Complex dice roll question, please read description for explanation?

OK, so my friend came up with a special rare occurrence rule for Risk.
If someone rolls 3(6's) in a single role, 6 times in a single game, then everyone quits and that person wins. What are the odds of this happening?

Constraints: only an attacker can ever have 3 dice at once, and probably only will have 3 about half the time, though I haven't calculated this- it's just a guess.

My guess is that any given player, in a game consisting of four players, will roll with 3 dice about 50-100 times. I would be satisfied with calculations for either or something else with better reasoning.

The odds of rolling 3 out of 3 dice a 6 is 1/(6^3) = 1/216. Would this essentially turn each 3-dice roll into a single 216-sided dice (statistically speaking)?

If there are 50 rolls of this, would it be (50/216)^6, 50/(216^6), or what?
and what if, instead of getting 6 at some point, you needed to get a 3x6 roll, 3 times in a row, or 6 times in a row? Those are side questions, the main one is the odds of getting 6 at some point in a game.

Thanks