He told them nothing they did not already know

An island of perfect logicians. Nobody may mention eye colour, and anyone who works out their own must leave that night. Some of them have blue eyes; each can see everyone else's. Nothing happens, for years. Then a visitor says, out loud, “at least one of you has blue eyes” — something every islander could already see. On the nth night, all n of them leave.

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The strip of numbers is the point. It lists every possible answer to “how many of us have blue eyes?” that has not yet been ruled out — not by any one islander, but by all of them together, in a way each knows the others have ruled out too.

Saying it out loud crosses off zero. That is the whole effect of the announcement.

Then nights do the rest. Each night that passes with nobody leaving is itself a public fact, and it rules out one more number: if there had been exactly one blue-eyed islander, that person would have seen no others, known the visitor meant them, and left on night one. Nobody left, so it wasn't one. Now it can't be two either, because those two would each have seen one other and worked it out on night two. The strip shortens from the left, one number per night.

A blue-eyed islander sees n−1 others, so only two numbers can describe their world: n−1 if their own eyes are not blue, n if they are. They can go home the moment one of those two is crossed off. That happens on night n. Every time.

Now the part that stings. Set the slider to 4 and ask what the visitor told anyone. Every islander can already see at least one blue-eyed person, so all four of them already knew the announcement was true — and each of them knew the others knew, and knew that too. Not one of them learned the fact itself. Be careful how far that goes: the chain runs exactly n−1 links deep and then breaks. At n = 2 each blue-eyed islander knows the fact but does not know the other knows it — they consider it possible their own eyes are not blue, in which case the other sees nobody blue at all. The missing top link is exactly what the announcement supplies, and it is why the answer is night n rather than any other night.

Press Whisper it to each to see that this is the real difference. The visitor takes each islander aside and tells them privately. Every single one now knows the fact — the content is delivered in full. But nobody knows that anyone else was told, so nobody can reason about what the others will do, no number is ever crossed off, and the island stays full forever. One exception, and the exhibit will show it to you: with exactly one blue-eyed islander, a whisper works. There is nobody else for the fact to be common with, so for that islander private and public are the same thing.

The rule, exactly. Let P be the set of totals still publicly possible, starting as {0, 1, …, N}. Saying it aloud deletes 0. An islander who sees t−1 blue-eyed others leaves when exactly one of {t−1, t} remains in P. Each night with no departures deletes from P every total at which somebody would have left, which gives departure night = n, for every n ≥ 1 This page runs that elimination; it does not look the answer up. Verified in node (improve/verify/116-common-knowledge.js, 14 checks):

Night n is derived, not asserted, for every n from 1 to 40. Nothing in the proof adds one to anything.
The induction is tight: stop the clock at night n−1 and the island is still full.
The answer depends only on how many have blue eyes, not on how many people live there, so the engine is eliminating worlds rather than counting nights.
Negative control. With no announcement nobody leaves in 400 nights, at any n — the same code, one bit different.
Negative control. Whispered privately, every islander learns the fact and still nobody ever leaves, for every n from 2 to 40.
The n = 1 exception is checked rather than waved away: a whisper does empty the island, on night 1.
Negative control. For every n ≥ 2 each islander already sees a blue-eyed person, and for n ≥ 3 each knows every other islander does too — the fact is mutual knowledge two levels deep before the visitor speaks.
Negative control (vacuity). With n = 0 the announcement would be false and nobody leaves, so the model is not just emptying the island on cue.

“Nobody ever leaves” is settled here rather than merely waited out: in this model the state is just P, so once a night passes with no departure and no deletion, every later night is identical to it. The page detects that fixed point and stops. That is a proof about the model on this page, not a full epistemic proof — it assumes the whisper reaches each islander separately, that nobody sees or hears anyone else being told, and that this is itself not known. What this leaves out: real people are not perfect logicians and do not reliably iterate this reasoning past two or three levels. The puzzle is a statement about what information does, not a prediction about anyone's behaviour.

He told them nothing they didn't know.

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