Yes, monks will leave, but which night depends on how many are actually enlightened, a number nobody in the story is told directly. The general rule: if exactly k monks have reached enlightenment, all k of them leave together on the k-th night after the announcement, and no one else ever leaves.

By induction: if k = 1, the one enlightened monk sees zero marks on anyone else. Before the announcement, he might have assumed someone else could be the “at least one.” Seeing nobody marked, he realizes it has to be him, and leaves that very first night. If k = 2, each of the two enlightened monks sees exactly one mark, on the other one. When night 1 passes and that other monk doesn't leave, that tells both of them “he didn't see zero marks, so there must be at least one more,” meaning it has to be the monk watching him. Both reach that conclusion simultaneously and leave together on night 2.

This continues the same way: a monk who sees k-1 marks reasons that if the true total were only k-1, those k-1 monks would have already left by night k-1. When night k-1 passes with nobody leaving, that rules out “k-1 total,” which means he himself must be the missing mark, pushing the true count to k. All monks in this position reach the same conclusion at the same time and leave on night k.

The announcement is doing real work here, even though “at least one” sounds like it tells the group nothing it couldn't already see for itself (with 2 or more enlightened monks, everyone can already see at least one mark). What it actually creates is common knowledge: not just that someone is enlightened, but that everyone knows everyone knows it, all the way up, which is exactly the ingredient the induction needs to get started. Without that public announcement, this same silent standoff could persist indefinitely, no matter how many monks are actually marked.

Logic Games