[CamperBob2]

If it has cycles of 50 or less only, then they're guaranteed to find their own name (they start the cycle on the box corresponding to themselves, so that cycle must contain them)

But there are 100 boxes, assigned at random. The prisoners can come up with a mapping function that predetermines which boxes they will open based on their names, but that function will have no relationship to the one (if any) that was used by the warden.

There is no way to guarantee that the first prisoner finds his name, and that seems to be true for all of the others. It must genuinely be a case where I haven't understood the problem correctly.

[function_seven]

If no cycle is longer than 50 boxes (~30% chance of that being true), then by starting with the box that matches your number, you have a 100% chance of navigating to the box containing your number before your 50-box limit is reached. It’s impossible to start in the wrong cycle, because that cycle contains neither the pointer or the value.

You have to find your pointer in a circular linked list. But it’s only singly-linked, so you start just in front of it and work around the links the long way. 30% of the time, all the lists are 50 elements or shorter, meaning everyone is guaranteed to succeed.

[CamperBob2]

But what if your name is in box #51, which you aren't allowed to open?

[bewaretheirs]

Then there's a cycle of length > 50 and everyone loses.

[function_seven]

Like sibling says, this only happens when there’s a cycle longer than 50. In that case, you’re all doomed.