100 Prisoners & the Boxes
Random guessing gives near-zero odds. One clever strategy gives 31%. How?
by MindYourPuzzle
One of the most beautiful results in probability — the answer feels impossible until you see the trick.
100 prisoners are numbered 1–100. In a room sit 100 boxes, each containing one prisoner's number in random order (one number per box).
One at a time, each prisoner may open up to 50 boxes, looking for their own number, then leaves the room exactly as they found it — no communication afterward.
If every single prisoner finds their own number, they all go free. If even one fails, they all die.
Random guessing gives success probability (½)¹⁰⁰ — essentially zero. But there's a strategy that works over 30% of the time.
Using the best strategy, what is the approximate probability (in %) that all 100 prisoners survive?
(Answer with the whole-number percentage.)