Interesting Theorems Generated from Puzzles

Aunt Agatha

Someone who lives in Dreadbury Mansion killed Aunt Agatha. Agatha, the butler, and Charles live in Dreadbury Mansion, and are the only people who live therein. A killer always hates his victim, and is never richer than his victim. Charles hates no one that Aunt Agatha hates. Agatha hates everyone except the butler. The butler hates everyone not richer than Aunt Agatha. The butler hates everyone Aunt Agatha hates. No one hates everyone. Agatha is not the butler.

Truthtellers and Liars

On a certain island the inhabitants are partitioned into those who always tell the truth and those who always lie. I landed on the island and met three inhabitants A, B, and C. I asked A, 'Are you a truthteller or a liar?' He mumbled something which I couldn't make out. I asked B what A had said. B replied, 'A said he was a liar'. C then volunteered, 'Don't believe B, he's lying!'