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100 Logician Puzzle Answer

Answer: All 5 logicians found out that they had a black dot on their back on the 5th day and they all committed suicide on that day!
To understand the answer, suppose you're one of the logicians living on the island.


One person with a black dot.

This is the simple case. If you don't see any dot, then it implies you have the dot. You will commit suicide that day.


Two persons with a black dot.

Suppose you have a black dot on you. As a result you only see one person with a dot. You wait and watch his actions--ie does he commit suicide that day?. If that was the only person with a dot then he will commit suicide that day (see Case 1). If he doesn't that means he is seeing a dot on someone and since you don't see any other dot he must be seeing a dot on you.

Since the other guy is also thinking the same way as you're, he is also checking to see if you would commit suicide on the first day or not. Since you don't, next day he too
determines that he has a black dot and both you commit suicide on the second day.


Three persons with a black dot.

If you have the black dot then you would see only two persons with a black dot. If they are they are the only ones who have the black dots, then we've proven with Case 2 that they kill themselves on the 2nd day. Since they don't, it implies that you have a dot. So on the third day all three of you commit suicide.


Four persons with a black dot.

If you had the black dot, then you would see only 3 dots. You wait and see if the three people with the black dot commit suicide on the third day or not. If they don't then you plus the other three commit suicide on the fourth day.


If there are five persons with a dot, they all commit suicide on the fifth day.


If there are n logicians with a dot, then they all commit suicide on the nth day.



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