INKWIRE(7) reasoning puzzles INKWIRE(7)
blue-eyes
The Ferry That Leaves at Midnight
A rule, a mirror-free island, and one sentence that changes everything.
NAME
Problem
On an island of perfectly logical people, each inhabitant has green or grey eyes and can see everyone's eyes but their own. There are no reflective surfaces and no discussion of eye colour is permitted.
The law is exact: anyone who deduces their own eye colour must leave on the midnight ferry that night, and anyone who guesses wrongly is punished. Nobody has left for a hundred years.
One morning a ferryman, who has counted for years, says to the assembled island: "I can see at least one person with green eyes." He is telling the truth. Suppose exactly 40 of the 300 islanders have green eyes. What happens, and when?
SIGNALS
Conditions and what each one forces
| # | Condition | What it forces |
|---|---|---|
| 1 | Every islander sees all other islanders' eyes | A green-eyed islander sees 39 green; a grey-eyed islander sees 40 green |
| 2 | No mirrors, no discussion of eye colour | Nobody can learn their own colour by observation or by asking; only by inference from others' behaviour |
| 3 | Leaving requires certainty, guessing is punished | Each islander leaves only when every other possibility is eliminated by the observed absence of departures |
| 4 | "At least one green" was true and already believed by everyone | The sentence's content is old; what is new is that everyone now knows that everyone heard it, and knows that they know, indefinitely |
| 5 | Exactly 40 green-eyed islanders | Induction gives 40 nights: on night k, all islanders who see exactly k−1 green eyes depart |
EXAMPLES
Method
Induction on the number of green-eyed islanders. The announcement's only effect is to create a base case for the chain of reasoning, which previously could not get started.
[+] Worked solution 4
Take the smallest case. If exactly one islander had green eyes, they would see none, hear the announcement, and know instantly that the announcement was about them. They leave on night one.
With two: each green-eyed islander sees one other green and reasons "if I were grey, that person would be the only one and would leave tonight". Nobody leaves on night one, so both conclude they are green and leave on night two.
By induction, with n green-eyed islanders all n leave on night n. Here n = 40, so on the fortieth midnight ferry all forty green-eyed islanders leave together, and the remaining 260 grey-eyed islanders stay — they can only conclude their own colour is grey after night 39 passes without departures, but for them that conclusion is consistent with either 40 greens (if they are grey) or 41 (if they are green), and night 39's silence rules out the lower counts, not theirs.
The precise statement of the last point: a grey-eyed islander sees 40 greens. They reason "if I am green there are 41 greens, and the departure would be on night 41; if I am grey there are 40, and it happens tonight." So they wait for night 40 and leave only if nothing happens. But night 40 is when the 40 greens leave, so the grey-eyed islanders learn their colour the following morning and depart on night 41.
Kindred entries
- [P] census Everyone already knows, and yet nothing happens until it is said aloud.
- [P] twelve-coin One coin is counterfeit and you do not know if it is heavy or light.
- index — /t/puzzle/
[+] Notes and objections 3
- Why does the number 40 matter if everyone can already see green eyes?
- Because the depth of the reasoning is what matters. A person seeing 39 green eyes must consider what happens if there are only 38, and the person in that hypothetical must consider 37, and so on down to zero. The announcement terminates that descent.
- Do the grey-eyed islanders also leave?
- Yes, one night later. Seeing 40 greens, they cannot distinguish "I am grey" from "I am green" until night 40 passes: if they were green the count would be 41 and nothing would happen on night 40. When the fortieth ferry takes 40 people, they know, and leave on night 41.
- What if the ferryman had said "at least two have green eyes"?
- Then the whole schedule shifts down by one: the induction would start from two instead of one, and the departures would occur on night 39 rather than night 40. The announced floor sets the base case.
Scope — Puzzle presented for practice; the induction above is exact for any count of green-eyed islanders.