Blue Eyed Islanders Puzzle And Blue Lights  

So two things related today only entirely incidentally by the color blue.

First thing is the Blue Eyed Islanders Puzzle, which I first chanced upon a year or two ago at xkcd and then here at Terry Tao’s site

I promptly proceeded to give up on the puzzle and read the solution, be thoroughly impressed by it, and forget it.

While reading ahead last week in Spivak, I noticed that problem 27 in Chapter 2 was almost exactly the same thing, albeit with a different scenario regarding math professors who resign from their posts once they become aware of errors in their published works and one visiting professor who is out to get revenge on all of the other professors for not offering his a position at the university. Kind of awesome.

In any case, this caused me to revisit the puzzle, and this time attempt to solve it in earnest (since I had entirely forgotten the solution, as so far as my conscious mind is aware).

Spoilers follow.

I did vaguely remember what I believed to be the solution, which was that on the first day after the revealing of the knowledge, no one killed themselves, and on the second day, everyone killed themselves.

This proved to be precisely incorrect, but broadly correct. I still had no idea why, until I showed the problem to Travis and we jointly attempted a solution.

Primary credit goes to him to suggesting that we regress the problem to just 1 blue eyed islander and 999 brown eyed islanders. In this case, the one blue eyed islander would kill himself the day immediately following the revealing of the knowledge that there is a blue eyed islander on the island, for in this case there is clear new knowledge imparted – the blue eyed islander did not know he had blue eyes, and now he does, as he sees everyone else on the island has brown eyes and consequently the visitor must be referring to him.

Now, in the case of 2 blue eyed islanders, it becomes quite interesting, in that each blue eyed islander is now put in the same position as a brown eyed islander had been in the case of 1 blue eyed islander, in that they see only 1 blue eyed islander and 998 brown eyed islanders, just as a brown eyed islander would have seen in the first scenario with only one blue eyed islander. Hence, they have no idea if they’re a brown eyed islander or a blue eyed islander just yet, since there is at least one other person with blue eyes the visitor could have been referring to.

However, here is the pivotal fact. Everyone on the island has now been imparted common knowledge at exactly the same time, and since they are all perfectly logical, they know exactly what would happen in each situation. So, in the two blue eyed islander scenario, each blue eyed islander is crucially debating whether there is one blue eyed islander or two blue eyed islanders (which, also, is what every brown-eyed islander was debating when there was only one blue eyed islander. Fortunately, to their collective great relief, the blue eyed islander killed himself the next day in that situation and they vindicated of their burden).

However, now, unlike before the visitor said anything, both blue eyed islanders know that the other blue eyed islander now knows that there is at least one blue eyed islander on the island. Hence, if one of the blue eyed islanders was actually brown eyed, he knows that the other blue eyed islander, seeing no other blue eyed islanders, would promptly kill himself the day following the visitor’s announcement.

When he does not, the other blue eyed islander is forced to the terrible conclusion that he, too, must have blue eyes, and that is why the other blue eyed islander did not kill himself – because he too was thinking exactly the same thing and debating whether or not there were one or two blue eyed islanders.

Now, unfortunately, they both become aware that there are indeed two blue eyed islanders, and both off themselves on the second day following the visitor’s announcement.

This is to the great relief again of every brown eyed islander on the island, for if the two blue eyed islanders did not off themselves, then that would mean that there were in fact three blue eyed islanders and the person who only saw two blue eyed islanders (i.e, every browned eyed islander in a two blue eyed islander situation) would be the third islander and consequently all three of them must kill themselves on the third day.

So I went about that one in a bit of a convoluted fashion, so let me repeat. If there were three blue eyed islanders, each would see two other blue eyed islanders, and be crucially debating whether he was the third blue eyed islander or if he had another eye color.

Since he knows that if there are two blue eyed islanders they will off themselves on the second day, when they do not, he is forced to acknowledge that he must be the third blue islander (note that every blue eyed islander goes through the exact same thought process), and hence all three of them kill themselves on the third day.

This is repeatable for 4 blue islanders, and the same for 5, etc. all the way up to 100.

At 100, all the blue eyed islanders wait 99 days, then kill themselves on the 100th day. Note that if it is known that there are only two eye colors, brown or blue, then there is no hope for anyone on the island and they are all doomed from the beginning, and everyone with brown eyes kills themselves on the 101st day. If this is not known and the possibility of red eyes, or gold eyes, etc, is available, then no one else kills themselves as they are all hoping they are extraordinarily lucky and don’t in fact have brown eyes like everyone else but perhaps are blessed with pretty pink eyes.

So, the fascinating question is this – since the visitor imparts no new knowledge, as they all knew that at least one person had blue eyes before he told them this, how could him stating this fact change anything?

The truth is he does, in fact, impart something, and what he imparts is the common knowledge to everyone involved that if there was only one blue eyed islander on the island, he would off himself the following day.

While this seems highly removed and irrelevant, it’s not, as the crucial question every one of the 100 blue eyed islanders is trying to answer (or, rather, hoping not to ever have to answer) is are there 99 blue eyed islanders on the island, or are there 100?

Now, this causes a downward spiral all the way to in fact 1 blue eyed islander, as if there were indeed 99 blue eyed islanders, then each of them would be asking if there were 98 (as they could only see 98 each – even though each blue eyed islander now can see 99, they know that each of the 99, were there only 99, would see 98 and be asking the same thing of 98, and so forth – common knowledge), and if there were 98, then each would be asking if there were 97, and so forth, all the way down to if there were 2, then each would be asking if there was 1.

Now, unlike before, that crucial first scenario can be answered. Before, if there was 1, he would have been without knowledge that he had blue eyes, and therefore wouldn’t ever do anything, and consequently two blue eyed people couldn’t do anything, and three couldn’t, all the way up to a hundred. Now, since everyone knows that he will kill himself precisely the next day, the scenario for 2 blue eyed islanders is also answered, and 3, and 4, all the way to 100.

And so what the visitor imparted upon everyone as new knowledge was the commonly shared knowledge that if there was only one blue eyed islander, he would kill himself the next day. This was something that had not been previously known, and the one fact that made all the difference.

BAM.

Anyway, on blue lights. I think I likely have some form of either mild or severe seasonal affective disorder (perhaps mild exacerbated by the fact that I have no great inclination for life and am presently concluded that life is objectively meaningless), and so right out of Harvard I purchased the Philips goLite BLU to use in the mornings both to make me happier and adjust my circadian rhythm. Unfortunately, I had absolutely no structure for introducing new habits then and thus entirely failed to follow up with the goLite after just a few days and promptly forgot about it for a year and a half.

Just recently a month or two ago I rediscovered it and decided to use it this winter, and I do believe the results have been stellar. If I go too long without it I start to have Absurd thoughts…but no Absurd thoughts when I use it, so far as I can tell.

The blue light is the bane of existentialism! Woohoo!