[jldugger]

Imagine Strategy A, where everyone started at box 1 and checked 1-50. Half of the prisoners would succeed, guaranteed, but half would fail, guaranteed, and prisoners would never win.

A slightly better Strategy B is to start at their own box and all subsequnt ones, wrapping around at 100. Now you have at least a small chance, but each prisoner winning independently is low.

What cycle method Strategy C does is group up failures. When prisoners fail because there's a cycle of size 51+, 51 fail. When there's a cycle of 99 boxes, all prisoners fail. But in exchange, there's more possibilities where there aren't cycles that long.

So it's not that the odds are better for an individual as much as the odds are better that the individual's successes coincide. In Strategy A, the success of an individual prisoner decreases the estimated success probability for subsequent prisoners. In Strategy B, that decreased probability is nullified by incrementing the starting location by one step. In strategy C, a prisoner's success or failure amplifies the estimated success probability for others. Not by changing any state, but revealing the state itself.