# Mock AIME II 2012 Problems/Problem 5

## Problem

A fair die with sides numbered through inclusive is rolled times. The probability that the sum of the rolls is is nonzero and is equivalent to the probability that a sum of is rolled. Find the minimum value of .

## Solution

Consider the generating function for a 12 sided die. When rolled n times, the generating function is . This polynomial is clearly symmetric, and the coefficient of is thus the same as the coefficient of .

Thus, the coefficient of is the same as the coefficient of . Note that , and thus the minimum value for n is 168. The minimum value of is thus , so the answer is .