Solution to The Impossible Bet | The 100 Prisoners Problem

The “impossible bet” in the 100 prisoners problem looks unwinnable because each person is limited to checking only 50 boxes, and naive random choices make the chance that all 100 find their own marked bills astronomically small—about 2^-100, essentially zero. Even small coordination tweaks (like making two people avoid picking the same box) only help marginally, and the advantage shrinks as the number of prisoners grows.

The winning insight is that the boxes aren’t just hiding information—they encode it. The room is identical every time: box numbers are fixed, and the bill inside each box points deterministically to another box. That means a prisoner can treat the contents as a set of directed links: if a prisoner opens box i and finds the bill labeled j, the prisoner should next open box j. Starting from the prisoner’s own number, this creates a “scavenger hunt” path that follows the chain formed by the bills’ labels.

These chains behave like loops in a finite directed graph. Every chain eventually cycles back to a previously visited box, and the longest possible chain can include all 100 boxes. The crucial win condition is whether the cycle closes within the first 50 steps. If a prisoner’s path returns to the starting box within 50 boxes, then the prisoner must have encountered the bill that points back to their own number—meaning the prisoner has found their own marked bill. If the cycle takes longer than 50, the prisoner runs out of allowed checks and fails.

Why this lifts the group’s odds so dramatically comes down to how random box arrangements generate chain lengths. For a random permutation of bills across boxes, the probability that the relevant cycle containing a given starting point stays within 50 is high enough that, when all 100 prisoners follow the same “follow the bill” strategy from their own starting boxes, the probability that everyone succeeds lands at a little over 30%. That’s far larger than the near-zero probability from independent random guessing, and it’s also better than the baseline of only two prisoners choosing randomly (which would give 25% for both to succeed).

The strategy’s real power isn’t that any one prisoner becomes more likely to succeed in isolation; it’s that outcomes become synchronized. Because every prisoner’s fate depends on the same underlying chain structure, successes and failures occur together: either the chain lengths stay short enough for all starting points to close within 50, or they don’t. In effect, the prisoners’ linked paths through the boxes force the entire group to win or lose together, turning an “impossible” collective bet into one with a meaningful chance of success.