Continue this number-triplet series

(4-6-4) (6-12-8) (8-12-6) (12-30-20)

What is the next number triplet - and why?

I'll add hints if nobody gets the answer first. In the past I've used this riddle in a game with my RPG gaming group.

Edit: The answer I'm seeking in this riddle is unique and well defined. It requires thinking out of the box a bit. I will add hints over time. One, actually, has already been given.

The last triplet is $(20, 30, 12)$. These are the number of faces, edges, and vertices of the five Platonic solids in ordered triples, in order of increasing face count: tetrahedron, cube, octahedron, dodecahedron, and icosahedron.

It doesn't seem like this riddle involves much outside-of-the-box thinking at all; it's just a case of knowing what to look for. I would consider this to be trivia more than it is something that can be reasoned out.

• Maybe you're right. It is a very fun puzzle for RPG games though. I've watched my players (with the dice set in front of them and far more hints) for a rather (embarrassingly) long time... – BmyGuest Nov 21 '14 at 0:24
• Haha, yes, when the dice are all Platonic solids, it can be embarrassing when they don't realize the answer is right in front of them on the table. – Joe Z. Nov 21 '14 at 0:24

I think that there are many possible solution for the next triplet.

(18, 48, 24)

Explanation:

I've seen number patterns like this where there is no math pattern. I think that the pattern is that the first and third numbers must both be less than the middle number.

• The solution I'm after has a unique answer. I agree that there might be different answers with different rule-sets, but I'm looking for a distinct one. I will add hints towards that goal over time. I've upvoted your answer because you've made a fair point. (Prior the first edit of my riddle.) – BmyGuest Nov 20 '14 at 23:55