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.

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.

Number of test cases (1-5):

Input data (separated by spaces or newlines):

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

Output:

(please click 'Go !')

(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.

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>
 
//#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.

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.
Peak memory usage was about 6 MByte.

(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

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

45% 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.

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
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The 306 solved problems (that's level 12) had an average difficulty of 32.5% 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.

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.
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