# Count On

## Explorer

#### Fair Dice

The most common type of die is the cube with 6 faces. However, there are some other shapes that make good dice too. To make sure one face is not preferred over any other face, we need all the edges to be the same length, and all the angles to be identical too.

A pyramid with a triangular base and triangular sides is called a tetrahedron from the Greek words for four tetra and faced hedron. If all the sides are the same length then all the angles will be equal too and the shape would make a good four-sided dice.

There are three other shapes that make good dice, and, surprisingly, only three, if we insist on all sides being exactly the same length and all angles equal. They are:

 the 8-sidedoctahedron the 12-sideddodecahedron the 20-sidedicosahedron
Can you...
1. say how many edges there are on each shape?
2. say how many faces there are on each shape?
3. say how many vertices (corners) there are on each shape?

There are two more shapes that also give fair dice. They also have the advantage that, instead of just working for 4, 6, 8, 12 or 20 numbers, they can be adapted for any number of faces. However, these shapes, unlike those above, are irregular.
The first is the prism.

An example is a pencil with 6 flat sides. We can roll the pencil and any one of the six faces is equally likely to show on top. We can also imagine a square pencil with 4 sides, or a pencil with 12 sides or 10 sides, or whatever number we like. If we look at the uncut end of the pencil, it should be a regular polygon on n sides.

The second choice is a spinner. This time, cut out of card a regular n-sided shape. If n is 4, this is a square, if n is 6 this is a hexagon and so on. Write a number near each of the edges on the top side of the card. Stick a cocktail stick or pencil stub through the center of the card so that the card can be spun. It will stop spinning and land with just one edge on the table.