<< 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.

# My code

… was written in C++ and can be compiled with G++, Clang++, Visual C++. You can download it, too.

#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.

# 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 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)*

# 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 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:

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p173.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 172 - Investigating numbers with few repeated digits | Counting the number of "hollow" square laminae ... - problem 174 >> |