<< 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?
My 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
proven code is still the fastest way to solve a problem ...
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 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)
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.
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 13, 2017 submitted solution
April 27, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler076
My code solves 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 rarely 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:
C# 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)
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 | |
the flashing problem is the one I solved most recently |
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I scored 13,486 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 75 - Singular integer right triangles | Prime summations - problem 77 >> |