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==Problem Statement==

Link: https://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?

==Explanation==

This problem extends the hollow square laminae concept from Problem 173. The key insight is that a square lamina with an inner hole of side length <math>s_h</math> and outer side length <math>s_h + 2k</math> requires <math>4k(s_h + k)</math> tiles, where <math>k</math> is the distance between the hole and the outer edge.

For a fixed <math>k</math>, the number of possible laminae is <math>\lfloor \frac{T}{4k} - k \rfloor</math> where <math>T = 10^6</math> is the maximum number of tiles. The maximum <math>k</math> occurs when <math>s_h = 1</math>, giving <math>k_{\max} = \lfloor \frac{\sqrt{T+1} - 1}{2} \rfloor</math>.

The answer is obtained by summing over all <math>k</math> from 1 to <math>k_{\max}</math>.

==Answer==

1572729

==Flags==

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Project Euler/173 - Revision history

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