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# Problem 12: Highly divisible triangular number

(see projecteuler.net/problem=12)

The sequence of triangle numbers is generated by adding the natural numbers.

So the 7th triangle number would be 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28.

The first ten terms would be:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...

Let us list the factors of the first seven triangle numbers:

1: 1

3: 1,3

6: 1,2,3,6

10: 1,2,5,10

15: 1,3,5,15

21: 1,3,7,21

28: 1,2,4,7,14,28

We can see that 28 is the first triangle number to have over five divisors.

What is the value of the first triangle number to have over five hundred divisors?

# Algorithm

Similar to other problems, my solution consists of two steps

1. precompute all possible inputs

2. for each test case: perform a simple lookup

It takes less than a second to find all such numbers with at most 1000 divisors.

Two "tricks" are responsible to achieve that speed:

You can get all divisors of x by analyzing all potential divisors i<=sqrt{x} instead of i<x.

Whenever we find a valid divisor i then another divisor j=frac{x}{y} exists.

The only exception is i=sqrt{x} because then j=i.

Somehow more subtle is my observation that when numbers have more than about 300 divisors,

the smallest one always end with a zero. I cannot prove that, I just saw it while debugging my code.

I decided to store all my results in a `std::vector`

called `smallest`

where

`smallest[x]`

contains the smallest triangle number with at least `x`

divisors.

While filling that container, the program encounters many "gaps":

e.g. 10 is the smallest number with 4 divisors and 28 is the smallest number with 6 divisors

but there is no number between 10 and 28 with *5* divisors.

Therefore 28 is the smallest number with *at least 5* divisors, too.

## Alternative Approaches

Prime factorization can find the result probably a bit faster.

# My code

… was written in C++ and can be compiled with G++, Clang++, Visual C++. You can download it, too.

#include <iostream>
#include <vector>
int main()
{
// find the smallest number with at least 1000 divisors
// (due to Hackerrank's input range)
const unsigned int MaxDivisors = 1000;
// store [divisors] => [smallest number]
std::vector<unsigned int> smallest;
smallest.push_back(0); // 0 => no divisors
// for index=1 we have triangle=1
// for index=2 we have triangle=3
// for index=3 we have triangle=6
// ...
// for index=7 we have triangle=28
// ...
unsigned int index = 0;
unsigned int triangle = 0; // same as index*(index+1)/2
while (smallest.size() < MaxDivisors)
{
// next triangle number
index++;
triangle += index;
// performance tweak (5x faster):
// I observed that the "best" numbers with more than 300 divisors end with a zero
// that's something I cannot prove right now, I just "saw" that debugging my code
if (smallest.size() > 300 && triangle % 10 != 0)
continue;
// find all divisors i where i*j=triangle
// it's much faster to assume i < j, which means i*i < triangle
// whenever we find i then there is a j, too
unsigned int divisors = 0;
unsigned int i = 1;
while (i*i < triangle)
{
// divisible ? yes, we found i and j, that's two divisors
if (triangle % i == 0)
divisors += 2;
i++;
}
// if i=j then i^2=triangle and we have another divisor
if (i*i == triangle)
divisors++;
// fill gaps:
// e.g. 10 is the smallest number with 4 divisors
// 28 is the smallest number with 6 divisors
// there is no number between 10 and 28 with 5 divisors
// therefore 28 is the smallest number with AT LEAST 5 divisors, too
while (smallest.size() <= divisors)
smallest.push_back(triangle);
}
unsigned int tests;
std::cin >> tests;
while (tests--)
{
unsigned int minDivisors;
std::cin >> minDivisors;
// problem setting asks for "over" x divisors => "plus one"
std::cout << smallest[minDivisors + 1] << std::endl;
}
return 0;
}

This solution contains 9 empty lines, 24 comments and 2 preprocessor commands.

# Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

This is equivalent to`echo "1 7" | ./12`

Output:

*Note:* the original problem's input `500`

__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 0.46 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 23, 2017 submitted solution

March 30, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler012

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.

# Links

projecteuler.net/thread=12 - **the** best forum on the subject (*note:* you have to submit the correct solution first)

Code in various languages:

Python: www.mathblog.dk/triangle-number-with-more-than-500-divisors/ (written by Kristian Edlund)

Haskell: github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p012.hs (written by Nayuki)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p012.java (written by Nayuki)

Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p012.mathematica (written by Nayuki)

C: github.com/eagletmt/project-euler-c/blob/master/10-19/problem12.c (written by eagletmt)

Go: github.com/frrad/project-euler/blob/master/golang/Problem012.go (written by Frederick Robinson)

Javascript: github.com/dsernst/ProjectEuler/blob/master/12 Highly divisible triangular number.js (written by David Ernst)

Scala: github.com/samskivert/euler-scala/blob/master/Euler012.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.

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