Further Reading for Teachers

Euler's Number

For each of the five Platonic Solids,

(number of faces) - (number of edges) + (number of vertices) = 2

This is part of a more general fact:
Any three-dimensional solid formed by putting together faces with straight edges (usually called a polyhedron) that would look like a balloon were you able to inflate it shares the property that

(number of faces) - (number of edges) + (number of vertices) = 2

Here is another example:

It has 6 faces, 12 edges and 8 vertices. Let's check: 6 - 12 + 8 = 2!!

Now, if your faceted solid were to look like a doughnut when inflated, you would get:

(number of faces) - (number of edges) + (number of vertices) = 0

An example is "the hollow brick:"

It has 16 faces (4 at the top, 4 at the bottom, 4 outer sides and the 4 sides of the hole), 32 edges and 16 vertices. Here: 16 - 32 + 16 = 0 as expected.

In general any faceted solid that would look like a doughnut with many holes were you able to inflate it will satisfy that

(number of faces) - (number of edges) + (number of vertices) =
2 × (1 - number of holes)

For example, this solid with 2 holes responds to the formula

(number of faces) - (number of edges) + (number of vertices) =
2 × (1 - 2) = 2 × (-1) = -1

It has 28 faces (7 at the top, 7 at the bottom, 6 on the outer sides and the 8 sides of the 2 holes), 58 edges and 28 vertices. We may verify that 28 - 58 + 28 = -2.