> If just one prisoner does not find his number, all prisoners die. You're trying to find a strategy that is successful for all prisoners. Following the chain

> If just one prisoner does not find his number, all prisoners die. You're trying to find a strategy that is successful for all prisoners. Following the chain reduces the problem to "is there a cycle of length >50", which can be calculated easier. Opening boxes randomly (or arbitrarily, as in "50 boxes starting with my number), by all prisoners, is 50%100 which as mentioned is vanishingly small. The trick is that chain-following actually gives information to each prisoner as they go, which helps leverage into a better win chance.