Problem:
Two fair dice, each with at least 6 faces, are rolled. On each face of each die is printed a distinct integer from 1 to the number of faces on that die, inclusive. The probability of rolling a sum of 7 is of the probability of rolling a sum of 10 and the probability of rolling a sum of 12 is . What is the least possible number of faces on the two dice combined?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Answer (B): Because each die has at least 6 faces, the number of ways to roll a total of 7 is 6 , namely with rolls , and . Because the probability of rolling a total of 7 is that of rolling a total of 10 , the number of ways to roll a 10 is . There are 9 possible combinations of two positive integers that total 10 , and thus one of these combinations cannot be made with the numbers on the two dice. Therefore one of the combinations and may be rolled, but not the other, but both and are possible. This implies one die has exactly 8 faces and the other die has 9 or more faces. If the other die has 9 faces, there are 72 possible rolls, of which 6 add to 12 . Then the probability of rolling a total of 12 would be , as required. There are faces on the two dice combined. (Dice with 8 and 12 faces also have the stated property, because in that case the probability of rolling a total of 12 would be .)
The problems on this page are the property of the MAA's American Mathematics Competitions