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:
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B.
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D.
E.
Solution:
Suppose the dice have and faces. So, without loss of generality, .
Since each die has at least faces, there will always be ways to sum to . As a result, there must be ways to sum to . There are at most nine distinct ways to get a sum of , which are possible whenever . To achieve exactly eight ways, must have faces, and . Let be the number of ways to obtain a sum of , then .
Since , . In addition to , we only have to test , of which both work. Taking the smaller one, our answer becomes .
Suppose the dice have and faces, and WLOG . Note that if since they are both , there is one way to make , and incrementing or by one will add another way. This gives us the probability of making a as
Cross-multiplying, we get
Simon's Favorite Factoring Trick now gives
This narrows the possibilities down to ordered pairs of , which are , and . We can obviously ignore the first pair and test the next two straightforwardly. The last pair yields the answer:
The answer is then .
The problems on this page are the property of the MAA's American Mathematics Competitions