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# Problem 6: Sum square difference

(see projecteuler.net/problem=6)

The sum of the squares of the first ten natural numbers is,

1^2 + 2^2 + ... + 10^2 = 385

The square of the sum of the first ten natural numbers is,

(1 + 2 + ... + 10)^2 = 55^2 = 3025

Hence the difference between the sum of the squares of the first ten natural numbers

and the square of the sum is 3025 - 385 = 2640.

Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum.

# Algorithm

A very simple problem:

- a `for`

-loop adds all natural numbers (`sum`

) and their squares (`sumSquared`

).

- finally `squaredSum=sum*sum`

- and print the difference between `squaredSum - sumSquared`

The only minor hiccup was to switch from `int`

to `long long`

(basically from 32 to 64 bits).

## Alternative Approaches

The series of sums of all natural numbers are the so-called Triangular numbers (en.wikipedia.org/wiki/Triangular_number).

They have a closed form, too:

sum{x}=frac{x(x+1)}{2}

And there is a closed form for the sum of squares as well (en.wikipedia.org/wiki/Square_pyramidal_number)

sum{x^2}=frac{x(x+1)(2x+1)}{6}

I can easily derive the formula for triangular numbers but the one for the sum of squares isn't

that obvious to me - I had to look it up.

Therefore it would feel like cheating when using the closed form in my code without further explanation ...

and the simple `for`

-loop is fast enough to pass all tests, too.

# My code

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

#include <iostream>
int main()
{
unsigned int tests;
std::cin >> tests;
while (tests--)
{
unsigned long long x;
std::cin >> x;
unsigned long long sum = 0; // 1 + 2 + ...
unsigned long long sumSquared = 0; // 1^2 + 2^2 + ...
for (unsigned long long i = 1; i <= x; i++)
{
sum += i;
sumSquared += i*i;
}
// chances are that your compiler (partially) unrolls this simple loop
// actually we don't need a loop for the sum (and the sum of squares)
// => see "Alternative" section above
// we had (1+2+...) instead of (1+2+...)^2
unsigned long long squaredSum = sum * sum;
std::cout << (squaredSum - sumSquared) << std::endl;
}
return 0;
}

This solution contains 5 empty lines, 4 comments and 1 preprocessor command.

# 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 1000" | ./6`

Output:

*Note:* the original problem's input `100`

__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 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 23, 2017 submitted solution

March 28, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler006

My code solves **2** out of **2** 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.

# Links

projecteuler.net/thread=6 - **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-problem-6/ (written by Kristian Edlund)

Haskell: github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p006.hs (written by Nayuki)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p006.java (written by Nayuki)

Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p006.mathematica (written by Nayuki)

C: github.com/eagletmt/project-euler-c/blob/master/1-9/problem6.c (written by eagletmt)

Go: github.com/frrad/project-euler/blob/master/golang/Problem006.go (written by Frederick Robinson)

Javascript: github.com/dsernst/ProjectEuler/blob/master/6 Sum square difference.js (written by David Ernst)

Scala: github.com/samskivert/euler-scala/blob/master/Euler006.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.

red problems are solved but exceed the time limit of one minute.

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

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I scored 12,626 points (out of 14300 possible points, top rank was 20 out ouf ≈60000 in July 2017) at Hackerrank's Project Euler+.

Look at my progress and performance pages to get more details.

My username at Project Euler is

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

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