Even Fibonacci numbers - problem 2 >> |
Problem 1: Multiples of 3 and 5
(see projecteuler.net/problem=1)
If we list all the natural numbers below 10 that are multiples of 3 or 5,
we get 3, 5, 6 and 9. The sum of these multiples is 23.
Find the sum of all the multiples of 3 or 5 below 1000.
My Algorithm
We are supposed to find of all multiples of 3 or 5 below the input number,
therefore we decrement it by one.
In general, the sum of all numbers between 1 and x is sum_{1..x}i=x * (x+1)/2
(see en.wikipedia.org/wiki/Triangular_number)
There are floor{x/3} numbers between 1 and x which are divisible by 3 (assuming floor{x/3} is an integer division).
e.g. the range 1..10 contains floor{10/3}=3 such numbers (it's 3, 6 and 9). Their sum is 3+6+9=18.
This can be written as 3/3 * (3+6+9) which is the same as 3 * (3/3+6/3+9/3)=3 * (1+2+3).
Those brackets represent sum_{1..3}i = sum_{1..10/3}i (or short: sum{10/3})
and thus our overall formula for the sum of all multiples of 3 becomes 3 * sum{x/3}.
The same formula can be used for 5:
The sum of all numbers divisible by 5 is 5 * sum{x/5}
However, there are numbers divisible by 3 and 5, which means they are part of both sums.
We must not count them twice, that's why we (in addition to the aforementioned sums)
compute the sum of all numbers divisible by 3*5=15 to correct for this error.
In the end we print sumThree + sumFive - sumFifteen
Alternative Approaches
Looping through all numbers from 1 and 1000 and checking each of those numbers
whether they are divisible by 3 or 5 easily solves the problem, too, and produces the result pretty much instantly.
Even more, the code will be probably a bit shorter.
However, Hackerrank's input numbers are too large for that simple approach (up to 10^9 with 10^5 test cases)
and will lead to timeouts.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho "1 100" | ./1
Output:
Note: the original problem's input 1000
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++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.
#include <iostream>
// triangular number: sum{x}=1+2+..+x = x*(x+1)/2
unsigned long long sum(unsigned long long x)
{
return x * (x + 1) / 2;
}
int main()
{
unsigned int tests;
std::cin >> tests;
while (tests--)
{
unsigned long long last;
std::cin >> last;
// not including that number
last--;
// find sum of all numbers divisible by 3 or 5
auto sumThree = 3 * sum(last / 3);
auto sumFive = 5 * sum(last / 5);
// however, those numbers divisible by 3 AND 5 will be counted twice
auto sumFifteen = 15 * sum(last / 15);
std::cout << (sumThree + sumFive - sumFifteen) << std::endl;
}
return 0;
}
This solution contains 7 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 22, 2017 submitted solution
March 23, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler001
My code solves 5 out of 5 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=1 - 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-1/ (written by Kristian Edlund)
C github.com/eagletmt/project-euler-c/blob/master/1-9/problem1.c (written by eagletmt)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p001.java (written by Nayuki)
Javascript github.com/dsernst/ProjectEuler/blob/master/1 Multiples of 3 and 5.js (written by David Ernst)
Go github.com/frrad/project-euler/blob/master/golang/Problem001.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p001.mathematica (written by Nayuki)
Haskell github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p001.hs (written by Nayuki)
Scala github.com/samskivert/euler-scala/blob/master/Euler001.scala (written by Michael Bayne)
Perl github.com/gustafe/projecteuler/blob/master/001-multiples-of-3-and-5.pl (written by Gustaf Erikson)
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 | |
[new] | the flashing problem is the one I solved most recently |
I stopped working on Project Euler problems around the time they released 617.
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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 !!!
Even Fibonacci numbers - problem 2 >> |