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.Perhaps you will enjoy hearing "Baby Got Back" while working on this puzzle.

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.)

Thanks are due to Daniel Velleman for inspiring the puzzle, Rob Helpy-Chalk for pointing out an inconsistency since corrected, and Dennis Clark and Supriya Sinhababu for playtesting.

## 7 comments:

The question to ask is, 'Do you like my butt?'

R1 = likes big & tells truth

R2 = likes small & tells lies

R3 = confused about these shiny devices & random

The room presently has two rappers, Ri and Rn. Suppose Ri=R1 & Rn=R2. Then both will give the same answer to the question, 'Do you like my butt?' If it's big, then both will answer, 'Yes,' because R1 will truthfully tell you yes, and R2 will lie and tell you, 'Yes,' also. If it's small, then R1 & R2 will both answer, 'No,' because R1 truthfully doesn't like big butts. R2 lies about liking them.

Now, the problem occurs with R3. The fact that one rapper left the room is a red herring. We actually /want/ them all to be in the room, because it makes it possible to discard the arbitrary answer. Given only two rappers, suppose R3=Ri or R3=Rn. Then R3 will answer identically, in which case it doesn't matter, or differently, which is unhelpful, because then we aren't sure which one is random. But we want to get the answer most efficiently, without possibility of error. So we don't care about two rappers in a room, unless R3 answers identically - in which case we know the correct answer by the above. We /want/ all three to be in a room, because then we look for two identical answers as correct, and drop the isolated answer. If all three answer identically, this doesn't affect the puzzle at all; we still have our butt sized correctly.

Solution: wait until the third rapper comes back. Ask: 'Do you like my butt?' R1&R2 will answer the same, so we get two 'Yes' responses and one random response; in this case, our butt is big. Otherwise, we get two 'No' responses and one random response; in this case, our butt is small. This covers the only two possibilities by efficiently asking one question.

Indeed, the question to ask is, 'Do you like my butt?' However, you don't need to wait for the third person to come back.

The third rapper doesn't need to know the size of your butt to judge it. To "like" a butt is a preference question, not a fact question.

Q: "Do you like my butt ?"

Suppose you butt is big. With combinatorics, we may deduce these 3 possible arrangements :

a) R1 answers yes ; R2 answers yes

b) R1 answers yes ; R3 answers yes

c) R2 answers yes ; R3 answers yes

Suppose you butt is small. With combinatorics, we may deduce these 3 possible arrangements :

d) R1 answers no ; R2 answers no

e) R1 answers no ; R3 answers yes

f) R2 answers no ; R3 answers yes

We know know how to decide whether your butt is big or small. If all participants answers yes (a, b or c), then your butt is big. If some participants disagree (e or f), then your butt is small. If all participants dislike yout butt (d), then your butt is small. However, you may not truly know whether rappers like or dislike your butt.

Question 1: Is the sky blue?

The big-butt lover will say Yes. The small-butt lover will say No. The guy who can't assess the size of butts will still know the answer to that question and say Yes.

If you get one no, you've correctly identified the liar and can follow-up by asking him if he likes your butt. Yes = big butt, No = small butt.

If you get two yes's it's a bit trickier. You know the liar is out the picture, but you can't identify the remaining two.

In that case, my second question would be "Do you like small butts?"

Our confused rapper will say yes, since he likes all butts. The other will say no.

Now you can ask our big butt lover if he likes yours, and you'll be set.

So that's either two or three questions total depending on the answer you get from question 1.

Monair has it!

Hard to say without knowing about the truthtelling status of the rapper who can't assess butt sizes, isn't it?

Yep, Monair is wrong. In his combinations there are more than 3 combinations as in his case it looks like R3 always says "yes" to all questions where he should be random.

Mohair is right. R3 gives random answers only if question regards butt size, and he gives right answer to all the other questions (I guess so, and I think that his truthtelling status should be clarified in puzzle conditions), including "do you like my butt?" question to which he must answer "yes" since he likes all butts.

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