<< problem 44 - Pentagon numbers | Goldbach's other conjecture - problem 46 >> |
Problem 45: Triangular, pentagonal, and hexagonal
(see projecteuler.net/problem=45)
Triangle, pentagonal, and hexagonal numbers are generated by the following formulae:
Triangle T_n=n(n+1)/2
→ 1, 3, 6, 10, 15, ...
Pentagonal P_n=n(3n-1)/2
→ 1, 5, 12, 22, 35, ...
Hexagonal H_n=n(2n-1)
→ 1, 6, 15, 28, 45, ...
It can be verified that T_285 = P_165 = H_143 = 40755.
Find the next triangle number that is also pentagonal and hexagonal.
My Algorithm
In Problem 42 and Problem 44 I already had to check a number whether it is a triangular or a pentagonal number.
My code generates all hexagonal numbers (starting with H_144). And stop as soon as I find a hexagonal number that is triangular and pentagonal, too.
By the way: every hexagonal number is triangular, too:
H_n=T_{2n-1}
Modifications by HackerRank
The problem was heavily modified by Hackerrank:
the program has to find all numbers, up to an input value, that are
- triangular and pentagonal or
- pentagonal and hexagonal
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 "100000 5 6" | ./45
Output:
(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. Or just jump to my GitHub repository.
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>
// note: isTriangular and isPentagonal based on Euler problems 42 and 44
bool isTriangular(unsigned long long x)
{
unsigned long long n = sqrt(2*x);
// if n is actually the right answer then t(n) = x
unsigned long long check = n * (n + 1) / 2;
return (x == check);
}
bool isPentagonal(unsigned long long x)
{
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
// 143 is the first number which is triangular, pentagonal and hexagonal
for (unsigned int i = 144; ; i++)
{
unsigned int hexagonal = i * (2*i - 1);
if (isPentagonal(hexagonal))
{
// found it !
std::cout << hexagonal << std::endl;
return 0;
}
}
#else
// hexagonal numbers grow the fastest, triangular the slowest
unsigned long long limit;
unsigned int a, b;
std::cin >> limit >> a >> b;
// triangular and pentagonal at the same time
if (a == 3 && b == 5)
{
// let's generate the sequence of all pentagonal numbers, check if triangular, too
for (unsigned long long i = 1; ; i++)
{
auto pentagonal = i * (3*i - 1) / 2;
if (pentagonal >= limit)
break;
if (isTriangular(pentagonal))
std::cout << pentagonal << std::endl;
}
}
// same idea for pentagonal and hexagonal numbers
if (a == 5 && b == 6)
{
// let's generate the sequence of all hexagonal numbers, check if pentagonal, too
for (unsigned long long i = 1; ; i++)
{
auto hexagonal = i * (2*i - 1);
if (hexagonal >= limit)
break;
if (isPentagonal(hexagonal))
std::cout << hexagonal << std::endl;
}
}
#endif
return 0;
}
This solution contains 12 empty lines, 11 comments and 5 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.
(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/euler045
My code solves 8 out of 8 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 42: Coded triangle numbers
Problem 44: Pentagon numbers
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=45 - 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-45-next-triangle-pentagonal-hexagonal-number/ (written by Kristian Edlund)
C github.com/eagletmt/project-euler-c/blob/master/40-49/problem45.c (written by eagletmt)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p045.java (written by Nayuki)
Javascript github.com/dsernst/ProjectEuler/blob/master/45 Triangular, pentagonal, and hexagonal.js (written by David Ernst)
Go github.com/frrad/project-euler/blob/master/golang/Problem045.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p045.mathematica (written by Nayuki)
Scala github.com/samskivert/euler-scala/blob/master/Euler045.scala (written by Michael Bayne)
Perl github.com/gustafe/projecteuler/blob/master/045-Triangular-Pentagonal-Hexagonal.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 44 - Pentagon numbers | Goldbach's other conjecture - problem 46 >> |