<< problem 179 - Consecutive positive divisors | Maximum product of parts - problem 183 >> |

# Problem 181: Investigating in how many ways objects of two different colours can be grouped

(see projecteuler.net/problem=181)

Having three black objects B and one white object W they can be grouped in 7 ways like this:

(BBBW), (B,BBW), (B,B,BW), (B,B,B,W), (B,BB,W), (BBB,W), (BB,BW)

In how many ways can sixty black objects B and forty white objects W be thus grouped?

# My Algorithm

My solution is somehow similar to the coin-change algorithm:

- each group can be treated as a sequence of black and white objects, BWBW is the same as BBWW

- all groups can be sorted by their size and, if multiple groups have the same size, by their lexicographical order

For `maxBlack = 3`

and `maxWhite = 1`

these groups exist:

(B,B,B,W), (B,B,BW), (B,W,BB), (B, BBW), (W, BBB), (BB, BW), (BBBW)

These are exactly the same as in the problem statement but in a different order.

Two outer loops iterate over all possible sequences of black and white objects.

The inner loops place them at every possible positions (until the number of available objects is exhausted).

The result will be found in `current[60][40]`

.

## Alternative Approaches

You can solve this problem with Dynamic Programming, too.

I wrote a simple prototype but it turned out to be much slower (8 seconds vs. 0.04 seconds).

# 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>
// limits

const unsigned int MaxBlack = 160;
const unsigned int MaxWhite = 160;
const unsigned int Modulo = 1000000007; // Hackerrank only
int main()
{
#define ORIGINAL
#ifndef ORIGINAL
unsigned int tests = 1;
std::cin >> tests;
while (tests--)
#endif
{
unsigned int maxBlack = MaxBlack;
unsigned int maxWhite = MaxWhite;
std::cin >> maxBlack >> maxWhite;
// static array size: actually it would be sufficient to use maxBlack instead of MaxBlack
// (and maxWhite instead of MaxWhite)
unsigned long long previous[MaxBlack + 1][MaxWhite + 1];
unsigned long long current [MaxBlack + 1][MaxWhite + 1];
// initialize
for (unsigned int i = 0; i <= maxBlack; i++)
for (unsigned int j = 0; j <= maxWhite; j++)
previous[i][j] = 0;
previous[0][0] = 1;
// all possible subsets
for (unsigned int useBlack = 0; useBlack <= maxBlack; useBlack++)
for (unsigned int useWhite = 0; useWhite <= maxWhite; useWhite++)
{
// skip empty subset
if (useBlack == 0 && useWhite == 0)
continue;
// put subset at every possible position
for (unsigned int i = 0; i <= maxBlack; i++)
for (unsigned int j = 0; j <= maxWhite; j++)
{
current[i][j] = 0;
// place it repeatedly
unsigned int k = 0;
while (i >= k * useBlack && j >= k * useWhite)
{
current[i][j] += previous[i - k * useBlack][j - k * useWhite];
k++;
}
#ifndef ORIGINAL
current[i][j] %= Modulo;
#endif
}
// copy for next iteration
for (unsigned int i = 0; i <= maxBlack; i++)
for (unsigned int j = 0; j <= maxWhite; j++)
previous[i][j] = current[i][j];
}
// print result
std::cout << current[maxBlack][maxWhite] << std::endl;
}
return 0;
}

This solution contains 10 empty lines, 10 comments and 6 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 "3 1" | ./181`

Output:

*Note:* the original problem's input `60 40`

__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 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

June 28, 2017 submitted solution

June 28, 2017 added comments

July 28, 2017 modified to solve Hackerrank, too

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler181

My code solves **2** out of **3** test cases (score: **9.09%**)

I failed **0** test cases due to wrong answers and **1** because of timeouts

# Difficulty

Project Euler ranks this problem at **70%** (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=181 - **the** best forum on the subject (*note:* you have to submit the correct solution first)

# 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 13,183 points (out of 15300 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 179 - Consecutive positive divisors | Maximum product of parts - problem 183 >> |