Problem 116: Red, green or blue tiles

(see projecteuler.net/problem=116)

A row of five black square tiles is to have a number of its tiles replaced with coloured oblong tiles chosen from red (length two), green (length three), or blue (length four).

If red tiles are chosen there are exactly seven ways this can be done.

red tiles

If green tiles are chosen there are three ways.

green tiles

And if blue tiles are chosen there are two ways.

blue tiles

Assuming that colours cannot be mixed there are 7 + 3 + 2 = 12 ways of replacing the black tiles in a row measuring five units in length.

How many different ways can the black tiles in a row measuring fifty units in length be replaced if colours cannot be mixed and at least one coloured tile must be used?

NOTE: This is related to Problem 117.

Algorithm

The row is filling with an arbitrary sequence of black and colored blocks.
The length of the sequence is the sum of black and colored blocks.
For a known number of black and colored blocks, this is:
{black+colored}choose{black} = dfrac{(black+colored)!}{black!colored!}

It took me a little bit to write a choose function such that the factorials don't overflow too soon.

Modifications by HackerRank

As always, Hackerrank's input is ridiculously high and my poor C++ data types can't handle such numbers.
[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 #ifdefs 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
 
// number of ways to choose n elements from k available
unsigned long long choose(unsigned long long n, unsigned long long k)
{
// n! / (n-k)!k!
unsigned long long result = 1;
// reduce overflow by dividing as soon as possible to keep numbers small
for (unsigned long long invK = 1; invK <= k; invK++)
{
result *= n;
result /= invK;
n--;
}
return result;
}
 
int main()
{
unsigned int tests;
std::cin >> tests;
while (tests--)
{
// length of row
unsigned int totalLength = 50;
std::cin >> totalLength;
 
unsigned long long sum = 0;
// try all four block lengths
for (unsigned int blockLength = 2; blockLength <= 4; blockLength++)
{
// maximum number of blocks of the current size
auto maxBlocks = totalLength / blockLength;
 
// insert 1 to maxBlocks colored blocks
for (unsigned int colored = 1; colored <= maxBlocks; colored++)
{
// remaining black tiles
auto black = totalLength - colored * blockLength;
// total number of black and colored tiles
auto tiles = black + colored;
// count combinations
auto combinations = choose(tiles, colored);
sum += combinations;
 
#ifndef ORIGINAL
// Hackerrank only
sum %= 1000000007;
#endif
}
}
 
std::cout << sum << std::endl;
}
 
return 0;
}

This solution contains 8 empty lines, 11 comments and 5 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):
Note: Enter the row size

This is equivalent to
echo "1 7" | ./116

Output:

(please click 'Go !')

Note: the original problem's input 50 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

May 14, 2017 submitted solution
May 14, 2017 added comments

Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler116

My code solved 1 out of 5 test cases (score: 0%)

I failed 0 test cases due to wrong answers and 4 because of timeouts

Difficulty

Project Euler ranks this problem at 30% (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=116 - 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-116-coloured-tiles/ (written by Kristian Edlund)
Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p116.java (written by Nayuki)
Scala: github.com/samskivert/euler-scala/blob/master/Euler116.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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The 133 solved problems had an average difficulty of 16.9% at Project Euler and I scored 11,174 points (out of 12300) at Hackerrank's Project Euler+.
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