I would guess you have translated this from another language, not quite clearly. Here is my version of what I think it means:

We have N balls that are distributed in some way among three boxes, each of which is large enough to hold all of the balls. We are allowed to choose any box and move to it as many balls as that box contains from another box, thus doubling the number of balls in the one box and reducing the number in the second; we can repeat this as many times as we wish. Prove that, regardless of the initial distribution of balls, it is always possible to end up with one box empty.

Please either confirm my interpretation or correct it; at several points I am departing from what you actually said.

I would start by "playing" with the idea for small N, to see if I could find a strategy to empty a box, as well as what might go wrong in trying to do so. (I haven't given any thought to it at all yet, as I don't want to waste time on a wrong interpretation; and my goal is just to suggest how one might approach it with no special knowledge, anyway.)

Now giving it a little thought, I see that one strategy may be to work backward from the goal, thinking about what must be true before the final step. You might even be able to show that starting with one box empty, you can reach any desired state by working backward.

May I also ask what topics were in that book that might be applicable? I presume it doesn't teach proof by induction ...