It’s extremely subtle……….I’ve never seen anything like it.
Anyhooz here’s the Hardest Logic Puzzle in the World:
A group of people live on an island. They are all perfect logicians — if a conclusion can be logically deduced, they will do it instantly. No one knows the color of their eyes. Every night at midnight, a ferry stops at the island. If anyone has figured out the color of their own eyes, they [must] leave the island that midnight.
On this island live 100 blue-eyed people, 100 brown-eyed people, and the Guru. The Guru has green eyes, and does not know her own eye color either. Everyone on the island knows the rules and is constantly aware of everyone else’s eye color, and keeps a constant count of the total number of each (excluding themselves). However, they cannot otherwise communicate. So any given blue-eyed person can see 100 people with brown eyes and 99 people with blue eyes, but that does not tell them their own eye color; it could be 101 brown and 99 blue. Or 100 brown, 99 blue, and the one could have red eyes.
The Guru speaks only once (let’s say at noon), on one day in all their endless years on the island. Standing before the islanders, she says the following:
“I can see someone with blue eyes.”
Who leaves the island, and on what night?
[Stipulations]
There are no mirrors or reflecting surfaces, nothing dumb, It is not a trick question, and the answer is logical. It doesn’t depend on tricky wording, and it doesn’t involve people doing something silly like creating a sign language or doing genetics. The Guru is not making eye contact with anyone in particular; she’s simply saying “I count at least one blue-eyed person on this island who isn’t me.”
And lastly, the answer is not “no one leaves.”
I think this/these stipulation[s] is[are] also necessary:
1) That everyone with blue eyes (at least) is wholly involved in figuring out if they have blue eyes and should comply (bear with me, this is different than you think)
Without this specification, there can be no implicit communication as to the understanding of others.
But to be fair, this is hardly the end of the specifications, and is why I so detest logic puzzles. An earlier poster had it right when they said that a logic puzzle is hardly about logic, but about communication between the puzzle maker and tester.
I think there is an issue with your stipulation about the islanders not “otherwise” communicating with each other because I think communication other than the exact count of islanders is exactly what happens.
To rectify this I could, for instance, instead demand there be such stipulations as:
2) The islanders don’t tell each other what their eye colors are;
3) Or otherwise sign;
[and because we're tired and want to stop this progression from escaping us we'll try a catch all...]
4) or intentionally let each others know;
Perhaps you might say someone demanding that there could be such a “outside mode of communication” isn’t “playing along”, but this is precisely what I mean. There must be a communication of what the solution might be for a guesser to play along with what might get him there. I would like to especially point out that one can get into the same trouble with rule 4 as you did with your stipulation (and for the same reason, and that this is unavoidable). I would say that an islander following stipulation 1 is implicitly breaking this or any such rule.
But we’re hardly done yet… I want to examine rule 1 a little closer and to do that I need to outline the “solution”.
1 blue eyed islander:
In the case of the 1 blue eyed islander, he sees that everyone else has brown eyes (and the guru green) so knows he must have blue eyes. He leaves. The others do not leave because they see him leave.
2 blue eyed islanders:
The two blue eyed islanders see that there is another blue eyed islander and so don’t leave on the first day. They then both leave on the second day, knowing that the other must have not left because they expected themselves to leave the first day*. No one else leaves, because they knew there were at least two others, and so waited for the third night**.
And so on for n blue eyed islanders.
So, first onto the problem with condition 1), and then onto those little stars.
Condition 1) at first seems as if it is a simple restatement of the condition that islanders know the count of other’s people eye color at all times. It is not. In addition, it also signifies that every islander (or at least ever islander with blue eyes) must be *carrying out* the logical processes and understand its implications.
But then, for any islander to have any certainty of what the other islanders know, he must also have the guarantee that every other islander is carrying out these processes. This leads us to:
5) Every islander knows that every (blue eyed) islander is wholly involved in deciding if they have blue eyes.
It’s actually a bit more involved and also involves knowing all other stipulations are also in effect, but I would like to keep this somewhat comprehensible.
And now that we realize this is a stipulation for the valid reasoning of an islander, an islander also needs to be assured of 5). Thus:
6) Every islander knows that every (blue eyed) islander knows that every (blue eyed) islander is wholly involved in deciding if they have blue eyes.
And so on. It is only important that every blue eyed islander has correctly proceeded for one to decide how to proceed.
Those stars:
*Conveniently, this directly extends from the previous statement, so it should flow nicely. For an islander with blue eyes to know that the others know he has blue eyes, he must assume that the islanders “correctly proceeded”. This, however, presumes that a solution exists, he knows it, all the islanders have followed it, that he knows all the other blue eyed islanders has followed it, and all the recursions that follow. Thus we do not get out of this recursion by simple invoking “correctly proceeding”.
**This necessity of knowing “blue eyeds proceed correctly” also applies to brown-eyed people and the Guru.
So, we have at the least problems with: islanders’ foreknowledge of their own disappearance, their intentionality, and modes of communication. The problem is logically indefinable, though it may indeed communicate a solution. The answer (as in all “mind puzzles”), is to communicate, *not* to “think out of the box”. You are no smarter for having figured it out, but perhaps, a little enriched.
So it is the Guru who leaves, on the first night, to travel to another island and impart this wisdom there.
Anyways, a full mapped out answer is here:
http://www.scatmania.org/wp-content/blue_eyes/#
Leave A Regular Reply: