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:

Number of test cases (1-5):

Input data (separated by spaces or newlines):

This is equivalent to
echo "1 1000" | ./6

Output:

(please click 'Go !')

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 solved 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.

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 already 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 133 solved problems had an average difficulty of 16.9% at Project Euler and I scored 11,174 points (out of 12300) at Hackerrank's Project Euler+.
more about me can be found on my homepage.
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