<< problem 173 - Using up to one million tiles how many different ... | Step Numbers - problem 178 >> |
Problem 174: Counting the number of "hollow" square laminae that can form one, two, three, ... distinct arrangements
(see projecteuler.net/problem=174)
We shall define a square lamina to be a square outline with a square "hole" so that the shape possesses vertical and horizontal symmetry.
Given eight tiles it is possible to form a lamina in only one way: 3x3 square with a 1x1 hole in the middle.
However, using thirty-two tiles it is possible to form two distinct laminae.
If t represents the number of tiles used, we shall say that t = 8 is type L(1) and t = 32 is type L(2).
Let N(n) be the number of t <= 1000000 such that t is type L(n); for example, N(15) = 832.
What is sum{N(n)} for 1 <= n <= 10?
My Algorithm
I copied most of my solution from the previous problem (problem 173).
A second pass iterates over all solutions and counts which number of tiles has between 1 and 10 solutions. This information is stored in result
.
Then we just have to look up the relevant element, e.g. result[1000000]
for the original problem.
Modifications by HackerRank
I could have written my solution without the result
array because only result[1000000]
was relevant.
However, Hackerrank queries thousands of other elements from result
.
Due to heavy I/O processing, which is kind of slow in C++, I barely solve the second test case.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho "1 100" | ./174
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.
#include <iostream>
#include <vector>
#include <algorithm>
int main()
{
const unsigned int limit = 1000000;
// count different layouts for a number of tiles
std::vector<unsigned int> solutions(limit + 1, 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;
solutions[sum]++;
}
// no more inner rings possible, abort
if (sum == 0)
break;
}
// pre-process all possible answers
std::vector<unsigned int> result(solutions.size());
unsigned int one2ten = 0;
for (unsigned int i = 0; i < solutions.size(); i++)
{
auto s = solutions[i];
if (s >= 1 && s <= 10)
one2ten++;
result[i] = one2ten;
}
// process test cases
unsigned int tests;
std::cin >> tests;
while (tests--)
{
unsigned int last = limit;
std::cin >> last;
// simple lookup
std::cout << result[last] << std::endl;
}
return 0;
}
This solution contains 10 empty lines, 10 comments and 3 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in 0.18 seconds on an 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=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 23, 2017 submitted solution
May 23, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler174
My code solves 2 out of 2 test cases (score: 100%)
Difficulty
Project Euler ranks this problem at 40% (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=174 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
Python github.com/nayuki/Project-Euler-solutions/blob/master/python/p174.py (written by Nayuki)
C++ github.com/Meng-Gen/ProjectEuler/blob/master/174.cc (written by Meng-Gen Tsai)
C++ github.com/roosephu/project-euler/blob/master/174.cpp (written by Yuping Luo)
C++ github.com/smacke/project-euler/blob/master/cpp/174.cpp (written by Stephen Macke)
C++ github.com/steve98654/ProjectEuler/blob/master/174.cpp
C++ github.com/zmwangx/Project-Euler/blob/master/174/174.cpp (written by Zhiming Wang)
C github.com/LaurentMazare/ProjectEuler/blob/master/e174.c (written by Laurent Mazare)
Java github.com/dcrousso/ProjectEuler/blob/master/PE174.java (written by Devin Rousso)
Java github.com/HaochenLiu/My-Project-Euler/blob/master/174.java (written by Haochen Liu)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p174.java (written by Nayuki)
Java github.com/thrap/project-euler/blob/master/src/Java/Problem174.java (written by Magnus Solheim Thrap)
Go github.com/frrad/project-euler/blob/master/golang/Problem174.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p174.mathematica (written by Nayuki)
Clojure github.com/guillaume-nargeot/project-euler-clojure/blob/master/src/project_euler/problem_174.clj (written by Guillaume Nargeot)
Perl github.com/gustafe/projecteuler/blob/master/174-counting-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 173 - Using up to one million tiles how many different ... | Step Numbers - problem 178 >> |