<< problem 43 - Sub-string divisibility | Triangular, pentagonal, and hexagonal - problem 45 >> |
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?
My 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.
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 toecho 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)
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 <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.
Benchmark
The correct solution to the original Project Euler problem was found in 0.05 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
(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
February 27, 2017 submitted solution
April 19, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler044
My code solves 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 rarely 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:
C# www.mathblog.dk/project-euler-44-smallest-pair-pentagonal-numbers/ (written by Kristian Edlund)
C github.com/eagletmt/project-euler-c/blob/master/40-49/problem44.c (written by eagletmt)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p044.java (written by Nayuki)
Javascript github.com/dsernst/ProjectEuler/blob/master/44 Pentagon numbers.js (written by David Ernst)
Go github.com/frrad/project-euler/blob/master/golang/Problem044.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p044.mathematica (written by Nayuki)
Scala github.com/samskivert/euler-scala/blob/master/Euler044.scala (written by Michael Bayne)
Perl github.com/gustafe/projecteuler/blob/master/044-Pentagon-numbers.pl (written by Gustaf Erikson)
Those links are just an unordered selection of source code I found with a semi-automatic search script on Google/Bing/GitHub/whatever.
You will probably stumble upon better solutions when searching on your own.
Maybe not all linked resources produce the correct result and/or exceed time/memory limits.
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 |
I stopped working on Project Euler problems around the time they released 617.
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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.
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 43 - Sub-string divisibility | Triangular, pentagonal, and hexagonal - problem 45 >> |