National Association of Math Circles Wiki
The Island of Blue and Brown Eyes

Blake Thornton is taking the lead on this activity for the 2012 Circle on the Road festival.

AUDIENCE: middle school

PLANNING ABSTRACT: We will investitage some logical puzzles about "common
information." The basic puzzle is this. There is a special island where
the inhabitants' religion forbids them to know thier eye color (which they
do know to only be either brown or blue). If an islander ever discovers his
eye color, he or she is required to leave the island. These amazing
islanders, of course, understand logic and never make any logical mistakes.
One day, a visitor comes to the island. Ignorant of their religion and
customs, this visitor says, "it sure is nice to see someone else with blue
eyes." What happens next?

APPRENTICES: Irina Mitrea, Nieves McNulty, Maria Lorenz

DISCUSSION:

The basic idea for this activity is from Tao's blog. This is a useful thing to read (including the comments):
http://terrytao.wordpress.com/2008/02/05/the-blue-eyed-islanders-puzzle/

So, the idea is to introduce the participants to the setting--the island with blue and brown eyes, their religion of not knowing eye color, etc. Middle school kids will need some time to adjust to this idea (why don't the islanders lie instead of killing themselves, etc.).

Then, we make the participants islanders. We ran this in our circle by putting dot stickers on their forheads for their eye color and this works fine for a very small number of particpants. With more participants, I think it will probably work better if we give them cards telling them how many people with blue eyes they see. We start with only 1 blue-eyed islander and work up from there.

It was pretty fun when we did this in our circle but it was also pretty chaotic.

There are a number of other similar puzzles or variations on the above puzzle that would also be fun to do. Here are a few of those.

http://www.math.ucla.edu/~tao/blue_variant.html
http://www.mytechinterviews.com/is-your-husband-a-cheat
http://plato.stanford.edu/entries/common-knowledge/

One question I always have when running a math circle is how much of the deep mathematics to present. In this case, the field seems to be related to formal logic, which I don't understand well enough to really reproduce. In any case, the question of how deep to go into this is probably worth having.

My idea for handouts is to provide the statement of the puzzles and links for further investigation.

What do you think???

-Blake

Thank you for the handout. I certainly like 6.1 and 6.2 the best! Is there
anyone else concerned about having to deal with the whole suicide issue
when talking with middle school children? Or is our audience consisting of
adults?

Irina

Yes! The suicide thing was dealt with in our circle by:
"The islander has to leave the island"

I'm okay with either leaving or committing suicide. In writing this up, I decided to leave in the suicide as originally stated on Tao's blog. If you're worried about it, I am more than happy to change it (I've found that most students are okay with this sort of thing and I've stopped worrying about it for the most part in our circle).

One more thing, I was actually concerned about 6.3 (Is your husband a cheat). I think there are quite a few sexist statements in that problem that I find a bit offensive. But, in the end, I'm okay with this problem since we are trying to teach the students to draw conclusions from a given set of facts and the problem is still a good logic puzzle. I am happy to remove this one too if you want.

Blake

AttachmentSize
Blake - Blue Eyed Islanders.pdf113.45 KB