<< problem 5 - Smallest multiple | 10001st prime - problem 7 >> |
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.
My 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
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.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho "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)
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.
Benchmark
The correct solution to the original Project Euler problem was found in less than 0.01 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=gnu++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:
C# www.mathblog.dk/project-euler-problem-6/ (written by Kristian Edlund)
C github.com/eagletmt/project-euler-c/blob/master/1-9/problem6.c (written by eagletmt)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p006.java (written by Nayuki)
Javascript github.com/dsernst/ProjectEuler/blob/master/6 Sum square difference.js (written by David Ernst)
Go github.com/frrad/project-euler/blob/master/golang/Problem006.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p006.mathematica (written by Nayuki)
Haskell github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p006.hs (written by Nayuki)
Scala github.com/samskivert/euler-scala/blob/master/Euler006.scala (written by Michael Bayne)
Those links are just an unordered selection of source code I found with a semi-automatic search script on Google/Bing/GitHub/whatever.
You will probably stumble upon better solutions when searching on your own. Maybe not all linked resources produce the correct result and/or exceed time/memory limits.
Heatmap
Please click on a problem's number to open my solution to that problem:
green | solutions solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too | |
yellow | solutions score less than 100% at Hackerrank (but still solve the original problem easily) | |
gray | problems are already solved but I haven't published my solution yet | |
blue | solutions are relevant for Project Euler only: there wasn't a Hackerrank version of it (at the time I solved it) or it differed too much | |
orange | problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte | |
red | problems are not solved yet but I wrote a simulation to approximate the result or verified at least the given example - usually I sketched a few ideas, too | |
black | problems are solved but access to the solution is blocked for a few days until the next problem is published | |
the flashing problem is the one I solved most recently |
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I scored 13526 points (out of 15700 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.
My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.
Look at my progress and performance pages to get more details.
Copyright
I hope you enjoy my code and learn something - or give me feedback how I can improve my solutions.
All of my solutions can be used for any purpose and I am in no way liable for any damages caused.
You can even remove my name and claim it's yours. But then you shall burn in hell.
The problems and most of the problems' images were created by Project Euler.
Thanks for all their endless effort !!!
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