<< problem 76 - Counting summations | Coin partitions - problem 78 >> |
Problem 77: Prime summations
(see projecteuler.net/problem=77)
It is possible to write ten as the sum of primes in exactly five different ways:
7 + 3
5 + 5
5 + 3 + 2
3 + 3 + 2 + 2
2 + 2 + 2 + 2 + 2
What is the first value which can be written as the sum of primes in over five thousand different ways?
My Algorithm
Instead of re-using the code from problem 76 (which was based on probem 31) I wrote this code from scratch
because the solution is actually much simpler than the previous challenges.
Main idea:
combinations(x) = combinations(x - prime1) + combinations(x - prime2) + ...
I subtract each prime and look up combinations(x - currentPrime)
and sum all those numbers
→ nice Dynamic Programming solution !
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This live test is based on the Hackerrank problem.
This is equivalent toecho "1 10" | ./77
Output:
(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.
The code contains #ifdef
s to switch between the original problem and the Hackerrank version.
Enable #ifdef ORIGINAL
to produce the result for the original problem (default setting for most problems).
#include <iostream>
#include <vector>
int main()
{
const unsigned int MaxNumber = 1000;
// store number of ways to represent a number as a sum of primes
std::vector<unsigned long long> combinations(MaxNumber + 1, 0);
// degenerated case
combinations[0] = 1;
// store all primes
std::vector<unsigned int> primes;
for (unsigned int i = 2; i <= MaxNumber; i++)
{
bool isPrime = true;
// test against all prime numbers we have so far (in ascending order)
for (auto p : primes)
{
// next prime is too large to be a divisor ?
if (p*p > i)
break;
// divisible ? => not prime
if (i % p == 0)
{
isPrime = false;
break;
}
}
// only primes after this point ...
if (!isPrime)
continue;
primes.push_back(i);
// now add all solutions
for (unsigned int pos = 0; pos <= MaxNumber - i; pos++)
combinations[pos + i] += combinations[pos];
}
//#define ORIGINAL
#ifdef ORIGINAL
// find first number with more than 5000 combinations
for (size_t i = 0; i < combinations.size(); i++)
if (combinations[i] > 5000)
{
std::cout << i << std::endl;
break;
}
#else
unsigned int tests = 1;
std::cin >> tests;
while (tests--)
{
// look up combinations
unsigned int n;
std::cin >> n;
std::cout << combinations[n] << std::endl;
}
#endif
return 0;
}
This solution contains 9 empty lines, 11 comments and 5 preprocessor commands.
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
March 17, 2017 submitted solution
May 2, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler077
My code solves 6 out of 6 test cases (score: 100%)
Difficulty
Project Euler ranks this problem at 25% (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=77 - 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-77-sum-of-primes-five-thousand-ways/ (written by Kristian Edlund)
Python github.com/hughdbrown/Project-Euler/blob/master/euler-077.py (written by Hugh Brown)
Python github.com/nayuki/Project-Euler-solutions/blob/master/python/p077.py (written by Nayuki)
Python github.com/sefakilic/euler/blob/master/python/euler077.py (written by Sefa Kilic)
Python github.com/smacke/project-euler/blob/master/python/77.py (written by Stephen Macke)
C++ github.com/HaochenLiu/My-Project-Euler/blob/master/077.cpp (written by Haochen Liu)
C++ github.com/Meng-Gen/ProjectEuler/blob/master/77.cc (written by Meng-Gen Tsai)
C++ github.com/zmwangx/Project-Euler/blob/master/077/077.cpp (written by Zhiming Wang)
Java github.com/dcrousso/ProjectEuler/blob/master/PE077.java (written by Devin Rousso)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p077.java (written by Nayuki)
Java github.com/thrap/project-euler/blob/master/src/Java/Problem77.java (written by Magnus Solheim Thrap)
Go github.com/frrad/project-euler/blob/master/golang/Problem077.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p077.mathematica (written by Nayuki)
Mathematica github.com/steve98654/ProjectEuler/blob/master/077.nb
Haskell github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p077.hs (written by Nayuki)
Haskell github.com/roosephu/project-euler/blob/master/77.hs (written by Yuping Luo)
Clojure github.com/rm-hull/project-euler/blob/master/src/euler077.clj (written by Richard Hull)
Scala github.com/samskivert/euler-scala/blob/master/Euler077.scala (written by Michael Bayne)
Perl github.com/gustafe/projecteuler/blob/master/077-Prime-summations.pl (written by Gustaf Erikson)
Rust github.com/gifnksm/ProjectEulerRust/blob/master/src/bin/p077.rs
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 !!!
<< problem 76 - Counting summations | Coin partitions - problem 78 >> |