<< problem 8 - Largest product in a series | Summation of primes - problem 10 >> |

# 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:

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

Output:

*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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