[ Mathematical logic (miscellaneous).]

Author: Tokarev S.i.

On an island, there live knights who always tell the truth, and liars who always lie. A traveler met three islanders and asked each of them: "How many knights are among your companions?". The first one answered: "None". The second one said: "One". What did the third one say?
If the first is a knight, then due to his words, the second and third are liars, which is impossible because of the statement of the second islander. Therefore, the first is a liar. If the second is a liar, then due to his words, the third is also a liar, but then the first would have told the truth, which he should not have. Therefore, the second is a knight. Due to his words, the third is also a knight. The third will honestly answer: "One".

## Answer

The third said "One".

We call a natural number "amazing" if it has the form $\mathrm{a}^{\mathrm{b}}+\mathrm{b}^{\mathrm{a}}$ (where a and $\mathrm{b}$ are natural numbers).

For example, the number 57 is amazing because $57=2^{5}+5^{2}$. Is the number 2006 amazing?

## Solution

Yes, it is: $2006=2005^{1}+1^{2005}$.

## Answer

Yes, it is.