Help with Independent Probability Problem

Question

There are n dice of different colors being rolled simultaneously. The numbers that show up on the faces are added up to compute X. How many different die rolls result in X being divisible by 3? (basically how many of the n tuples with each number being between 1 and 6 are such that the sum of these n numbers is divisible by 3)

I was having some problem figuring out how exactly to answer this question. We discussed how to solve this problem using a known number of dice, but I can't figure out how I would accomplish this using a variable number of dice, as it is hard to calculate the possibility of all numbers that the dice can sum to and be divisible by 3. Any help would be appreciated.

Answer 1

Hints:

• How many $n$-tuples are possible in total?
• What proportion have a sum which is a multiple of $3$? (Easier than you might think at first - consider the effect of the last dice to roll)