<< problem 172 - Investigating numbers with few repeated digits | Counting the number of "hollow" square laminae ... - problem 174 >> |
Problem 173: Using up to one million tiles how many different "hollow" square laminae can be formed?
(see projecteuler.net/problem=173)
We shall define a square lamina to be a square outline with a square "hole" so that the shape possesses vertical and horizontal symmetry.
For example, using exactly thirty-two square tiles we can form two different square laminae:
With one-hundred tiles, and not necessarily using all of the tiles at one time, it is possible to form forty-one different square laminae.
Using up to one million tiles how many different square laminae can be formed?
My Algorithm
Each tiling consists of multiple "rings". The first example has two rings, the second only one.
For each ring with side/edge length x we need 4(x-1) tiles.
My program iterates over all possible rings and tries to insert as many smaller rings as possible until one million is exceeded.
The smallest ring has an edge length of 3 tiles. A ring inside another ring has an edge length which is 2 tiles shorter.
Modifications by HackerRank
My brute-force approach solves the original problem in less than 0.01 seconds but can't solve situations with more than 10^9 tiles in a reasonable time.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho 100 | ./173
Output:
Note: the original problem's input 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)
My code
… was written in C++ and can be compiled with G++, Clang++, Visual C++. You can download it, too. Or just jump to my GitHub repository.
#include <iostream>
int main()
{
unsigned int limit = 1000000;
std::cin >> limit;
// result
unsigned int count = 0;
// start with smallest outer ring
for (unsigned int outer = 3; ; outer++)
{
unsigned int sum = 0;
// add as many inner rings as possible
for (unsigned int inner = outer; inner >= 3; inner -= 2)
{
// tiles needed to create one ring whose sides are "inner" tiles long
unsigned int ring = 4 * (inner - 1);
// runnng out of tiles ?
if (sum + ring > limit)
break;
// add valid ring
sum += ring;
count++;
}
// no more inner rings possible, abort
if (sum == 0)
break;
}
std::cout << count << std::endl;
return 0;
}
This solution contains 7 empty lines, 7 comments and 1 preprocessor command.
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 22, 2017 submitted solution
May 22, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler173
My code solves 3 out of 10 test cases (score: 22.22%)
I failed 0 test cases due to wrong answers and 7 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=173 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
Python github.com/hughdbrown/Project-Euler/blob/master/euler-173.py (written by Hugh Brown)
Python github.com/nayuki/Project-Euler-solutions/blob/master/python/p173.py (written by Nayuki)
C++ github.com/Meng-Gen/ProjectEuler/blob/master/173.cc (written by Meng-Gen Tsai)
C++ github.com/roosephu/project-euler/blob/master/173.cpp (written by Yuping Luo)
C++ github.com/steve98654/ProjectEuler/blob/master/173.cpp
C github.com/zmwangx/Project-Euler/blob/master/173/173.c (written by Zhiming Wang)
Java github.com/dcrousso/ProjectEuler/blob/master/PE173.java (written by Devin Rousso)
Java github.com/HaochenLiu/My-Project-Euler/blob/master/173.java (written by Haochen Liu)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p173.java (written by Nayuki)
Java github.com/thrap/project-euler/blob/master/src/Java/Problem173.java (written by Magnus Solheim Thrap)
Go github.com/frrad/project-euler/blob/master/golang/Problem173.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p173.mathematica (written by Nayuki)
Mathematica github.com/steve98654/ProjectEuler/blob/master/173.nb
Haskell github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p173.hs (written by Nayuki)
Clojure github.com/guillaume-nargeot/project-euler-clojure/blob/master/src/project_euler/problem_173.clj (written by Guillaume Nargeot)
Perl github.com/gustafe/projecteuler/blob/master/173-hollow-square-laminae.pl (written by Gustaf Erikson)
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 | |
[new] | the flashing problem is the one I solved most recently |
I stopped working on Project Euler problems around the time they released 617.
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I scored 13526 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 172 - Investigating numbers with few repeated digits | Counting the number of "hollow" square laminae ... - problem 174 >> |