Saturday, February 5, 2011

A Puzzle About Your Butt

Nineteen years ago yesterday, Sir Mix-a-Lot's "Baby Got Back" was released. In commemoration of this significant cultural event, I've put together a little puzzle.
You are curious whether your butt is big or small. Unfortunately, you lack the ability to accurately assess the size of butts. Fortunately, there are three rappers before you. You are of their preferred gender, so they are willing to collectively entertain exactly one yes-or-no question from you, to which they will each give an answer.

One rapper likes big butts and cannot lie. One rapper likes small butts and always lies. One rapper likes all butts but shares your inability to assess butt size, and will answer yes or no at random if asked whether a butt is big or small. You do not know which rapper is which. All the rappers know all other facts relevant to the situation, including everyone's identity and butt preferences.

Before you are able to ask your question, one rapper receives a booty call (the size of the booty is unknown to you) and leaves the room. The other two rappers remain and are willing to pronounce on your question. You still do not know who any of the rappers are.

To determine the size of your butt, what question should you ask them? (You may assume that all butts can be classified as either big or small and ignore contextual factors, e.g. from the presence of Oakland booty.)
Perhaps you will enjoy hearing "Baby Got Back" while working on this puzzle.

Thanks are due to Daniel Velleman for inspiring the puzzle, and Dennis Clark and Supriya Sinhababu for playtesting. [Edited: and Rob Helpy-Chalk for pointing out an inconsistency in the puzzle that is now fixed.]
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