Island of the Logicians Riddle
Posted by Rob Herman at August 8th, 2006
Another riddle for today. If you’re more interested in the board games than the riddles, my apologies. I haven’t been able to do much board gaming lately due to various irritating life factors. Following GenCon I hope to change that around.
In many ways this is the opposite of the Rope Riddle yesterday. The rope riddle has a very elusive solution, but the solution is crystal clear once you have it. By contrast, the solution to this one is pretty easy to get, but almost paradoxical in its nature. You may have seen a version of this riddle before. It’s the lead-up to a new riddle Thursday, invented by intrepid reader and riddler John Rhoadhouse. (Thursday’s is fierce and even more paradoxical than this one!)
On a certain island live 100 logicians. Some of the logicians have brown eyes and some blue. There are no mirrors or other reflective surfaces, and the logicians never discuss one another’s eye color. Thus, every logician knows the color of everyone’s eyes but his own.
Every night at midnight, a ferry comes to the island. Any logician who knows his eye color gets onto this ferry, leaving forever for the logicians’ paradise. The other logicians will wake up to find them gone. However, because of the lack of mirrors, this has never happened as far as the logicians know.
One day, in a crash of lightning, a stone tablet from The Powers Of Truth lands on the island. Its inscription: “At least one of you has blue eyes.”
Question 1: Who leaves the island, and on what day?
Question 2: The tablet doesn’t say anything that the logicians didn’t already know. How, then, does it help them to leave the island?
P.S. Without loss of generality, in the case where none of the logicians have blue eyes, the tablet says “At least one of you has brown eyes.”
I really can’t see how the first step would work unless there is only one blue-eyed logician. After noting all other eyes were brown, the tablet would affirm without a doubt that he had blue eyes. In the case where all the blue-eyed logicians leave the island, are you saying the tablet changes to read “brown eyes”, or does it remain how it was created?
Not sure how to explain this clearly, but I believe the answer is:
drumroll……..
Setting the day that the tablet fell from the sky as day one, all the blue-eyed logicians leave on day (actually night) X where X is the number of blue-eyed logicians. All the brown-eyed logicians leave on day (night) X+1
I see no relation to the rope riddle (which, by the way, I got almost immediately and don’t understand why everyone thinks it’s hard… unlike all the other riddles, which I found difficult-to-impossible).
Beaker is correct and to answer Garys question the tablet does not change what it says. the most interseting part is how do they get to leave when the tablet doesn’t tell them anything that each of them did not already know (assume at least two of the logicians have blue eyes)
On another note: I would like to thank Rob for granting me the XP needed to level up to “intrepid” reader. Also when Rob first told me this riddle I thought he said a _fairy_ came to the island every night to take the logicians away. It makes alot more sense now.
Shruti> The only relation to the rope riddle is that they’re very different. As soon as you know the answer to the rope riddle it makes a lot of sense. But as Beaker found, you can have the answer to this riddle and still not completely understand it.
John> You know how NPR makes people’s titles a part of their name? How, for example, you never hear the words “Muqtada al-Sadr” except as part of the phrase “Shi’ite cleric Muqtada al-Sadr”? I was thniking of soing something similar with readers. You get to be intrepid reader John Rhoadhouse. If anyone else wants to reserve an adjective, let me know.
ooh…can I have an adjective? I am not too particular, anything nice will do.
Sure. How would you like “erudite”?
If you’re taking requests, I’d like ambivalent.
Gary, I’m not sure how I feel about that.
Sorry, but I’m not quite sure I see the necessity of Beaker’s answer. How is it that a Blue-eyed person knows he has blue-eyes if there’s anyone else who has them also. As far as I read, it only stated that somebody has them.
hmmm… hope I’m not inadvertantly lowering the average IQ here… hmm… dough-nuts.
Let’s say there are 2 people with blue eyes, the rest brown. The blue-eyed people are thinking “There are either 1 or 2 people with blue eyes”, the brown-eyed people are thinking “there are either 2 or 3 with blue eyes.” On day 2 it becomes clear to both blue-eyed people that they must be one of 2, because if the only other blue-eyed person they could see were the only one, that person would have left on the very first night. Likewise if there are 3 people with blue eyes, it becomes clear on the day 3, because if there were only 2 they would have figured it out on day 2.
The pronouncement gives you a place to start counting.