<< problem 38 - Pandigital multiples | Champernowne's constant - problem 40 >> |

# Problem 39: Integer right triangles

(see projecteuler.net/problem=39)

If p is the perimeter of a right angle triangle with integral length sides, \{ a,b,c \}, there are exactly three solutions for p = 120.

\{ 20,48,52 \}, \{ 24,45,51 \}, \{ 30,40,50 \}

For which value of p <= 1000, is the number of solutions maximised?

# My Algorithm

Euclid's formula generates all triplets \{ a,b,c \}, see en.wikipedia.org/wiki/Pythagorean triple

Assuming a <= b <= c:

a = k * (m^2 - n^2)

b = k * 2mn

c = k * (m^2 + n^2)

perimeter = a + b + c

Integer numbers m, n, k produce all triplets under these conditions:

- m and n are coprime → their Greatest Common Divisor is 1

- m and n are not both odd

And we can conclude:

a must be positive (as well as b and c) therefore m > n

Furthermore:

perimeter = k * (m^2 - n^2) + k * 2mn + k * (m^2 + n^2)

= k * (m^2 - n^2 + 2mn + m^2 + n^2)

= k * (2m^2 + 2mn)

= 2km * (m+n)

which gives an approximation of the upper limit: 2m^2 < MaxPerimeter

My program evaluates all combinations of m and n. For each valid pair all k are enumerated,

such that the perimeter does not exceed the maximum value.

A simple lookup container `count`

stores for each perimeter the number of triangles.

Following this precomputation step I perform a second step:

extract those perimeters with more triangles than any smaller perimeter.

The value stored at `best[perimeter]`

equals the highest `count[i]`

for all `i <= perimeter`

.

The actual test cases are plain look-ups into `best`

.

# 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 "1 120" | ./39`

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)*

# My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.

#include <iostream>
#include <set>
#include <vector>
// greatest common divisor

unsigned int gcd(unsigned int a, unsigned int b)
{
while (a != 0)
{
unsigned int c = a;
a = b % a;
b = c;
}
return b;
}
int main()
{
const unsigned int MaxPerimeter = 5000000;
// precomputation step 1:
// count all triplets per perimeter (up to upper limit 5 * 10^6)
// [perimeter] => [number of triplets]
std::vector<unsigned int> count(MaxPerimeter + 1, 0);
// note: long long instead of int because otherwise the squares m^2, n^2, ... might overflow
for (unsigned long long m = 1; 2*m*m < MaxPerimeter; m++)
for (unsigned long long n = 1; n < m; n++)
{
// make sure all triplets a,b,c are unique
if (m % 2 == 1 && n % 2 == 1)
continue;
if (gcd(m, n) > 1)
continue;
unsigned int k = 1;
while (true)
{
// see Euclidian formula above
auto a = k * (m*m - n*n);
auto b = k * 2*m*n;
auto c = k * (m*m + n*n);
k++;
// abort if largest perimeter is exceeded
auto perimeter = a + b + c;
if (perimeter > MaxPerimeter)
break;
// ok, found a triplet
count[perimeter]++;
}
}
// precomputation step 2:
// store only best perimeters
unsigned long long bestCount = 0;
std::set<unsigned int> best;
best.insert(0); // degenerated case
for (unsigned int i = 0; i < count.size(); i++)
if (bestCount < count[i])
{
bestCount = count[i];
best.insert(i);
}
// processing input boils down to a simple lookup
unsigned int tests;
std::cin >> tests;
while (tests--)
{
unsigned int maxPerimeter;
std::cin >> maxPerimeter;
// find the perimeter with the largest count
auto i = best.upper_bound(maxPerimeter);
// we went one step too far
i--;
// print result
std::cout << *i << std::endl;
}
return 0;
}

This solution contains 10 empty lines, 15 comments and 3 preprocessor commands.

# Benchmark

The correct solution to the original Project Euler problem was found in 0.11 seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.

Peak memory usage was about 21 MByte.

(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 25, 2017 submitted solution

April 18, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler039

My code solves **7** out of **7** 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 rarely an option.

# Links

projecteuler.net/thread=39 - **the** best forum on the subject (*note:* you have to submit the correct solution first)

Code in various languages:

C#: www.mathblog.dk/project-euler-39-perimeter-right-angle-triangle/ (written by Kristian Edlund)

Haskell: github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p039.hs (written by Nayuki)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p039.java (written by Nayuki)

Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p039.mathematica (written by Nayuki)

C: github.com/eagletmt/project-euler-c/blob/master/30-39/problem39.c (written by eagletmt)

Go: github.com/frrad/project-euler/blob/master/golang/Problem039.go (written by Frederick Robinson)

Javascript: github.com/dsernst/ProjectEuler/blob/master/39 Integer right triangles.js (written by David Ernst)

Scala: github.com/samskivert/euler-scala/blob/master/Euler039.scala (written by Michael Bayne)

Those links are just an unordered selection of source code I found with a semi-automatic search script on Google/Bing/GitHub/whatever.

You will probably stumble upon better solutions when searching on your own. Maybe not all linked resources produce the correct result.

# 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 |

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I scored 13,386 points (out of 15600 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.

# Copyright

I hope you enjoy my code and learn something - or give me feedback how I can improve my solutions.

All of my solutions can be used for any purpose and I am in no way liable for any damages caused.

You can even remove my name and claim it's yours. But then you shall burn in hell.

The problems and most of the problems' images were created by Project Euler.

Thanks for all their endless effort !!!

<< problem 38 - Pandigital multiples | Champernowne's constant - problem 40 >> |