Challenge Level

Take a look at some ordinary dice.

What do you notice about the way the numbers are arranged?

Now look at these three dice in a row:

The numbers on the tops of the dice read $6$, $1$ and $5$.

What do the numbers on the top add up to?

What is the sum of the numbers on the bottoms of the dice?

Let's try that again.

This time the numbers on the top read $1$, $4$ and $3$.

Can you work out the total?

Can you work out the numbers on the bottom and their total?

Try out some arrangements yourself. Each time record the sum of the numbers on the top and the sum of the numbers on the bottom.

Do you notice a relationship between the 'top sum' and the 'bottom sum'?

Can you explain it?

I experimented with arrangements where the top sum is a multiple of three, and find that in each case the bottom sum is also a multiple of three. Is it always true?

I try to arrange the dice so that the top and bottom sums are both multiples of four, but can't seem to be able to do it. Can you? Can you explain what you find out?

On the other hand, if I arrange