#2             76. NEVER SAY DIE                       


Three dice are stacked to form a tower as illustrated at right. We can see only seven of the faces, but of course someone walking around the pile could see 13.

However, five of the faces remain hidden – the two pairs of touching faces, and the face on the bottom of the pile.

What is the sum of the dots on the five hidden faces? And in general, what is the smallest amount of information we need to know about the placing of the dice, in order to determine this sum?
HINT 1

To solve this problem, you need to understand how a die (plural ‘dice’) is structured. Find an actual die, and study the way the dots are placed.
HINT 2

What is the sum of the dots on opposite faces?
SOLUTION

Opposite faces of a die add to seven. Hence the required sum is 7 + 7 + 2 = 16.

In fact, the only information we need to determine this sum is the number of dots on the top face.
EXTENSIONS

1. What is the sum of all the numbers on a die? If the opposite faces are to add to a common number, must it be 7? Why?

2. Make up some dice problems of your own which use this property. You could try stacks of different heights, or other arrangements of the dice.