A logician met three people, A, B, and C. He knew that one was a knight, one was a liar, and one was a knave, but he did not know which was which. The logician asked A, "What type is B?" A either said that B was a knight, B was a liar, or B was a knave. Then, he asked B, "What type is C?" B either said that C was a knight, C was a liar, or C was a knave. Finally, he asked C, "What type is A?" C either said that A was a knight, A was a liar, or A was a knave. The logician now knew what type each person was.

The next day, the logician met a friend. He told his friend about his conversation with A, B, and C. The friend asked, "What type did A say B was?" The logician told him. The friend was not able to figure out what type any of A, B, and C was.

The day after that, the logician met another friend. He told the second friend the same puzzle he told the first friend. He also said that the first friend asked what type A said B was, but that the first friend could not solve what any of them was. He did not tell the second friend what type A said B was. The second friend asked, "What type did B say C was?" The logician told him. The second friend could not figure out what type either A, B, or C was.

The day after that, the logician met a third friend. He told the third friend the same puzzle he told the other two friends. He talked about the first friend not being able to solve what any of A, B, and C was. However, he did not talk about the second friend. The third friend asked, "What type did C say A was?" The logician told him. The third friend could not figure out what type any of A, B, and C was.

What are A, B, and C, and what did they say?