Problem 75: Singular integer right triangles

(see projecteuler.net/problem=75)

It turns out that 12 cm is the smallest length of wire that can be bent to form an integer sided right angle triangle in exactly one way, but there are many more examples.

12 cm: (3,4,5)
24 cm: (6,8,10)
30 cm: (5,12,13)
36 cm: (9,12,15)
40 cm: (8,15,17)
48 cm: (12,16,20)

In contrast, some lengths of wire, like 20 cm, cannot be bent to form an integer sided right angle triangle, and other lengths allow more than one solution to be found;
for example, using 120 cm it is possible to form exactly three different integer sided right angle triangles.

120 cm: (30,40,50), (20,48,52), (24,45,51)

Given that L is the length of the wire, for how many values of L <= 1,500,000 can exactly one integer sided right angle triangle be formed?

Algorithm

Euclid's formula produces all triplets a, b, c such that a^2 + b^2 = c^2 (see en.wikipedia.org/wiki/Pythagorean_triple)
All basic triplets can be generated by:
a = m^2 - n^2
b = 2mn
c = m^2 + n^2
where m > n and (m+n) mod 2 == 1 and m, n are coprime (i.e. gcd(m,n) = 1)

To find all "non-basic" triplets: multiply a, b, c by any integer k > 1.

My pre-computation step 1 counts how many combinations exists for every perimeter a+b+c below 1500000.
The function gcd is known from previous problems.

In step 2, only those perimeters which are unique are copied to a container named once.

The final step is running the tests: find the smallest perimeter exceeding the user input (=1500000 for the original problem).
Its index is the desired result.

My code

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

#include <iostream>
#include <vector>
#include <algorithm>
#include <cmath>
 
// find greatest common divisor
template <typename T>
T gcd(T a, T b)
{
while (a != 0)
{
T c = a;
a = b % a;
b = c;
}
return b;
}
 
int main()
{
// pre-computation step 1: find all triangle combinations up to 1500000
const unsigned int MaxLength = 5 * 1000 * 1000;
// [length] => [number of valid combinations]
std::vector<unsigned int> combinations(MaxLength, 0);
 
for (unsigned int m = 2; m < sqrt(MaxLength); m++)
for (unsigned int n = 1; n < m; n++)
{
// only valid m and n
if ((m + n) % 2 != 1)
continue;
if (gcd(m, n) != 1)
continue;
 
// compute basic triplet
auto a = m*m - n*n;
auto b = 2*m*n;
auto c = m*m + n*n;
auto sum = a + b + c;
 
// and all of its multiples
unsigned int k = 1;
while (k*sum <= MaxLength)
{
combinations[k*sum]++;
k++;
}
}
 
// pre-computation step 2: extract those with exactly one combination
std::vector<unsigned int> once;
for (size_t i = 0; i < combinations.size(); i++)
if (combinations[i] == 1)
once.push_back(i);
 
// running the test-cases is a simple look-up
unsigned int tests = 1;
std::cin >> tests;
while (tests--)
{
unsigned int limit = 1500000;
std::cin >> limit;
 
// find first triangle perimeter exceeding 1500000 with exactly one combination
auto pos = std::upper_bound(once.begin(), once.end(), limit);
// count how many one-combo-triangles are smaller
auto result = std::distance(once.begin(), pos);
// and print that number
std::cout << result << std::endl;
}
}

This solution contains 8 empty lines, 11 comments and 4 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" | ./75

Output:

(please click 'Go !')

Note: the original problem's input 1500000 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 0.15 seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.
Peak memory usage was about 26 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

March 12, 2017 submitted solution
May 3, 2017 added comments

Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler075

My code solves 7 out of 7 test cases (score: 100%)

Difficulty

Project Euler ranks this problem at 25% (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=75 - 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-75-lengths-of-wire-right-angle-triangle/ (written by Kristian Edlund)
Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p075.java (written by Nayuki)
Go: github.com/frrad/project-euler/blob/master/golang/Problem075.go (written by Frederick Robinson)
Scala: github.com/samskivert/euler-scala/blob/master/Euler075.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.
red problems are solved but exceed the time limit of one minute.

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
251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275
276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300
301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325
326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350
351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375
The 206 solved problems had an average difficulty of 27.5% at Project Euler and
I scored 12,626 points (out of 14300 possible points, top rank was 20 out ouf ≈60000 in July 2017) at Hackerrank's Project Euler+.
Look at my progress and performance pages to get more details.

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

more about me can be found on my homepage, especially in my coding blog.
some names mentioned on this site may be trademarks of their respective owners.
thanks to the KaTeX team for their great typesetting library !