Four mathematicians have the following conversation:
Alice: I am insane.
Bob: I am pure.
Charlie: I am applied.
Dorothy: I am sane.
Alice: Charlie is pure.
Bob: Dorothy is insane.
Charlie: Bob is applied.
Dorothy: Charlie is sane.
You are also given that:
● Pure mathematicians tell the truth about their beliefs.
● Applied mathematicians lie about their beliefs.
● Sane mathematicians beliefs are correct.
● Insane mathematicians beliefs are incorrect.
Describe the four mathematicians.

Suppose that Dorothy is telling the truth. Then she is sane (and hence a pure mathematician) and Charlie is sane. Bob says that Charlie is insane, so Bob is not telling the truth. Since he says he’s pure, he’s actually applied and hence Charlie is telling the truth. But then he’s applied and sane applied mathematicians do not tell the truth. This is a contradiction.

Therefore, Dorothy is not telling the truth. Thus, both Charlie and Dorothy are insane (hence Dorothy is pure). Consequently, Bob is telling the truth. Therefore Bob is pure (and sane, since he is a truth teller). Since Charlie says that Bob is applied, he is not telling the truth. Since we know that he is insane, he must be pure.

Since Alice says that Charlie is pure, she is telling the truth. Therefore she is insane and consequently applied.

In summary:

Alice is an insane applied mathematician.

Bob is a sane pure mathematician.

Charlie is an insane pure mathematician.

Dorothy is an insane pure mathematician.