There is a story about a math dept of 20 professors who all seem to know about the infidelities of other wives, but if any one discovered that his own wife was unfaithful, he would shoot her the next morning.
One day a visiting professor happens to mention to everyone that one (or more) of the wives was unfaithful. This might seem inconsequential, because they all already knew that. Or it seems that way. But these were mathematicians, and 20 days later, they all shot their wives.
You can find the explanation at The blue-eyed islanders puzzle, Common knowledge (logic), or Muddy Children Puzzle.
A mathematician is apt to find the argument convincing, but no one else.
I am reminded of this as I read puzzle about 10 pirates dividing 100 coins. A solution is given, and proved correct, except that you cannot imagine any pirates using it. Most of them would die, if they tried.
According to the proof, the first pirate will take 96 coins for himself. The pirate can vote to reject this offer, throw the first pirate overboard, and let the second pirate divide the coins among the remaining 9 pirates. But a clever induction argument claims to prove that the pirates will vote to accept the division of coins!
In the comments at the above link, I argue that the induction argument is flawed, and that pirates will vote to reject such lousy deals. Everyone else disagreed with me. Read for yourself, and form your own opinion.