Problem 3.3. Several letters A and several letters B were sitting on a pipe. After some A fell and some B disappeared, only three letters remained on the pipe, and the following dialogue occurred between them:

First letter: “I am the only one of my kind here.”

Second letter: “There are definitely fewer than two A letters here.”

Third letter: “There is only one B letter among us.”

It turned out that each letter told the truth if it was A, and lied if it was B. Determine which letter is which.
Answer: B, A, B.

Solution. Suppose the third letter is A. Then she told the truth, and the letter B is only one. If the first letter is A, then she lied, which is impossible. If the first letter is B, then she told the truth, which is also impossible.

Therefore, the third letter is B, and she lied. This means that at least one of the first two letters is also B. If the second letter is B, then she told the truth, which is impossible. Therefore, the second letter is A, and the first letter is B. It is easy to see that the order of the letters B, A, B satisfies all the conditions of the problem.