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?

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.

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.

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 "1 120" | ./39

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 0.11 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 never 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:

Python: 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)

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 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:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75
76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125
126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150
151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175
176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200
201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225
226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250
The 160 solved problems had an average difficulty of 21.8% at Project Euler and I scored 11,807 points (out of 13100) at Hackerrank's Project Euler+.
My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.
more about me can be found on my homepage.
some names mentioned on this site may be trademarks of their respective owners.
thanks to the KaTeX team for their great typesetting library !