# Convex and concave polygons

Given \(n\) congruent squares, we can ask how many convex shapes can be formed. In this case, the answer is easy, because the only convex shapes possible are rectangles. To find the number of rectangles, first list all the proper divisors of \(n.\) Now each of those is the width of some rectangle. But a rectangle which is 2x3, is the same as a rectangle which is 3x2, for example. That is, factors come in pairs. Thus, if we let \(d(n)\) denote the number of proper divisors of \(n,\) as is commonly done, then the number of rectangles is

$$\left\lceil \dfrac{d(n)}{2} \right\rceil$$Here's the OEIS entry for the number of distinct convex shapes that can be formed with \(n\) congruent isosceles right triangles. Reflections are not counted as different. We could ask kids to find the number of convex polygons that can be made for a specific number of congruent shapes. For example, how many convex polygons can be made by composing 10 congruent equilateral triangles?