<< 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?

# 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 !

# 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.

# 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 to`echo "1 10" | ./77`

Output:

*(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

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 never 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:

Python: www.mathblog.dk/project-euler-77-sum-of-primes-five-thousand-ways/ (written by Kristian Edlund)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p077.java (written by Nayuki)

Go: github.com/frrad/project-euler/blob/master/golang/Problem077.go (written by Frederick Robinson)

Scala: github.com/samskivert/euler-scala/blob/master/Euler077.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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My username at Project Euler is

**stephanbrumme**while it's stbrumme at Hackerrank.

<< problem 76 - Counting summations | Coin partitions - problem 78 >> |