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# Problem 142: Perfect Square Collection

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.

# My Algorithm

I have to solve six equations:
(1) a^2 = x + y
(2) b^2 = x - y
(3) c^2 = x + z
(4) d^2 = x - z
(5) e^2 = y + z
(6) f^2 = y - z

And this condition has to be obeyed:
(7) x > y > z > 0

If I add (1) and (2), then I have x:
(8) a^2 + b^2 = 2xx = frac{1}{2}(a^2 + b^2)

After reordering (1) I get the value of y:
(9) y = a^2 - x

The same trick applied to (3):
(10) z = c^2 - x

All equations can be solved if I find valid values for a^2, b^2 and c^2.
However, when computing d^2, e^2 and f^2 based on (4), (5) and (6) then a few "fake" solutions appear where
d^2, e^2 and f^2 are not perfect squares.

My program has three nested loops that iterate over possible values a, b and c.
A few optimizations are required to finish the algorithm in a reasonable amount of time:

• x must be an integer and from (8) follows that a^2 + b^2 must be even, that means a and b are both odd or both even
• all numbers are positive therefore from (10) follows that c^2 > x which is the same as c > sqrt{x}
• instead of checking a number on-the-fly whether it is a perfect square, I precompute a small look-up table

## Alternative Approaches

There are many faster solutions. My short program finishes in less than 0.1 seconds - that's why I don't care about more optimizations.

# Interactive test

This feature is not available for the current problem.

# My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too. Or just jump to my GitHub repository.

       #include <iostream>
#include <bitset>
#include <cmath>

int main()
{
// in my tests I found that all tested squares are well below one million
const unsigned int Limit = 1000000;

// record all square numbers
// [n] => [true, if n is a perfect square]
std::bitset<Limit> isSquare; // default is false / zero
for (unsigned int i = 1; i*i < isSquare.size(); i++)
isSquare[i*i] = true;

// substitute in (3):
// (9) z = c^2 - x
for (unsigned int a = 3; ; a++)
{
// a and b must be both odd or both even
unsigned int minB = (a % 2 == 0) ? 2 : 1;
for (unsigned int b = minB; b < a; b += 2) // keep parity, skip every second number
{
// from (8): compute x
auto x = (a*a + b*b) / 2;
// from (9): compute y
auto y = a*a - x;
// ensure x > y
if (x <= y)
break;

for (unsigned int c = (unsigned int)sqrt(x) + 1; ; c++)
{
// from (10): compute z
auto z = c*c - x;
// ensure y > z
if (y <= z)
break;

// check whether d^2, e^2 and f^2 are perfect squares
if (isSquare[x - z] &&
isSquare[y + z] &&
isSquare[y - z])
{
// found a solution, stop program
std::cout << x + y + z << std::endl;
return 0;
}
}
}
}

// never reached
}


This solution contains 6 empty lines, 14 comments and 3 preprocessor commands.

# Benchmark

The correct solution to the original Project Euler problem was found in 0.06 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=gnu++11 -DORIGINAL)

See here for a comparison of all solutions.

Note: interactive tests run on a weaker (=slower) computer. Some interactive tests are compiled without -DORIGINAL.

# Changelog

June 27, 2017 submitted solution

# Difficulty

Project Euler ranks this problem at 45% (out of 100%).

# Heatmap

Please click on a problem's number to open my solution to that problem:

 green solutions solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too yellow solutions score less than 100% at Hackerrank (but still solve the original problem easily) gray problems are already solved but I haven't published my solution yet blue solutions are relevant for Project Euler only: there wasn't a Hackerrank version of it (at the time I solved it) or it differed too much orange problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte red problems are not solved yet but I wrote a simulation to approximate the result or verified at least the given example - usually I sketched a few ideas, too black problems are solved but access to the solution is blocked for a few days until the next problem is published [new] the flashing problem is the one I solved most recently

I stopped working on Project Euler problems around the time they released 617.
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The 310 solved problems (that's level 12) had an average difficulty of 32.6% at Project Euler and
I scored 13526 points (out of 15700 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.

My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.

Look at my progress and performance pages to get more details.

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