I need your help with a math problem!

If n traditional six-sided dice are tossed, the probability of obtaining a sum of 1994 is greater than zero and is equal to the probability of obtaining a sum of S.  What is the smallest possible value of S?

This problem apparently appeared on the 1994 AHSME (American High School Mathematics Examination, which has now been replaced with the AMC 10 and the AMC 12). I have an answer (cube root of 37,259,704; I’m writing it like this so as not to spoil your fun!) but would love to hear someone else’s approach.  Please leave a comment or send an e-mail if you have a solution.

Updated:  Upon further reflection and consultation with Dr. Hockema and Mr. Thomas (East Jackson High School), I think my original answer is incorrect.  Now, I believe the answer is the cube root of 38272753, although I want to think about it some more.

I was amazed to see diagonals of Pascal’s triangle appear in my investigation.  Very, very cool!

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