Imagine taking your shower in a public bath house. Two blue towels are hanging on hooks next to each other. You can't remember which is yours.

You ask the other shower taker, "Who's towel is whose? Who's towel is who's? Whose towel is whose? Whose towel is who's?"

Does 'whom' apply here at all?

Naturally, I would ask "Which towel is your's and which is mine?" But that wouldn't be as much fun, would it?