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:

Number of test cases (1-5):

Input data (separated by spaces or newlines):

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

Output:

(please click 'Go !')

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:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75
76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125
126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150
151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175
176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200
The 126 solved problems had an average difficulty of 16.0% at Project Euler and I scored 11,074 points (out of 12500) at Hackerrank's Project Euler+.
more about me can be found on my homepage.
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