<< 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.

# 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.

# 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 "1 100" | ./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)*

# Benchmark

The correct solution to the original Project Euler problem was found in 0.18 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 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:

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p174.java (written by Nayuki)

# 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 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.

red problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte.

*Please click on a problem's number to open my solution to that problem:*

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I scored 13,183 points (out of 15300 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.

Look at my progress and performance pages to get more details.

My username at Project Euler is

**stephanbrumme**while it's stbrumme at Hackerrank.

# 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 >> |