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[ed boy]

Alice, Bob and Charlie are well-known expert logicians.
(i) The King places a hat on each of their heads. Each of the logicians can see the others’ hats, but not his or her own.
The King says “Each of your hats is either black or white, but you don’t all have the same colour hat”.
All four are honest, and all trust one another.
The King now asks Alice “Do you know what colour your hat is?”.
Alice says “Yes, it’s white”.
What colour are the others’ hats?

(ii) The King now changes some of the hats, and again says “Each of your hats is either
black or white, but you don’t all have the same colour hat”. He now asks Alice “Do you
know what colour your hat is?”.
Alice replies “No”
Can Bob work out what colour his hat is?

(iii) The King now changes some of the hats, and then says “Each of your hats is either
black or white. At least one of you has a white hat.”
He now asks them all “Do you know what colour your hat is?”. They all simultaneously
reply “No”.
What can you deduce about the colour of their hats?

(iv) He again asks “Do you know what colour your hat is?” Alice says “No”, but Bob
and Charlie both say “Yes” (all three answer simultaneously).
What colour are their hats?

[Darvi]

i: Black.
ii: Yes. It's the opposite of Charlie's
iii: I think that setup is impossible
iv: See iii

[Max White]

Alice, Bob and Charlie are well-known expert logicians.
(i) The King places a hat on each of their heads. Each of the logicians can see the others’ hats, but not his or her own.
The King says “Each of your hats is either black or white, but you don’t all have the same colour hat”.
All four are honest, and all trust one another.
The King now asks Alice “Do you know what colour your hat is?”.
Alice says “Yes, it’s white”.
What colour are the others’ hats?

(ii) The King now changes some of the hats, and again says “Each of your hats is either
black or white, but you don’t all have the same colour hat”. He now asks Alice “Do you
know what colour your hat is?”.
Alice replies “No”
Can Bob work out what colour his hat is?

(iii) The King now changes some of the hats, and then says “Each of your hats is either
black or white. At least one of you has a white hat.”
He now asks them all “Do you know what colour your hat is?”. They all simultaneously
reply “No”.
What can you deduce about the colour of their hats?

(iv) He again asks “Do you know what colour your hat is?” Alice says “No”, but Bob
and Charlie both say “Yes” (all three answer simultaneously).
What colour are their hats?

Well i)
The only way Alice can be sure she has white is because the kind said "but you don’t all have the same colour hat", and because Alice said she has white, the only way she could have known this is by the other two both having black, forcing her into white. I like the colour white.

Now ii)
Yes he can! Because Alice must see a black and a white, and the same rules are enforced, he must have the opposite colour to Alice.

Onto iii)
They are at least two white.

And finaly iv)
Alice has black.



What I'm left wondering is, if they all trust each other, and they can see each others hats, could they not just tell each other? As nice as logical observation is, in this example, because there are three of them, any one could have the other two agree on a colour, so the peer review process seems a preferable solution.

[Darvi]

At  iii then the guy with the black hat would know that he has a black hat.

Oh wait. He didn't state that the hats have to be a different color :V

[Max White]

Yea, new rule set, there could be two or three white. Although the next question proves there is a black.

[Darvi]

Also, you got the second answer slightly wrong. He has the opposite color to Charlie. Assuming that Alice doesn't have the same color as Charlie that is.

[Max White]

Nope, Alice sees a black and a white, that is why she isn't sure. However, what ever Alice has, Charlie can see two of those, Alices and Bob's. Because Alice and Bob have taken those colours, he is forced into the opposite.

Wait, unless he has the same as Alice. In that case Charlia would not be able to tell. So somebody can tell, either Bob or Charlie, but we don't know who! Crap! I got that one wrong!!!!

EDIT: But wait, he can see Charlie, and would know he is the opposite to Charlie. FUUUUUUUUU! You beat me there Darvi.

[Darvi]

We know that charlie has a different color than Bob. That's why Alice isn't sure. But bob knows that, so he and Charlie can deduce their color from that.

E: heheheh.

[Max White]

Damn you logic! I fail...

Still, let me ask the next question. A scientist wishes to test if humans have a sense of magnetic navigation. To do this, he wants to put a person in a perfectly cubic room, in a spinning chair, have them spun until dizzy, then try to guess what way north is. However, for this test to work, all four walls must be identical so that the subject can't identify north from memory.

The problem here is that one of the walls has a door in it. So, what is the best way to make the door indistinguishable?

[Darvi]

Put a door in every wall?

[Max White]

You would be surprised how few people get that one. 

[ed boy]

What I'm left wondering is, if they all trust each other, and they can see each others hats, could they not just tell each other? As nice as logical observation is, in this example, because there are three of them, any one could have the other two agree on a colour, so the peer review process seems a preferable solution.

Because it is unimaginably rude to speak in the presence of the king without him adressing you first. The rules of etiquette forbid it.

The Millennium school has 1000 students and 1000 student lockers. The lockers are in a line in a long corridor and are numbered from 1 to 1000.
Initially all the lockers are closed (but unlocked).
The first student walks along the corridor and opens every locker.
The second student then walks along the corridor and closes every second locker, i.e. closes lockers 2, 4, 6, etc. At that point there are 500 lockers that are open and 500 that are closed.
The third student then walks along the corridor, changing the state of every third locker.
Thus s/he closes locker 3 (which had been left open by the first student), opens locker 6 (closed by the second student), closes locker 9, etc.
All the remaining students now walk by in order, with the nth student changing the state of every nth locker, and this continues until all 1000 students have walked along the corridor.
(i) How many lockers are closed immediately after the third student has walked along the corridor?
(ii) How many lockers are closed immediately after the fourth student has walked along the corridor?
(iii) After the hundredth student has walked along the corridor, what is the state of locker 1000?
(iv) At the end (after all 1000 students have passed), which lockers are open and which are closed?

[Darvi]

i: Half of 'em
ii: The ones   divisible by 12 are open. The ones divisible by 6 (but not by 12) are   closed. The ones divisible by 2 (but not by 6) are open. Odd numbers are   open, always. so 1/2-1/6+1/12= (6-2+1)/12 = 5/12 are closed.
iii: Open.
iv: uh... at least half of them are closed.                 

[Urist Imiknorris]

For part iv, the perfect squares (1, 4, 9, and so on) are all open, and the rest closed, so that would be 31.

[Darvi]

Damn, I was just about to google how many there are.