<< problem 134 - Prime pair connection | Singleton differences - problem 136 >> |

# Problem 135: Same differences

(see projecteuler.net/problem=135)

Given the positive integers, x, y, and z, are consecutive terms of an arithmetic progression, the least value of the positive integer, n,

for which the equation, x^2 - y^2 - z^2 = n, has exactly two solutions is n = 27:

34^2 - 27^2 - 20^2 = 12^2 - 9^2 - 6^2 = 27

It turns out that n = 1155 is the least value which has exactly ten solutions.

How many values of n less than one million have exactly ten distinct solutions?

# My Algorithm

Let's assume y = a, x = a + b and z = a - b. Then I have to solve:

(a + b)^2 - a^2 - (a - b)^2

= a^2 + 2ab + b^2 - a^2 - a^2 + 2ab - b^2

= 4ab - a^2

= a(4b - a)

All variables are positive integers, therefore 1 <= a < n.

The value inside the brackets has to be positive, too, and b must be lower than a such that \lceil frac{a}{4} \rceil <= b < a

Finally, iterating over all `solutions`

and counting how often 10 occurs given the desired result.

## Modifications by HackerRank

Just print how many solutions exists for a given input. The upper limit is 8 million (inclusive) opposed to one million (exclusive) of the original problem.

# 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>
//#define ORIGINAL

int main()
{
#ifdef ORIGINAL
unsigned int limit = 1000000; // "less than one million"
#else
unsigned int limit = 8000001; // up to 8 million (inclusive)
#endif
// precompute solutions
std::vector<unsigned int> solutions(limit, 0);
for (unsigned int a = 1; a < limit; a++)
for (auto b = (a + 3) / 4; b < a; b++)
{
auto current = a * (4*b - a);
if (current >= limit)
break;
solutions[current]++;
}
#ifdef ORIGINAL
// count all with exactly 10 solutions
unsigned int count = 0;
for (auto s : solutions)
if (s == 10)
count++;
std::cout << count << std::endl;
#else
// look up number of solutions
unsigned int tests;
std::cin >> tests;
while (tests--)
{
unsigned int pos;
std::cin >> pos;
std::cout << solutions[pos] << std::endl;
}
#endif
return 0;
}

This solution contains 8 empty lines, 4 comments and 8 preprocessor commands.

# Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

This live test is based on the Hackerrank problem.

This is equivalent to`echo "1 1155" | ./135`

Output:

*(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.

Peak memory usage was about 6 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

May 19, 2017 submitted solution

May 22, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler135

My code solves **16** out of **16** test cases (score: **100%**)

# Difficulty

Project Euler ranks this problem at **45%** (out of 100%).

Hackerrank describes this problem as **medium**.

*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=135 - **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-135-same-differences/ (written by Kristian Edlund)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p135.java (written by Nayuki)

# 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 134 - Prime pair connection | Singleton differences - problem 136 >> |