Problem 9: Special Pythagorean triplet

(see projecteuler.net/problem=9)

A Pythagorean triplet is a set of three natural numbers, a < b < c,
for which, a^2 + b^2 = c^2

For example, 3^2 + 4^2 = 9 + 16 = 25 = 5^2.

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

Algorithm

I loop through all pairs a<b and compute c=sqrt{a^2+b^2}.
If c is an integer and a+b+c<=3000 then the largest product abc is stored.

Modifications by HackerRank

For some sums a+b+c multiple solutions might exist and the largest product abc should be returned.
It is necessary to have a pre-computation step of all perimeters' solutions to handle the huge amount of test cases.

My code

… was written in C++ and can be compiled with G++, Clang++, Visual C++. You can download it, too.

#include <iostream>
#include <vector>
#include <cmath>
 
int main()
{
// precompute all pairs a<b<c where a+b+c <= 3000
const int MaxPerimeter = 3000;
// -1 means "no triplets" for that perimeter
const int NoSolution = -1;
 
// cache[0] remains unused
std::vector<int> cache(MaxPerimeter + 1, NoSolution);
 
// scan all pairs a<b
for (int a = 1; a < MaxPerimeter; a++)
for (int b = a + 1; b < MaxPerimeter - a; b++)
{
// find c
int c2 = a*a + b*b;
// approximate square root as integer
int c = sqrt(c2);
// was it the correct square root ?
if (c*c != c2)
continue;
 
// check summing condition
int sum = a + b + c;
if (sum > MaxPerimeter)
break;
 
// better solution than before ?
if (cache[sum] < a*b*c)
cache[sum] = a*b*c;
}
 
unsigned int tests;
std::cin >> tests;
while (tests--)
{
unsigned int n;
std::cin >> n;
// just lookup results (-1 if no solution)
std::cout << cache[n] << std::endl;
}
return 0;
}

This solution contains 6 empty lines, 10 comments and 3 preprocessor commands.

Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

Number of test cases (1-5):

Input data (separated by spaces or newlines):

This is equivalent to
echo "1 50" | ./9

Output:

(please click 'Go !')

Note: the original problem's input 1000 cannot be entered
because just copying results is a soft skill reserved for idiots.

(this interactive test is still under development, computations will be aborted after one second)

Benchmark

The correct solution to the original Project Euler problem was found in 0.02 seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=c++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

February 23, 2017 submitted solution
March 29, 2017 added comments

Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler009

My code solved 8 out of 8 test cases (score: 100%)

Difficulty

Project Euler ranks this problem at 5% (out of 100%).

Hackerrank describes this problem as easy.

Note:
Hackerrank has strict execution time limits (typically 2 seconds for C++ code) and often a much wider input range than the original problem.
In my opinion, Hackerrank's modified problems are usually a lot harder to solve. As a rule thumb: brute-force is never an option.

Links

projecteuler.net/thread=9 - the best forum on the subject (note: you have to submit the correct solution first)

Code in various languages:

Python: www.mathblog.dk/pythagorean-triplets/ (written by Kristian Edlund)
Haskell: github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p009.hs (written by Nayuki)
Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p009.java (written by Nayuki)
Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p009.mathematica (written by Nayuki)
C: github.com/eagletmt/project-euler-c/blob/master/1-9/problem9.c (written by eagletmt)
Go: github.com/frrad/project-euler/blob/master/golang/Problem009.go (written by Frederick Robinson)
Javascript: github.com/dsernst/ProjectEuler/blob/master/9 Special Pythagorean triplet.js (written by David Ernst)
Scala: github.com/samskivert/euler-scala/blob/master/Euler009.scala (written by Michael Bayne)

Heatmap

green problems solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too.
yellow problems score less than 100% at Hackerrank (but still solve the original problem).
gray problems are already solved but I haven't published my solution yet.
blue problems are already solved and there wasn't a Hackerrank version of it (at the time I solved it) or I didn't care about it because it differed too much.

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

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The 126 solved problems had an average difficulty of 16.0% at Project Euler and I scored 11,074 points (out of 12500) at Hackerrank's Project Euler+.
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