<< 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 !

# 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 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:

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.

red problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte.

*Please click on a problem's number to open my solution to that problem:*

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I scored 12,983 points (out of 15100 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.

Look at my progress and performance pages to get more details.

My username at Project Euler is

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

# 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 >> |