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 A Barbershop Problem

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TheSleepingDragon

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PostSubject: A Barbershop Problem   6/27/2010, 02:14

Try this one on for size.

In a small village on an island off the coast of Borneo there's a rather small and quaint barbershop. Strangely successful for a remote barbershop (comparatively speaking) an anthropologist recently went to study what this small village's fascination for being clean cut was. Upon returning, he concluded that it was because of the village's very strange rules. However, one rather puzzling conundrum continued to confuse and befuddle him.

"You see, the village had very strict rules concerning the barbershop. First, all men had to be clean shaven at least once a week, every day preferably (the women visibly shun men with even slight amounts of facial hair). But that's not the strange part. The strange part is that the barber shaves only men who live in the village who don't shave themselves, and it's village law that no man in the village shaves himself and that all men who live in the village are shaved by the barber. Nobody who lives in the village is allowed to use barbershops outside the village either; it's either local or exile. Not only that, but the barber must be a resident of the village."

While admittedly strange there seemed to be nothing wrong to the researcher's interviewer. When questioned as to the nature of the puzzle the researcher eyed his interviewer quizzically and responded, "Well who shaves Mr. Teno, the village's barber?"

So I ask you the question:

Who shaves the barber?


Last edited by TheSleepingDragon on 6/27/2010, 02:45; edited 4 times in total (Reason for editing : Clarity and grammatical issues)
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PostSubject: Re: A Barbershop Problem   6/27/2010, 09:46

I've heard this one before and the answer was that the barber was a woman. But since your version adds the barber is "Mr." I would have to assume, that the barber is either A.) too young to have facial hair, or B) doesn't grow any. Otherwise I'm stuck, foregoing any other assumptions or another premise you left out.
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TheSleepingDragon

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PostSubject: Re: A Barbershop Problem   6/27/2010, 14:41

Good catch Socrates!

Now try a slightly different version:

In a small village on an island off the coast of Borneo there's a rather small and quaint barbershop. Strangely successful for a remote barbershop (comparatively speaking) an anthropologist recently went to study what this small village's fascination for being clean cut was. Upon returning, he concluded that it was because of the village's very strange rules. However, one rather puzzling conundrum continued to confuse and befuddle him.

"You see, the village had very strict rules concerning the barbershop. First, all men had to be clean shaven at least once a week, every day preferably (the women visibly shun men with even slight amounts of facial hair). But that's not the strange part. The strange part is that the barber shaves only men who live in the village who don't shave themselves, and it's village law that no man in the village shaves himself and that all men who live in the village are shaved by the barber. Nobody who lives in the village is allowed to use barbershops outside the village either; it's either local or exile. Not only that, but the barber must be a resident of the village. Finally some interesting genetics factor in; specifically that all the men have an active gene that forces the growth of facial hair."

While admittedly strange there seemed to be nothing wrong to the researcher's interviewer. When questioned as to the nature of the puzzle the researcher eyed his interviewer quizzically and responded, "Well who shaves Mr. Teno, the village's barber?"

So I ask you the question:

Who shaves the barber?
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PostSubject: Re: A Barbershop Problem   6/27/2010, 23:31

The barber is a resident now, but wasn't born in the village, excluding him from the forced growth of facial hair. He is still young (and foreign) so he has none yet.
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ChaosInTheCrypt



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PostSubject: Reply   6/28/2010, 00:16

I would assume the village forced the Barber family to move into the village, maybe with an incentive or a threat). So another Barber would cut the babers hair.
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TheSleepingDragon

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PostSubject: Re: A Barbershop Problem   6/28/2010, 00:47

Great answers by all.

Now I'll just be straightforward about the whole problem. Here's what I've been trying to say while being playful about it:

You already know that the barber only shaves men who live in the village who don't shave themselves.
Instead of a village in Borneo let's imagine an ideal village consisting of only men in a universe where that village is the only settlement that exists. All people who exist in that universe live in the village, and therefore are men by definition of the village. All the men grow facial hair, without exception.

Under these conditions, now I ask the question:

Who shaves the barber?
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PostSubject: Re: A Barbershop Problem   6/28/2010, 21:58

A very well trained chimpanzee.
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TheSleepingDragon

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PostSubject: Re: A Barbershop Problem   6/28/2010, 22:53

*Nods* Quite apropos Socrates.

However the appropriate answer would be that such a barber can neither shave nor not shave himself. That is, such a barber cannot exist (even in our ideal, imagined world). Which is what I've been trying to go for this entire, complicated, time.

The puzzle was supposed to illustrate the problem of Russell's Paradox.

First, a little back story.
Back when mathematicians were looking to put mathematics on firm axiomatic footing, it was decided that set theory would be the basis of all mathematics. An interesting choice to say the least, with far reaching consequences.

Set theory deals with sets; and sets were defined as "a collection of objects"
The terms collection and object are left undefined, which technically means we mathematicians don't know what we're talking about but the spirit of what we're talking about is preserved in the English definitions of the words "collection" and "object".

A set B could be defined as follows:
B = {1, 2, 3, F, King, Squirrel, Jujube, 9, Pi}

Now to the meat of Russell's Paradox:

It is first important to note that the definition of sets, as given, does not exclude the existence of the following set:
B = { 1, 2, 3, B }
That is, the set B contains itself. It is a strange concept to be sure, but not against the rules as far as we know (yet at any rate).

Russell's Paradox:

Let S be the set of all sets that do not contain themselves.
Now answer the following question:
Does S contain itself?

If S does not contain itself, then by definition of S, it does contain itself.
If S does contain itself, then by definition of S, it does not contain itself.

We have a major contradiction born from a flaw in the definition of a set, and since the definition of a set was flawed, so too was the entirety of the foundation of mathematics.

Years later, Russell would fix the problem by tacking an additional clause onto the definition of a set:

A set is a collection of objects that does not contain itself.
So far we haven't had problems with it, but remember the moral of the story:

Watch your definitions.
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PostSubject: Re: A Barbershop Problem   7/1/2010, 01:57

This was an excellent thread. It will be locked and made permanent once the posts stop for future members to read.
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Kenny Ruff



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PostSubject: Ok....   7/1/2010, 02:05

So....in any case, in the second version, the barber could have been a tranny...no way around that...


For the third one with the universe that only contains men, I have concluded that he could have had a horrible accident and got third degree burns, thus preventing him from growing hair. As for the whole logic behind this...that is no fun...my points still stand.
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