<< problem 114 - Counting block combinations I | Red, green or blue tiles - problem 116 >> |

# Problem 115: Counting block combinations II

(see projecteuler.net/problem=115)

NOTE: This is a more difficult version of Problem 114.

A row measuring n units in length has red blocks with a minimum length of m units placed on it,

such that any two red blocks (which are allowed to be different lengths) are separated by at least one black square.

Let the fill-count function, F(m, n), represent the number of ways that a row can be filled.

For example, F(3, 29) = 673135 and F(3, 30) = 1089155.

That is, for m = 3, it can be seen that n = 30 is the smallest value for which the fill-count function first exceeds one million.

In the same way, for m = 10, it can be verified that F(10, 56) = 880711 and F(10, 57) = 1148904, so n = 57 is the least value for which the fill-count function first exceeds one million.

For m = 50, find the least value of n for which the fill-count function first exceeds one million.

# Algorithm

... almost the same as problem 114 !

Only the `main`

function had to adjusted to find in a simple loop the smallest value to exceed one million.

## Modifications by HackerRank

My approach needs a bit of memory. Hackerrank has inputs up to 10^18 which clearly exceed the RAM size of a desktop PC.

[TODO] find closed formula

# 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>
#define ORIGINAL
// memoized solutions

const long long Unknown = -1;
std::vector<long long> solutions;
// find result for row with a certain length

unsigned long long count(unsigned long long space, unsigned int minBlockLength)
{
// finished ?
if (space == 0)
return 1;
// already know the answer ?
if (solutions[space] != Unknown)
return solutions[space];
// one option is to leave the next cell black
auto result = count(space - 1, minBlockLength);
// insert red blocks at the current position with all possible spaces
for (auto block = minBlockLength; block <= space; block++)
{
// how much is left after inserting ?
auto next = space - block;
// must be followed by a black cell
if (next > 0)
next--;
// count all combinations
result += count(next, minBlockLength);
}
// memoize result
solutions[space] = result;
return result;
}
int main()
{
unsigned int tests;
std::cin >> tests;
while (tests--)
{
// minimum length of each red block
unsigned int minBlockLength = 3;
// what number should be exceeded ?
unsigned long long limit = 1000000;
std::cin >> minBlockLength >> limit;
// cached results
solutions.clear();
solutions.resize(limit + 1, Unknown); // not enough for Hackerrank
for (unsigned int totalLength = 1; ; totalLength++)
{
auto combinations = count(totalLength, minBlockLength);
if (combinations > limit)
{
std::cout << totalLength << std::endl;
break;
}
}
}
return 0;
}

This solution contains 12 empty lines, 13 comments and 3 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 3 1000000" | ./115`

Output:

*Note:* the original problem's input `50 1000000`

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.

Peak memory usage was about 10 MByte.

(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

May 13, 2017 submitted solution

May 13, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler115

My code solved **1** out of **15** test cases (score: **0%**)

I failed **10** test cases due to wrong answers and **4** because of timeouts

# Difficulty

Project Euler ranks this problem at **35%** (out of 100%).

Hackerrank describes this problem as **medium**.

*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=115 - **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-115-separated-blocks/ (written by Kristian Edlund)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p115.java (written by Nayuki)

Scala: github.com/samskivert/euler-scala/blob/master/Euler115.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 114 - Counting block combinations I | Red, green or blue tiles - problem 116 >> |