Probability question relating to harmonic series

Hello all,

I'm stuck on another simple question, I think the solution is based somewhere on reversing Euler's approximation for harmonic series, yet I'm unable to make the link between the math and the question.

Question:

McBurgerQueen wants to run a competition where the grand prize is a very expensive car. In order to enter the competition customers must posses 1 of each coupon that comes attached in a random manner to their McBurger.

However McBurgerQueen wants to ensure that everyone entering the competition purchase on average at least 12 burgers.

At least how many different coupons should there be?

I've run a simulation iterating from 2 to 10 coupons over 10^7 selections in each, and it seems that ~6 coupons is the smallest number required, to average 12 burgers and to posses all 6 unique coupons (5 averages ~11 and 7 averages ~14). I'd really like to know what the pure mathematical way is to solve this problem. any assistance will be greatly appreciated. (Nod)

Other ways to think of the problem:

1. How many times on average should a 6 sided dice be rolled so as to get each of the 6 sides at least once?

2. I have a fair n-sided dice, which on average takes 12 rolls to get each side at least once, how many sides does the dice have?