Ah, okay, that makes more sense. So the solution is to impose a random ordering on the (1-100 random numbers assigned to the) boxes that ensures you will return to the first one you picked, and that the one that points to your initial pick (or your initial pick itself) has your name in it.

This ordering then partitions the boxes into cycles where each prisoner is in their cycle, and has a low chance of having 50-sized cycles.

Still mesmerized at how the choice of query can improve your chances like that without accumulating information.