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

# Problem 138: Special isosceles triangles

(see projecteuler.net/problem=138)

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.

# My code

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

The code contains `#ifdef`

s 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;
// add solution
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.

# 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 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)*

# Benchmark

The correct solution to the original Project Euler problem was found in less than 0.01 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

July 10, 2017 submitted solution

July 10, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler138

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.

# Links

projecteuler.net/thread=138 - **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-138-special-isosceles-triangles/ (written by Kristian Edlund)

Go: github.com/frrad/project-euler/blob/master/golang/Problem138.go (written by Frederick Robinson)

# 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 or the memory limit of 256 MByte.

*Please click on a problem's number to open my solution to that problem:*

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I scored 13,183 points (out of 15300 possible points, top rank was 17 out of ≈60000 in August 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.

# 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 137 - Fibonacci golden nuggets | Pythagorean tiles - problem 139 >> |