For a little Friday Fun I have a logic puzzle for you. I’ll give you the puzzle at the beginning of the post, detour to some unrelated topics (to act as a spoiler barrier), and then explain the puzzle in the latter part of the post. I would encourage you to stop reading and think about the puzzle first — it’s quite a challenge. (I couldn’t solve it.)
The puzzle involves an island with a population of 100 blue-eyed people, 100 brown-eyed people, and a very strange social practice. The logic involved is downright nefarious, and even after reading the explanation, I had to think about it for a bit to really see it. (I still think it’s twisted.)
To be honest, I’m kinda writing this to make sure I understand it!
Randall Munroe (who writes xkcd and some cool books) calls this “The Hardest Logic Puzzle in the World.” That might be a little hyperbolic, but you couldn’t prove it by me. I can’t remember when I first encountered it, but I do remember I walked away shaking my head.
Recently, I started following Terry Tao’s blog, and his “Selected Articles” list includes a post about this puzzle. I read it, and this time I stuck with it. I didn’t solve it, but I did sort of kind of maybe in a way get a faint hint of a small glimmer. Maybe.
After reading the solution, I think I get it. To test myself, I’ll try to explain it later in the post. (Spoilers!) For now, here’s the puzzle (which combines Monroe’s and Tao’s formulations; I don’t like Monroe’s Guru or Tao’s ritual suicide):
As already mentioned, there is an island with 200 perfectly logical people on it, 100 of whom have brown eyes, and 100 of whom have blue eyes. They all belong to a sect that forbids any discussion of eye color. In particular, self-knowledge of one’s own eye color requires self-exile the following dawn. As such, all reflective surfaces are banned. No one may know their own eye-color, although of course they know everyone else’s.
One day a visitor comes to the island. Everyone attends a ceremony presenting the visitor with honors. Asked to speak, the visitor commits the faux pas of saying, “It’s rare to see a blue-eyed person in this part of the world.” Note the visitor does not single anyone out and is not looking at anyone in particular. There is no clue who the visitor means.
The puzzle is this: What, if anything, happens next? Specifically, do any islanders self-exile? Does the visitor’s statement change anything?
One theory is that the islanders don’t learn anything they don’t already know (there is a blue-eyed person on the island), so nothing changes.
Another theory is that the visitor has provided a new piece of information, and something does change.
I’ll give you a hint: the first theory is wrong.
The islanders are perfectly logical and will always make any logical deduction possible. They also always adhere completely to the requirements of their sect. Anyone who deduces their own eye color — and if logic permits, they will — self-exiles the following dawn.
The puzzle is purely logical and does not depend on any trickery or word-play. There are no secrets; every islander can see all the others. Assume the visitor immediately flees in shame without further comment.
Stop now and give it some thought. I’ll explain below.
Guess who won? (You get just one guess.)
They did it tournament style, pairing the actors in a face off.
Round two pitted George Lazenby against Pierce Brosnan, which is maybe a bit weird. Lazenby only did the one film (it was his first film, too), whereas Brosnan did four (and was very well known by 1995).
I might have pitted Lazenby against Timothy Dalton, who wasn’t as well known and who only did two Bond films. That said, as much as I love On Her Majesty’s Secret Service (it has Diana Rigg in it!), I think Lazenby loses to either Brosnan or Dalton.
[There is also that the location used — that restaurant on the mountain top with only cable car access — it’s a real place in Switzerland, and I’ve been there. (Back in high school. Long story. But it made that film one of my favorite Bond films.)]
Round three apparently surprised people. Roger Moore (41%) lost out to Timothy Dalton (49%). Doesn’t surprise me at all. Moore is, for my money, the worst Bond (don’t get me wrong; I loved Moore as The Saint).
In the final playoff, Guesswho won (44%), with “surprise” runner up Dalton (32%) placing, and Brosnan (23%) just showing. Again, no surprise to me. Dalton was a “truer” Bond — so, for that matter, is Craig.
(Bond has an inner element of cruelty, which is why friendly quipping Roger Moore was such a bad Bond (and Brosnan to a lesser extent). It’s why Connery, Dalton, and Craig, are so good.)
The power of nitrogen bonds is… impressive. Various nitrogen compounds have been used in rocket fuels. (One of the more interesting books nonfiction I’ve ever read is Ignition!: An Informal History of Liquid Rocket Propellants by John Drury Clark, who was not just a rocket scientist, but a rocket fuel scientist.)
Some nitrogen compounds are so unstable it’s said they’ll explode if you just look at them. Or even if you don’t.
Back to the island (of blue-eyed people)…
Let’s review the situation: 100 brown-eyed people know there are 100 blue-eyed people and (as far as they know) 99 brown-eyed people.
Similarly, the 100 blue-eyed people know there are 100 brown-eyed people and (as far as they can count) 99 blue-eyed people.
It would seem the visitor saying there is a blue-eyed person adds nothing — everyone knows there is a blue-eyed person. (Half the island thinks there are 99, half the island thinks there are 100.)
But consider the situation with 100 brown-eyed people and only one blue-eyed person. While 100 people know there is a blue-eyed person, the blue-eyed person does not. The visitor’s statement in this case obviously adds new information for that one blue-eyed person.
For simplicity, let’s call the number of blue-eyed people N. In the case where N=1, the visitor obviously adds new information. The blue-eyed person reasons that, “Since everyone I can see has brown eyes, I must have blue eyes.”
Therefore, the next morning, that person exiles.
That seems pretty clear. Now consider the case where N=2, and there are two blue-eyed people on the island.
They both know there is one blue-eyed person. (The brown-eyed people all know there are two.) Each reasons: “If that person sees only brown eyes, they now know they have blue eyes and must exile.” Each expects the other to exile.
When that doesn’t happen, seeing brown eyes everywhere else, both realize they have blue eyes. Both exile the second morning.
(The brown-eyed folks know there are two, and expect both to exile the second morning.)
The logic continues with N=3. Each blue-eyed person sees two others and expects them to leave on the second morning per the logic for N=2.
When that doesn’t happen, each of the three realize they have blue eyes, and all three exile the third morning.
As before, the brown-eyed people see one more blue-eyed person than the blue-eyed people do, so the brown-eyed people expect the blue-eyed people to leave on the Nth day.
When N people do leave on the Nth day, it confirms there were N blue-eyed people and everyone left must have brown eyes. (Which, of course, means they all have to leave the next morning. The visitor cleared the island.)
The logic works for any N , because brown-eyed people always see one more blue-eyed person than blue-eyed people do.
The visitor clearly adds new information in the case of N=1, and the necessary logic based on that propagates upward as N increases.
Randall Munroe has a solution page if you want to explore this further. There’s also a good discussion in the comments of Terry Tao’s post about it. Lastly, the Wikipedia article about common knowledge (logic) discusses it.
Hope you enjoyed the puzzle. A nice little distraction for a Friday.
Stay safe, my friends! Wear your masks — COVID-19 is airborne!