A box B gets a marble if the phase P is a factor of the box, right? If B/P = integer, then that box gets a marble.
The box that gets the most marbles would be the one with the most factors, not just prime factors. So box 16 = 2*2*2*2 has factors 2, 4, 8, 16. I would say that the box with the most non unique prime factors would have the most. We aren't talking about the size of the factor, just the number of different factors that can be made. That's my first guess.