<< problem 5 - Smallest multiple 10001st prime - problem 7 >>

# Problem 6: Sum square difference

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

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:

Number of test cases (1-5):

Input data (separated by spaces or newlines):

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

# Hackerrank

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 never an option.

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)
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.

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

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The 160 solved problems had an average difficulty of 21.8% at Project Euler and I scored 11,807 points (out of 13100) at Hackerrank's Project Euler+.
My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.
 << problem 5 - Smallest multiple 10001st prime - problem 7 >>
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