One hundred numbered prisoners each get fifty box-openings to find their own number; if a single prisoner fails, everyone dies. Guessing randomly, survival odds are (1/2)¹⁰⁰ — effectively zero. A strategy with no communication at all raises that to 31%, and the reason lives in the hidden loop structure of shuffles.
The slips secretly chain the boxes into loops: box 14 points to box 60, 60 points somewhere else, and eventually some box points back to 14 — it must, because no two boxes point to the same place. A shuffle isn't a mess; it's a drawer of closed necklaces.
Starting at the box with your own number isn't a guess — it plants you on your own loop. The slip bearing your number is the one that points at your starting box, so it's the loop's last stop: following slips must bring you home. The only question is whether your loop is short enough to lap within 50.
So the whole prison lives or dies on one number — the longest loop. At most one loop can be longer than 50, and it shows up far more often than a gut says it should.