<< 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.

# 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

# 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>
// 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.

# 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 to`echo "100000 5 6" | ./45`

Output:

*(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 less than 0.01 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/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 never 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:

Python: www.mathblog.dk/project-euler-45-next-triangle-pentagonal-hexagonal-number/ (written by Kristian Edlund)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p045.java (written by Nayuki)

Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p045.mathematica (written by Nayuki)

C: github.com/eagletmt/project-euler-c/blob/master/40-49/problem45.c (written by eagletmt)

Go: github.com/frrad/project-euler/blob/master/golang/Problem045.go (written by Frederick Robinson)

Javascript: github.com/dsernst/ProjectEuler/blob/master/45 Triangular, pentagonal, and hexagonal.js (written by David Ernst)

Scala: github.com/samskivert/euler-scala/blob/master/Euler045.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 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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My username at Project Euler is

**stephanbrumme**while it's stbrumme at Hackerrank.

<< problem 44 - Pentagon numbers | Goldbach's other conjecture - problem 46 >> |