How many pentominoes?
This unit should follow Shapes of numbers.
Here is a proof that there are exactly twelve pentominoes.
We first introduce some terminology. A "stem" of a pentomino is a square or group of squares that connect by at least two sides.
A "leaf" of a pentomino is a square that connects by only one side. So a stem is obtained by removing all the leaves.
There is no pentomino without a leaf, and there is only one with one leaf.
So we have these stems, shown in grey (the white squares are examples of leaves):
From each of these stems, we will build as many pentominoes as we can, and we will then count them. This kind of proof is called "proof by cases" or "proof by exhaustion". It doesn't mean that we get exhausted! It means that we have found all that there are.
Case 1.
There is only one pentomino; it has a stem of four and one leaf:
Case 2: A stem of one--
There is only one pentomino; it has a stem of one and four leaves:
Case 3: A stem of two:
There are three pentominoes with a stem of two (and thus three leaves):
So we now need to show that there are exactly seven pentominoes with stems of three. Stems of three are of two kinds, straight and crooked:
Case 4. 
Let's start with straight stems. There are four pentominoes with straight stems of three:
Case 5.
Now crooked stems of three; there are three pentominoes:
So 1 + 1 + 3 + 4 + 3 = 12. There are exactly twelve pentominoes total!
