Coolest variation of this: If the prisoners have a spy (like a friendly warden) who is allowed to enter the room first and swap the contents of just two boxes, the prisoners' survival probability jumps from 31% to 100%.

By swapping two items, the spy can break any cycle longer than 50 into two smaller cycles. Since every prisoner is then guaranteed to be on a loop of length 50 or less, everyone is mathematically certain to find their number.