# The Island in the Miracle Sea. (Christmas edition)

To all of you who love math like me, I have this puzzling riddle that I hope you find interesting :

On Christmas Eve just after midnight, Santa was riding his sleigh over the Miracle Sea when suddenly something went wrong and he crashed on a lonely Island.

Although this Island wasn't connected to the outer world, it wasn't completely deserted. It was populated by elves that was both intelligent and rational. But they had one flaw.

Among the elves it was considered a huge shame to have blue eyes. If an elf discovered that she/he had blue eyes, she/he would flee the island in shame the next midnight. You would think that since the elves had lived on the island for generations, there would be no blue eyed elves left, but since there didn't exist any mirrors on the island and the fact that all the elves was so polite, that they would never mention each others eye color, there was actually still some with blue eyes.

Now.. Santa was invited in by the elves and his sleigh was repaired. After three Christmas hours with the elves, which for Santa on Christmas Eve is almost nothing, he was ready to move on. But when he sat up in his sleigh he made the mistake of shouting "Ho! Ho! Ho! It's is so nice to see both blue and brown eyed elves living together. Merry Christmas to all of you!" And then he flew away. The 43'th midnight after Santas visit some of the elves left the island.

How many blue eyed elves was on the island, when Santa crashed?

Merry Christmas to all of you and this exercise is from "Kalkulus" by Tom Lindstrøm, which is a great book :D

This problem is one of my favorite problems also, I personally was not able to solve it the first time I saw it.

There is an idea that is very hard to refute, if there are more than two elves with blue eyes Santa is really not giving any information to anyone.

However lets look at the following idea. Suppose there was exactly one person with blue eyes, that person would flee the island immediately after one day.

Suppose there are two persons with blue eyes, pick one of those persons ,that elf would see only one elf with blue eyes and he would think to himself "poor elf he will leave today". But then the day would pass and that guy would still be there. The only possible explanation for that would be he has blue eyes too, so after two days they will leave.

When there are three elves with blue eyes select one of those guys, he will see only two blue eyed elves and say "Those two elves over there, I feel sorry for them, they will leave in exactly two days" but then the two days will come and he will be utterly surprised, and that night the three of them will leave the island.

Using induction suppose after $n$ days, if there are $n$ blue eyed elves, all the blue eyed elves will leave. Suppose there are $n+1$ elves with blue eyes on the island. Pick one of them, he sees $n$ elves with blue eyes and he will say: "those poor elves, after $n$ days they will leave the island". But then the $n$ days will come, and he will be entirely shocked, and after $n+1$ days the $n+1$ blue eyed chaps will have to leave.

So how could he have found those elves there? The reasoning I give for it is at first there was a person with blue eyes, then another person with blue eyes came and they had no problems with this, because when they where alone they did not know that they where was a person with blue eyes in his vicinity (which only includes himself). Of course when there are two of these blue-eyed elves they obviously know there is an elf with blue eyes on the island, but they only know of one. when Santa says there is one person with blue eyes, if there was only one such person with blue eyes, that person would have to leave that night.

So the key to how a collection of elves could have been formed is that they where added one by one.