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 probability of obtaining the sum and the probability of obtaining the sum are equal because any combination of which sum to can be replaced with to get the sum , 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 , and, examining the formula, this 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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