Problem 44: Pentagon numbers

(see projecteuler.net/problem=44)

Pentagonal numbers are generated by the formula, P_n=n(3n-1)/2. The first ten pentagonal numbers are:
1, 5, 12, 22, 35, 51, 70, 92, 117, 145, ...

It can be seen that P_4 + P_7 = 22 + 70 = 92 = P_8. However, their difference, 70 - 22 = 48, is not pentagonal.

Find the pair of pentagonal numbers, P_j and P_k, for which their sum and difference are pentagonal and D = |P_k - P_j| is minimised; what is the value of D?

Algorithm

The function isPentagonal(x) plays a core role is my solution:
under the assumption that x is pentagonal, that means x = P(n):
P(n) = n (3n - 1) / 2 → pentagonal formula from problem statement
P(n) = frac{3}{2} n^2 - frac{n}{2}
2 P(n) = 3n^2 - n
0 = 3n^2 - n - 2P(n)

Solving this quadratic equation (coefficients a=3, b=-1, c=-2P(n)):
n_{1,2} = dfrac{-b \pm sqrt{b^2 - 4ac}}{2a}

n_{1,2} = dfrac{1 \pm sqrt{1 + 24P(n)}}{6}

n must be a positive integer, therefore the only possible solution is
n = dfrac{1 + sqrt{1 + 24P(n)}}{6}

= dfrac{1 + sqrt{1 + 24x}}{6}

When generating pentagonal numbers, my code checks P(n)-P(n-d) and P(n)+P(n-d) for all d < n whether both are pentagonal, too.

Modifications by HackerRank

The Hackerrank problem is a bit easier because the distance is predefined by the user.

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 <cmath>
 
bool isPentagonal(unsigned long long x)
{
// see explanation above
unsigned long long n = (1 + sqrt(24*x + 1)) / 6;
 
// if x was indeed a pentagonal number then our assumption P(n) = x must be true
auto p_n = n * (3 * n - 1) / 2;
return p_n == x;
}
 
int main()
{
//#define ORIGINAL
#ifdef ORIGINAL
 
const unsigned int HugeNumber = 999999999;
unsigned int best = HugeNumber;
unsigned int n = 2; // start with P(2)
unsigned int last = 1; // P(1)
 
while (best == HugeNumber)
{
// next pentagonal number
auto p_n = n * (3 * n - 1) / 2;
// difference to closest pentagonal number larger than our best result ?
if (p_n - last > best)
break;
 
// check all pairs P(n) and P(n - distance) where 1 <= distance < n
for (unsigned int distance = 1; distance < n; distance++)
{
// compute P(n - distance) pentagonal number
auto x = n - distance;
auto p_x = x * (3 * x - 1) / 2;
 
// their sum and difference
auto sum = p_n + p_x;
auto difference = p_n - p_x;
 
// too far away ?
if (difference > best)
break;
 
// yes, found something
if (isPentagonal(sum) && isPentagonal(difference))
best = difference;
}
 
// check next pentagonal number
last = p_n;
n++;
}
std::cout << best << std::endl;
 
#else
 
unsigned int maxIndex;
unsigned int distance;
std::cin >> maxIndex >> distance;
 
// iterate over all pairs at a given distance
for (unsigned long long n = distance + 1; n <= maxIndex; n++)
{
auto p_n = n * (3 * n - 1) / 2;
 
auto x = n - distance;
auto p_x = x * (3 * x - 1) / 2;
 
// check sum and difference
auto sum = p_n + p_x;
auto difference = p_n - p_x;
 
// yes, found something
if (isPentagonal(sum) || isPentagonal(difference))
std::cout << p_n << std::endl;
}
#endif
 
return 0;
}

This solution contains 17 empty lines, 14 comments and 5 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.

Input data (separated by spaces or newlines):
Note: This live test requires Hackerrank-style input: enter a max. pentagonal index and a distance, then the program will display all pentagonal indices

This is equivalent to
echo 8 | ./44

Output:

(please click 'Go !')

Note: the original problem's input 9 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 0.05 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

February 27, 2017 submitted solution
April 19, 2017 added comments

Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler044

My code solved 6 out of 6 test cases (score: 100%)

Difficulty

Project Euler ranks this problem at 5% (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 never an option.

Similar problems at Project Euler

Problem 45: Triangular, pentagonal, and hexagonal

Note: I'm not even close to solving all problems at Project Euler. Chances are that similar problems do exist and I just haven't looked at them.

Links

projecteuler.net/thread=44 - 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-44-smallest-pair-pentagonal-numbers/ (written by Kristian Edlund)
Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p044.java (written by Nayuki)
Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p044.mathematica (written by Nayuki)
C: github.com/eagletmt/project-euler-c/blob/master/40-49/problem44.c (written by eagletmt)
Go: github.com/frrad/project-euler/blob/master/golang/Problem044.go (written by Frederick Robinson)
Javascript: github.com/dsernst/ProjectEuler/blob/master/44 Pentagon numbers.js (written by David Ernst)
Scala: github.com/samskivert/euler-scala/blob/master/Euler044.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 already 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.

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The 133 solved problems had an average difficulty of 16.9% at Project Euler and I scored 11,174 points (out of 12300) at Hackerrank's Project Euler+.
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