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Posted by Chris on September 21, 2011 – 5:23 pm

Three intelligent women, Alice, Barbara and Carol, sit down to try out a test in logical reasoning. They are arranged so that each can see each others hat, which is either red or green, but none of them can see her own hat. They are told that at least one of the hats is red. If any one of them can logically deduce the colour of her hat, she is to declare it immediately. Carol alone decides to play this game with her eyes closed. After a little time Carol, correctly declares her hat colour. What is Carol’s hat colour (and why)?

This post is under “Logic, Tom” and has 12 respond so far.

### 12 Responds so far- Add one»

1. 1. Eketahuna Said：

Carol’s hat it Red.

If Carol’s hat was green and Barbara’s hat was green, then Alice would know her hat was red and call it.
If Carol’s hat was green and Alice’s hat was green, then Barbara would know her hat was red and call it.
If Carol’s hat was green and Alice and Barbara both had red hats, then both Alice and Barabara would see a red and a green and know that the other must also see a red and green, otherwise they would see two green and know they had a red and call it, so they must have a red themselves and call it, which didn’t happen.
Therefore Carol’s hat must be red and the other’s must be green, so they both see a red and green, knowing the red is Carol’s they must have either a green or red as there is a MINIMUM of 1 red.

2. 2. Wizard of Oz Said：

Agree that Carol is red, but other two could also be red.

3. 3. JB Said：

All three hats are red! Assuming they all know that the other two women are also intelligent, Carol would know that if there were only 1 red hat, the one wearing it would know instantly, seeing only green hats. Therefore, there must be at least 2 red hats since nobody spoke up immediately. And if she was wearing a green hat, the other two would also know that there should be at least two red hats, but would only be seeing 1 and thus know that they are wearing the second red hat. Since nobody spoke up fast, each must be looking at two red hats, their thought processes taking a little longer to arrive at the proper conclusion that they must all be wearing red.

4. 4. rak Said：

all hats are red

5. 5. Chris Said：

My brain is quite incapable of following Eketahuna’s argument – I simply can’t hold the whole argument chain in my head in one lump. I didn’t even learn the 12 times table until I was about 15 years old, and except for a few special cases, I never have been able to factorise quadratic equations in my head. Don’t misunderstand me, I envy people who can do those mental
gymnastics.

The last time I posted this (on the old ToM site), I gave a different solution to the one that I now give. I think it is very straighforward, and doesn’t require a brain the size of a planet to follow (given that it’s been written down, so you can review every step as often as you please)

Writing the hat colours in the sequence Alice, Barbara, Carol (A,B and C from now). Initially the possible arrangements are (as GGG is forbidden):
GGR(1), GRG(2), GRR(3), RGG(4), RGR(5), RRG(6), RRR(7).
I’ll use g or r to show that the observer cannot actually see the colour.
If A saw GG, then we must have rGG (4), then A would immediately say R.
If B saw GG, then we must have GrG (2), then B would immediately say R.
Neither A or B called, so all three can eliminate RGG(4) and GRG(2) leaving:
GGR(1), GRR(3), RGR(5), RRG(6), RRR(7).

A cannot immediately distinguish gGR(1) from rGR(5), nor gRR(3) from rRR(7).
rRG(6) is unique for A, but A didn’t say R.
B cannot immediately distinguish GgR(1) from GrR(3), nor RgR(5) from RrR(7).
RrG(6) is unique for B, but B didn’t say R.

Everyone will be able to work that out, and eliminate RRG(6) and so realise that the only remaining possibilities are: GGR(1), GRR(3), RGR(5), RRR(7). At this point, whilst A and B are deep in thought, C could say “my hat is red”.

Case GGR(1): A thinks gGR(1) or rGR(5). B thinks GgR(1) or GrR(3).
Case GRR(3): A thinks gRR(3) or rRR(7). B thinks GgR(1) or GrR(3).
Case RGR(5): A thinks gGR(1) or rGR(5). B thinks RgR(5) or RrR(7).
Case RRR(7): A thinks gRR(3) or rgRR(7). B thinks RgR(5) or RrR(7).

I can see no way for A or B proceed further. Whatever “what-if” I choose, I don’t seem to end up with a definite outcome for A or B.

So, I conclude that A and B could be wearing any colour hat. or that I’m simply too lazy to make sure that A or B really are stuck.

6. 6. hamsterofdeath Said：

i forgot the women’s names, so i call them a b and c.
c has her eyes closed.

i’ll solve this one with a table:
woman -> a b c -> result below
color -> g g g -> universe crashes
color -> r g g -> a would announce “red”
color -> g r g -> b would announce “red”
color -> r r g -> both a and b wait for each other’s announcement. since both don’t say anything, both know that the other sees one red and one green and announce “red” at the same time, assuming the same brain speed.

so in each case where c has a g hat, a and b announce their colors. so i don’t even have to think about the other cases – in all of them, c = red.

7. 7. Chris Said：

Hi hamsterofdeath. You’d muddled up the ladies. I’ve fixed it to use the same names as in the posted problem.

Your answer is like the one I posted on the old ToM site, so it must be right

You’ve also implied that there is no way to say (for the posed problem) what colour hats A and B were wearing

Hi JB. Your second sentence isn’t true if Carol is wearing the only red hat. So you can’t conclude that there are (at least) two red hats.

Hi rak. Can you prove your assertion?

Hi Wiz. I’m sure you were there

Hi Me. You should have stayed with the method that hamsterofdeath used. But you made a good start.

8. 8. JB Said：

Chris,

I realized immediately after I posted that my answer was not correct as long as Carol’s eyes were closed. If they had been open and nobody immediately knew the color of their hat, then my answer would have been correct since, if there was only 1 red hat, its wearer would know instantly after seeing two blue hats.

9. 9. Eketahuna Said：

Hi hamsterofdeath (and anyone else saying they must all be red) , you forgot g g r

a and b would both see green and red and not know theirs.

from each point of view they would see g and r and think, ‘My hat could be green or red, as there is AT LEAST one red, and I can see the one red and so can a/b, therefore neither of us can make a deduction …’

10. 10. Eketahuna Said：

Oops, I think I stuffed that explanation up (too early in the morning)

I agree with Wiz that the other two can be either is what I’m trying to say, which is how I concluded, it’s just the 1st part of my last sentence in post one was where I went wrong in my typing

11. 11. Urban Cowboy Said：

Either Alice or Barbra sees that the other two hats are green and declares her hat is red, as one hat must be red. Based on this knowledge, Carole figures out that her hat is green because otherwise why would one of the girls say her hat is red.

12. 12. Chris Said：

Hi Urban Cowboy. Neither A nor B said anything.

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