# Numbered Logicians near the Blue Eyes island

There is an island near the famous Blue Eyes island where 100 perfect logicians live, each with a number between 1 and 100 (repetitions allowed) tattooed on their forehead. They know all the numbers of the other inhabitants except their own. If ever one of them learns their own number, tradition dictates that they krill themselves. That is, the next daily all-island meeting, at precisely 12:00 noon, they must dump a bucket of raw krill on their heads.

It so happens that half the inhabitants have the number $50$, while the other half have the number $51$. One day, a travelling knight (truth-teller) is addressing all the inhabitants, and mentions:

I was surprised to find only two distinct numbers among the people here, and that those two numbers were consecutive.

Immediately, all the inhabitants deduce that they have either a $50$ or a $51$ on their forehead. But that is not narrowed down enough to commit "sushi-cide". Do any of the inhabitants end up krilling themselves, and if so when?

Yes.

One important difference between this problem and the blue-eyes problem is that here, even if there was one 50 and ninety-nine 51s, the person with the 50 would not know their own number (it could also be 52). This means that the crucial chain of deductions cannot start from there.

However, there is a place to start this chain. If there was one 2 and ninety-nine 1s, then the person with the 2 would know their own number. So, if nothing happens after one day, it becomes common knowledge that this is not the distribution.

If there were two 2s and ninety-eight 1s, then both 2s would know their numbers after a day passed, and they would krill themselves on the second day. So if two days pass with no krillings, then it becomes common knowledge that this is not the distribution of numbers either.

After 99 days pass, it becomes common knowledge that the numbers are not 1s and 2s. Then, if there was one 3 and ninety-nine 2s, the person with the 3 would know that they didn't have a 1.

After 198 days, it is common knowledge that the numbers are not 2s and 3s. After the 4851st (99*49) days 49s are ruled out. A similar chain of deductions starting at 100 has also ruled out 52 and higher numbers, so it is common knowledge that the numbers are 50 and 51.

Finally, it takes 49 more days to eliminate all the possibilities except fifty 50s and fifty 51s. On the 4901st day, all of the islanders krill themselves.

• But don't all of the islanders already know that the numbers are 50s and 51s on the first day? – Ben Frankel Sep 20 '15 at 4:01
• They do, but it is not common knowledge. Common knowledge requires that everyone knows a piece of information, and everyone knows that everyone knows that information, and everyone knows that everyone knows that everyone knows that information, and so on. – f'' Sep 20 '15 at 4:27
• In the blue eyes problem, all islanders already know that there are 99, 100, or 101 with blue eyes, but they still have to go through the cases with a lower number of blue eyes before they can deduce their own eye colors. – f'' Sep 20 '15 at 4:29
• Since everybody can see more than one each of 50 and 51 it is common knowledge (after the visitor speaks) that all numbers are 50 and 51. – Ross Millikan Sep 20 '15 at 5:22
• @RossMillikan No, it's not. I know that you know that I know that the numbers are 50 and 51. However I don't know that you know that I know that you know that I know - with the "you know that I know" repeated 25 times - that the numbers are 50 and 51. – Taemyr Sep 23 '15 at 7:46