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.]

15 comments:

Rob Helpy-Chalk said...

Ok, here's my try:

If you ask one of the remaining rappers “does the other rapper like my butt.” If the answer is “yes” infer that your butt is small. If it is “no,” infer that your butt is big. You have a 2/3rds chance of being correct

Lets label the rappers (1), (2) and (3). Rapper (1) likes big butts and cannot lie, etc.

If you ask “Does the other rapper like my butt” there are twelve possibilities: rapper (1) answers about (2), rapper (2) answers about (1) etc. Also, your butt can be either big or small.

1. Rapper (1) asking about rapper (2)’s view of your big butt: Rapper (2) would not like your butt. He would lie, but it doesn’t matter. Rapper (1) tells the truth, and says “No.” You infer that your butt is big. CORRECT.

2. Rapper (1) asking about rapper (2)’s view of your small butt. Rapper (2) likes it, rapper (1) says yes. You infer that your butt is small. CORRECT.

3. Rapper (2) asking about rapper (1)’s view of your big butt. Rapper 1 likes it, but rapper 2 lies, and says he doesn’t. You infer that your butt is big. CORRECT.

4. Rapper (2) asking about rapper (1)’s view of your small but. Rapper (1) hates the scrawny thing. Rapper (2) lies. You infer that your butt is small. CORRECT.

5. Rapper (2) asking about rapper (3)’s view of your big butt. Rapper (3) likes it, because he likes all butts. Rapper (2) lies and says he doesn’t like it. You infer that your butt is big. CORRECT.

6. Rapper (2) asking about rapper (3)’s view of your small butt. Rapper (3) likes it, because he likes all butts. Rapper (2) lies and says he doesn’t like it. You infer that your butt is big. INCORRECT.

7. Rapper (3) asking about rapper (2)’s view of your big butt. Rapper (3) guesses the size of your butt randomly. Filtering this through (2)’s opinion doesn’t change the random nature. ½ CORRECT, ½ INCORRECT.

8. Rapper (3) asking about rapper (2)’s view of your small butt. Like (7). ½ CORRECT, ½ INCORRECT.

9. Rapper (3) asking about rapper (1)’s view of your big butt. Like (7). ½ CORRECT, ½ INCORRECT.

10. Rapper (3) asking about rapper (1)’s view of your small butt. Like (7). ½ CORRECT, ½ INCORRECT.

11. Rapper (1) asking about rapper (3)’s view of your big butt. (3) likes it, because he likes all butts. (1) tells the truth. You infer your butt is small. INCORRECT.

12. Rapper (1) asking about rapper (3)’s view of your small butt. (3) likes it, because he likes all butts. (1) tells the truth. You infer your butt is small. CORRECT.

Brock said...

Unless I'm mistaken, the simple question "Do you like my butt?" will give you the answer, because you'll either get two "yes" answers if your butt is big, or at least one "no" if your butt is small.

However, that may count as more than one question because of the differing referent of "you" for each rapper.

Is the identity relation for questions based on content or character, to use Kaplan's distinction?

Rob Helpy-Chalk said...

Wait, you don't even have to address all the rappers with a general "you." If you just pick one and say "Do you like my butt" and infer that your butt is big if they say "yes" you still have a 2/3rds chance of being correct.

It can't be that easy.

Also, having one rapper leave is red herring, right. It makes it look like you are dealing with a Monty Hall situation, but really the odds work the same way as if you were asking one of three present rappers.

yoyo said...

My answer was the same as rob is close, but it should be phrased "if I asked the other guy, would he say he likes my booty"?

Yoyo said...

Actually, why not ask "how did the other easier come to his conclusion about my booty" might work too, since while you are limited to one question you are nt limited to receiving a single statement in response

Rob Helpy-Chalk said...

On FB, people are pointing out that paragraph 1 says that all three rappers can correctly assess butt size, but paragraph 2 says the third rapper can't. But that is just a minor mistake, in the wording of the first paragraph right?

JR said...

Yoyo: It says the question must be yes-no so asking how the rappers reached their conclusion doesn't work.

BruceMcF said...

Courtesy of the Fast Show, we know that the correct question is, "Does my bum look big in this?"

Anonymous said...

See: Baby Got Back (folks version):

http://www.youtube.com/watch?v=Ltjbnyvq_SI

Neil Sinhababu said...

Thanks for pointing out the inconsistency, Rob. I've modified the first paragraph accordingly.

Brock, "Do you like my butt?" is the intended solution! Congratulations.

low-tech cyclist said...

But "Do you like my butt?" is an incomplete solution, as Rob points out.

If your butt is big, Rappers 1 and 2 will both respond 'Yes', while Rapper 3 will answer at random.

If your butt is small, Rappers 1 and 2 will both respond 'No', while Rapper 3 will answer at random.

If no rappers left, then at least 2 yesses would mean your butt's big; at least 2 no's would mean your butt's small.

The problem is that in the absence of either Rapper 1 or Rapper 2, both big and small butts can produce Y/N answer pairs, depending on what Rapper 3 says at random, so you can't determine your butt size from such a pair. (Contra Rob, the departure of one of the rappers is what makes the problem hard.)

As Rob points out, you've got a 2/3 chance of getting a Y/Y or N/N answer pair, in which case you know your butt size, but the other 1/3 are Y/N pairs, and you can't determine your butt size.

AJD said...

So the issue is, (3) doesn't answer *every* question randomly. Only questions *about butt size*. The unstated stipulation, I guess, is that (3) gives true answers to questions that aren't about butt size (such as, e.g., "Do you like my butt?").

Neil Sinhababu said...

AJD has this right. 3 is the guy who's like, "hooray butts!" but doesn't really care or think about what size they are.

I thought this was intuitive, but given the confusion surrounding it maybe I should've just made him respond yes to every question. That was actually the original formulation, but I changed it because it permitted too many alternative solutions.

low-tech cyclist said...

I noticed that it wasn't clear how Rapper 3 answered other questions, but in the absence of any other info about how Rapper 3 answered questions, I just took the question of whether one's butt was big or small to be a for-instance of how he would answer questions in general.

Now that that's cleared up, it's easy.

Connor said...

The problem is you have no definitive point of reference. So to get one you measure your booty then show the data and let the rappers look at your booty. Then ask them to estimate their previous ten or so booty call bootys to your measurements. If one rappers data fluctuates substantially you can assume he is the 3rd rapper and ignore his data for now. The remaining rappers data should be rather consistant. Compare his data to the 3rds highest and lowest to find where you standf. If the 3rd rapper is the one who left just compare the average data of the 1st and 2nd with yours to see how you measure up.