<< problem 43 - Sub-string divisibility Triangular, pentagonal, and hexagonal - problem 45 >>

# Problem 44: Pentagon numbers

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 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:

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

# Hackerrank

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.

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.

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

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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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