Problem | Submit a solution | Printable page |

I place three dice in a row. The numbers on the tops of the dice read $6$, $1$ and $5$. The sum of the numbers on the top is $6+1+5=12$. What is the sum of the numbers on the bottoms of the dice?


I place the dice in a row, and this time the numbers on the top read $1$, $4$ and $3$. Write down the sum of the numbers on the top, and the sum of the numbers on the bottom.


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 experiment 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 that when the top sum is a multiple of three, the bottom sum is also a multiple of three?

I try to arrange to 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 explain why not?

On the other hand, if I arrange four dice in a row it is easy to make the top and bottom sums both multiples of four. Can I arrange four dice so that the top and bottom sums are both multiples of three?