One assumption not explicitly explained is that the prisoners can identify the boxes by their order, since they are arranged in a line. So, the prisoners first agree among themselves that box number x should correspond to which prisoner, and vice versa. So now, each box contains a name, which points to another box at a certain position, which contains another name, ad infinitum, until the prisoner finds his name. It is guaranteed that he will, because of the relationship between permutations and cycles.

It is interesting that this question is regarded as the most math intensive. I myself was spoiled the solution a few years ago, but I have the impression that the challenge is in observing the correct algorithm rather than calculating the probability. Maybe the author felt that one needs mathematical insights in order to see the solution out of the blue.