<< problem 115 - Counting block combinations II | Red, green, and blue tiles - problem 117 >> |
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.
If green tiles are chosen there are three ways.
And if blue tiles are chosen there are two ways.
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.
My 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
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 7" | ./116
Output:
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)
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
// 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.
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
May 14, 2017 submitted solution
May 14, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler116
My code solves 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 rarely 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:
C# 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)
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 115 - Counting block combinations II | Red, green, and blue tiles - problem 117 >> |