[JMStewy]

The key insight is that this strategy causes there to be a strong correlation between the success of different prisoners.

As an example, let's say prisoner #10 opens his box and finds #21, then finds #5 in that box, then #84, then #51, then finally succeeds and finds his number 10 in box #51. These boxes form a cycle 10-21-5-84-51 (and then back to 10). Anyone who opens any of these boxes will eventually see the same set numbers that #10 did, so that means that we know prisoners #5, #10, #21, #51 and #84 will all succeed in finding their number by starting at their own box.

Compare that to the situation where they just randomly look in boxes and each independently have only a 50% chance to succeed - then the odds that those 5 prisoners all succeed would be (1/2)^5 = 1/32, or only about 3%. In every cycle containing n prisoners, instead of their success rate being (1/2)^n, now they all succeed together or all fail together.

Now that the success of every prisoner in the same cycle is perfectly correlated with each other, the prisoners' overall chance of success depends only on whether the random permutation created by the warden has any cycles >50 in length. If so, then everyone in that cycle will fail. If not, then all the cycles across the 100 boxes are short enough that all 100 prisoners will succeed.