<< problem 75 - Singular integer right triangles | Prime summations - problem 77 >> |

# Problem 76: Counting summations

(see projecteuler.net/problem=76)

It is possible to write five as a sum in exactly six different ways:

4 + 1

3 + 2

3 + 1 + 1

2 + 2 + 1

2 + 1 + 1 + 1

1 + 1 + 1 + 1 + 1

How many different ways can one hundred be written as a sum of at least two positive integers?

# Algorithm

Only very few adjustments to problem 31:

- replace anything related to coins by the numbers 1..100

- finally subtract 1 because the sum has to consist of at least two numbers (not just one)

- for more details on the algorithm itself, please read my explanation of problem 31

I am aware that there are more efficient solutions (even much shorter solutions !) but re-using old,

proven code is still the fastest way to solve a problem ...

# My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.

#include <iostream>
#include <vector>
typedef std::vector<unsigned long long> combinations;
int main()
{
const unsigned int MaxNumber = 1000;
// remember combinations for all combinations from 1 up to 1000
std::vector<combinations> history;
// store number of combinations in [x] if only summands up to x+1 are allowed:
// [0] => combinations if only 1s are allowed
// [1] => 1s and 2s are allowed, nothing more
// [2] => 1s, 2s and 3s are allowed, nothing more
// ...
// [99] => all but 100 are allowed
// [100] => using all numbers if possible
unsigned int tests = 1;
std::cin >> tests;
while (tests--)
{
// number that should be represented as a sum
unsigned int x = 100;
std::cin >> x;
// initially we start at zero
// but if there are previous test cases then we can re-use the old results
for (unsigned int j = history.size(); j <= x; j++)
{
combinations ways(MaxNumber, 0);
// one combination if using only 1s
ways[0] = 1;
// use larger numbers, too
for (unsigned int i = 1; i < MaxNumber; i++)
{
// first, pretend not to use that number
ways[i] = ways[i - 1];
// now use that number once (if possible)
auto current = i + 1;
if (j >= current)
{
auto remaining = j - current;
ways[i] += history[remaining][i];
}
// only for Hackerrank
// (it prevents printing huge numbers)
ways[i] %= 1000000007;
}
// store information for future use
history.push_back(ways);
}
// look up combinations
auto result = history[x];
// the last column contains the desired value
auto combinations = result.back();
// but it contains one undesired combination, too: the single number MaxNumber itself
// (which fails to be "the sum of two (!) numbers", it's just one number)
// therefore subtract 1
combinations--;
combinations %= 1000000007; // only for Hackerrank
std::cout << combinations << std::endl;
}
return 0;
}

This solution contains 13 empty lines, 23 comments and 2 preprocessor commands.

# Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

This is equivalent to`echo "1 6" | ./76`

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)*

# 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 13, 2017 submitted solution

April 27, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler076

My code solved **6** out of **6** test cases (score: **100%**)

# Difficulty

Project Euler ranks this problem at **10%** (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=76 - **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-76-one-hundred-sum-integers/ (written by Kristian Edlund)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p076.java (written by Nayuki)

Go: github.com/frrad/project-euler/blob/master/golang/Problem076.go (written by Frederick Robinson)

Scala: github.com/samskivert/euler-scala/blob/master/Euler076.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 already 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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<< problem 75 - Singular integer right triangles | Prime summations - problem 77 >> |