You might be familiar with the triangular numbers : the number of objects that form equilateral triangles like this: The sequence is 3, 6, 10, 15, 21, ... The first one, 3, is 1 + 2, the second one, 6, is 1 + 2 + 3, the third one, 10, is 1 + 2 + 3 + 4 and so on. For a triangle with a base of \(n\) blobs the number of blobs in the triangle is \(n * (n + 1) / 2\) (which can be proved by various means and was famously figured out by the mathematician Gauss while a school boy). But what about the hollow triangular numbers? The ones where you take the middle blobs out of the triangles and just leave the border. Like this: The pattern there is very simple. They are all multiples of three. And the formula would be that the hollow triangle with a base of size \(n\) has \(3 * (n-1)\) blobs in it (and since that's a multiple of three so is the number of blobs in a hollow triangle). Here are three ways to prove that that formula is correct. Subtracting the middle Because we a