1994 AHSME Problems/Problem 30
Problem
When standard 6-sided dice are rolled, the probability of obtaining a sum of 1994 is greater than zero and is the same as the probability of obtaining a sum of . The smallest possible value of is
Solution
Let be the number on the th die. There is a symmetry where we can replace each die's number with . Note that applying the symmetry twice we get back to where we started since , so this symmetry is its own inverse. Under this symmetry the sum is replaced by . As a result of this symmetry the sum the sum have the same probability because any combination of which sum to can be replaced with which sum to , and conversely. In other words, there is a one-to-one mapping between the combinations of dice which sum to and the combinations which sum to .
When we seek the smallest number , which clearly happens when is smallest. Therefore we want to find the smallest which gives non-zero probability of obtaining . This occurs when there are just enough dice for this sum to be possible, and any fewer dice would result in being impossible. Thus and . The answer is .
See Also
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