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 1230 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 20030916 10:24:16 