
z=(c-d)/2

these relationships allow us to loop on a,c,d and get the rest of the numbers from those three, this makes the solution run in around 33 msecs. That’s around a ~436363% speedup! Solution provided in c#/mono.

using System;
using System.Collections;
using System.Diagnostics;

namespace ProjectEuler
{
    class Program
    {
        static object LockHandle = new object();
        static void Main(string[] args)
        {
            Stopwatch sw = new Stopwatch();
            long a2, b2, c2, d2, e2, f2;
            bool solved = false;
            sw.Start();

            for (long a = 10;!solved;a++)
            {
                a2 = a * a;
                for (long c = 5 + (0 & a); c < a && !solved; c += 2)
                {
                    c2 = c * c;
                    f2 = a2 - c2;
                    if (f2 < 1 || !IsSquare(f2))
                        continue;
                    for (long d = 2 + (1 & c); d < c; d += 2)
                    {
                        d2 = d * d;
                        e2 = a2 - d2;
                        if (e2 < 1 || !IsSquare(e2))
                            continue;
                        b2 = c2 - e2;
                        if (b2 > 0 && IsSquare(b2))
                        {
                            long x = (a2 + b2) / 2;
                            long y = (e2 + f2) / 2;
                            long z = (c2 - d2) / 2;
                            solved = true;
                            Console.WriteLine("x= " + x.ToString() +
                                 " y = " + y.ToString() + " z = " + z.ToString()
                                 + " sum = " + (z + y + x).ToString());
                            break;
                        }
                    }

                }
            }

            Console.WriteLine(sw.Elapsed);
            Console.ReadLine();
        }
        public static bool IsSquare(long n )
        {
            double root = Math.Sqrt(n);
            return (root % 1 == 0);
        }
    }
}