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==Problem Statement==

Link: https://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: a 3×3 square with a 1×1 hole in the middle. However, using thirty-two tiles it is possible to form two distinct laminae.

If <math>t</math> represents the number of tiles used, we shall say that <math>t = 8</math> is type <math>L(1)</math> and <math>t = 32</math> is type <math>L(2)</math>.

Let <math>N(n)</math> be the number of <math>t \le 1000000</math> such that <math>t</math> is type <math>L(n)</math>; for example, <math>N(15) = 832</math>.

What is <math>\sum_{n=1}^{10} N(n)</math>?

==Explanation==

This problem builds on Problem 173. A square lamina with inner hole side <math>s_h</math> and thickness <math>k</math> uses <math>t = 4k(s_h + k)</math> tiles.

The type <math>L(n)</math> of <math>t</math> is the number of distinct laminae that can be formed using exactly <math>t</math> tiles. For each valid lamina (with <math>t \le 10^6</math>), count how many distinct <math>(s_h, k)</math> pairs produce that exact <math>t</math>. Then tally how many <math>t</math> values have each type <math>n</math> for <math>n = 1</math> through <math>10</math>, and sum those counts.

==Answer==

20956691

==Flags==

{{ProjectEulerFlag}}

Project Euler/174 - Revision history

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