This month's problem is a logic puzzle. Four mathematicians make the following statements:

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."

Describe the four mathematicians, 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.

Source: Mathematics and Informatics Quarterly (via Tom Halverson)