<< problem 137 - Fibonacci golden nuggets Pythagorean tiles - problem 139 >>

Problem 138: Special isosceles triangles

Consider the isosceles triangle with base length, b = 16, and legs, L = 17.

By using the Pythagorean theorem it can be seen that the height of the triangle, h = sqrt{172 - 82} = 15, which is one less than the base length.

With b = 272 and L = 305, we get h = 273, which is one more than the base length, and this is the second smallest isosceles triangle with the property that h = b \pm 1.

Find sum{L} for the twelve smallest isosceles triangles for which h = b \pm 1 and b, L are positive integers.

My Algorithm

Even though I felt right from the start that brute force can solve this problem, I tried it anyways.
And within a few seconds the function bruteForce displayed these 7 solutions:
17, 305, 5473, 98209, 1762289, 31622993, 567451585

I wasn't willing to dig deep into mathematical territory and decided to look hard at those numbers ...
and yes, a pattern revealed:
17
305 = 17*18 - 1
5473 = 305*18 - 17
98209 = 5473*18 - 305
1762289 = 98209*18 - 5473
31622993 = 1762289*18 - 98209
567451585 = 31622993*18 - 1762289

That's a surprisingly simple series:
x_0 = 1
x_1 = 17
x_n = x_{n-1} * 18 - x_{n-2}

My code looks a bit messy because of the Hackerrank modifications.

Alternative Approaches

You can apply Pell's equation to discover this sequence without "taking a hard look".
And many Fibonacci numbers pop up along the way ...

Modifications by HackerRank

Not only 12 but up to 10^18 triangles have to be found.
My code precomputes the first 100000 triangles (enough to pass 3 out of 4 test cases) and then
performs a simple lookup.

All numbers have to be modulo 10^9+7.

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 2" | ./138

Output:

Note: the original problem's input 12 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. Or just jump to my GitHub repository.

The code contains #ifdefs to switch between the original problem and the Hackerrank version.
Enable #ifdef ORIGINAL to produce the result for the original problem (default setting for most problems).

       #include <iostream>
#include <vector>
#include <cmath>

#define ORIGINAL

// find the first sevens solutions in a few seconds
void bruteForce(long long limit)
{
for (long long base = 2; base < limit; base += 2)
{
// both right-angled triangles with side lengths L, h, half
auto half = base / 2;

// two potential solutions
auto height1 = base - 1;
auto height2 = base + 1;

// compute L1 and L2
auto triangle1 = half * half + height1 * height1;
long long hypo1 = sqrt(triangle1);
if (hypo1 * hypo1 == triangle1)
std::cout << hypo1 << " => b=" << base << " h=" << height1 << " diff=" << (base - height1) << std::endl;

auto triangle2 = half * half + height2 * height2;
long long hypo2 = sqrt(triangle2);
if (hypo2 * hypo2 == triangle2)
std::cout << hypo2 << " => b=" << base << " h=" << height2 << " diff=" << (base - height2) << std::endl;
}
}

int main()
{
// precompute the first 10^6 solutions
std::vector<unsigned long long> solutions;

// the first seven solutions reveal a pattern ...
// 17
// 305 = 17*18 - 1
// 5473 = 305*18 - 17
// 98209 = 5473*18 - 305
// 1762289 = 98209*18 - 5473
// 31622993 = 1762289*18 - 98209
// 567451585 = 31622993*18 - 1762289

#ifdef ORIGINAL
const unsigned int MaxPrecompute = 12;
#else
const unsigned int MaxPrecompute = 1000000;
#endif

// first solution is 17
long long current  = 17;
solutions.push_back(current);
// add it to the total
long long sum      = current;

// previous solution ("zero-th" solution is 1
long long previous =  1;
for (unsigned int i = 2; i <= MaxPrecompute; i++)
{
// one more step ...
auto next = current * 18 - previous;

#ifndef ORIGINAL
// prevent negative numbers
const long long Modulo = 1000000007;
if (next < 0)
next += Modulo;
#endif

// next => current => previous
previous  = current;
current   = next;

sum += current;

#ifndef ORIGINAL
// keep only residue
sum     %= Modulo;
current %= Modulo;
#endif

// store solution
solutions.push_back(sum);
}

unsigned int tests = 1;
std::cin >> tests;
while (tests--)
{
unsigned long long smallest = 12;
std::cin >> smallest;

auto index = smallest - 1;
// not pre-computed ? => abort
if (index >= solutions.size())
return 0;

// look up result
std::cout << solutions[index] << std::endl;
}

return 0;
}


This solution contains 19 empty lines, 24 comments and 11 preprocessor commands.

Benchmark

The correct solution to the original Project Euler problem was found in less than 0.01 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=gnu++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

July 10, 2017 submitted solution

Hackerrank

My code solves 4 out of 5 test cases (score: 66.67%)

I failed 0 test cases due to wrong answers and 1 because of timeouts

Difficulty

Project Euler ranks this problem at 45% (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.

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 black problems are solved but access to the solution is blocked for a few days until the next problem is published [new] the flashing problem is the one I solved most recently

I stopped working on Project Euler problems around the time they released 617.
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The 310 solved problems (that's level 12) had an average difficulty of 32.6% at Project Euler and
I scored 13526 points (out of 15700 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.

 << problem 137 - Fibonacci golden nuggets Pythagorean tiles - problem 139 >>
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