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Short Summary
Main recommendation: The prisoners should each follow the "loop strategy": open the box with their own number, take the slip inside, then go to the box matching that slip's number and keep repeating until they find their own number. This gives the group about a 31 percent chance of all surviving, versus virtually zero with random searching. The video explains why the strategy works through permutation cycles (loops) and the math behind the probability.
Key Points
- With random searching, each prisoner has a 50 percent chance, so the chance that all 100 succeed is one half to the power of 100, a vanishingly small number.
- The boxes form closed loops; a prisoner who starts at their own box is guaranteed to be on the loop containing their slip, because every slip sits inside a box.
- All prisoners succeed only if no loop is longer than 50, since a longer loop means the prisoners on it run out of the 50 boxes they may open before reaching their slip.
- The probability of a loop of length k is 1/k, so the chance of a loop longer than 50 is 1/51 + 1/52 + ... + 1/100 ≈ 0.69, leaving about a 31 percent chance of success.
- The outcomes are linked: the prisoners either all succeed or most fail together, rather than each having an independent 50 percent chance.
- The success probability approaches a limit of about 30.7 percent as the number of prisoners grows, calculated as 1 minus the natural log of 2.
Details Worth Noting
- Concrete instructions: Start at the box bearing your own number. Open it, read the number on the slip inside, go to the box with that number, and repeat. Stop when you find the slip bearing your own number. Each prisoner may open at most 50 boxes, so the strategy only works if no loop exceeds 50 boxes.
- The prisoners can only coordinate this strategy beforehand, and the video notes the solution was left as an exercise by puzzle creator Peter Bro Miltersen (referred to in the transcript as Peter Borgstrom Milterson), who didn't think of the strategy himself until a colleague pointed it out.
- A sympathetic guard could guarantee success by swapping the contents of two boxes to split any loop longer than 50. A malicious guard who anticipates the strategy can be countered by renumbering the boxes, which restores the random-arrangement odds of about 31 percent.
- For 1,000 prisoners (each allowed 500 boxes) the success chance is about 30.74 percent, only half a percentage point lower than with 100 prisoners, and for 1 million it is about 30.685 percent; time to open boxes would be the practical obstacle.
Bottom Line
The counterintuitive result is that a cooperative cycle-following strategy turns near-impossible odds into roughly a one-in-three chance, and that this figure holds regardless of how many prisoners are involved.