You can see three sides of each of two dice, and notice that the total number of pips showing is a perfect cube.

What are the numbers of pips on each face that you see?

Opposite faces on a die add up to 7. Therefore, when three faces of a die are visible, exactly one number from each of the following three pairs is visible:
1,6
2,5
3,4

The eight possible combinations correspond to the eight vertices of the cube where three faces meet.

The lowest total for 1 die is therefore 6 and the highest is 15. So for two dice the lowest total is 12 and the highest is 30.

The only perfect cube in the range of 12-30 is 27. This is three short of the maximum of 30.

The only way to lose 3 points compared to showing 4, 5 and 6 on both dice is to have one of them show a 2 instead of a 5. (Any showing a 1 instead of a 6 would be too much lost; a showing of 3 instead of 4 would have to be accompanied by a loss of 2 somewhere, which is not possible.)

So one die shows 4, 5 and 6 while the other shows 4, 2 and 6.

Posted by Charlie on 2003-09-16 10:24:16