a logic problem (Hat Problem)

**sogart** · 05-19-2007 10:01 AM

If this is not for the philosophy area please feel free to move elsewhere...

this is an interesting problem I stumbled upon today. Maybe some of you have heard about it. Obviously the answer is all over the internet, but just try to figure it out for yourself. Its a completely logical answer, no cheating or strange things, just simple probability theory. enjoy (tell me what you think)

from the internets:

If you think that there's no new mathematics under the sun, consider the recently discovered "Hat Problem":

A team of three people are put into a room together. A hat, either red or blue, is placed on each. You can't see the color of your own hat, only the colors of the other two people's hats. No communication is allowed among teammates once they get their hats. After seeing the other hats, the three of you are separated and each is asked "What is the color of your hat?" You can answer "red" or "blue", or you can abstain from replying. If at least one person on the team guesses the right answer and nobody guesses wrong, then your team wins.

The challenge for the team: agree, before you go into the room, on a strategy that will maximize your chance of success.
How in the world can you possibly do better than 50%? None of you knows anything about your own hat, and you can't communicate. At best, it seems, you could conspire in advance to let one team member guess (a 50-50 gamble) and the other two abstain. If more than one of you guesses, the odds only get worse.
Or so it seems. But there is a way to do better --- in fact, to increase your chance of victory to 75%! Think about it for a while if you would like to figure it out for yourself. It's a clever, honest, totally counterintuitive method. It's extensible to more people and more complex problems. And the winning strategy is intimately related to ideas from information and coding theory.

**Patrick** · 05-19-2007 06:13 PM

EDIT I looked up the actual answer after writing this, then realised I'd actually guessed it right

, so I've blanked out my answer for people who want to guess themselves.

That's a really good problem... never heard it before...

(HIGHLIGHT to see answer)
I would probably say that each person would automatically abstain from voting, unless the two hats they saw were the same colour. There is no chance of there not being at least one person who sees two hats of the same colour, so at least one person will be guessing. The person/people who see two similar colours will vote that his hat is the opposite colour, since it isn't too likely that all three hats were the same colour.

That's what I would do, anyway.

**sogart** · 05-19-2007 06:26 PM

cool, I am unlucky because I read it all at once and I always like a good logic problem. Anyways... good for you.

PS: I lived in Aberdeen for 4 yrs (Uni). is it like 20 hours of day now over there?

**Patrick** · 05-19-2007 06:34 PM

Originally Posted by sogart

PS: I lived in Aberdeen for 4 yrs (Uni). is it like 20 hours of day now over there?

I'll PM you

**Patrick** · 05-20-2007 07:02 PM

Is anyone else gonna try this question? I feel kinda guilty for killing the thread...

**sourcejedi** · 05-20-2007 07:46 PM

Originally Posted by Pensive Patrick

Is anyone else gonna try this question?

I've seen the question and the solution twice, and it annoyed me because I got it wrong and then couldn't work out why I was wrong :-).

[SPOILER, this doesn't make sense unless you know the solution]
Another counterintuitive fact about the winning strategy is that if you take a given person, they guess wrongly *more* often than they guess right. But the group wins more overall - and thats because when one person guesses wrong, everyone else does in the same game, but when they win the game its because exactly one person has guessed, and they're right.

Pat: don't worry about the death of threads. It just happens.

**Umbrella** · 05-20-2007 08:05 PM

I know that problem. The solution is insanely difficult. Figuring it out by yourself is really quite something, congrats.

**Neruo** · 05-20-2007 10:42 PM

I shouldn't have read the answer so quickly. Really cool ´riddle´.

Makes sense.

**sogart** · 05-21-2007 12:15 PM

Originally Posted by sourcejedi

I've seen the question and the solution twice, and it annoyed me because I got it wrong and then couldn't work out why I was wrong :-).

[SPOILER, this doesn't make sense unless you know the solution]
Another counterintuitive fact about the winning strategy is that if you take a given person, they guess wrongly *more* often than they guess right. But the group wins more overall - and thats because when one person guesses wrong, everyone else does in the same game, but when they win the game its because exactly one person has guessed, and they're right.

Pat: don't worry about the death of threads. It just happens.

I think you are talking about a similar but different problem. In this one you can't win if everyone answers wrong, and the strategy doesn't have to work 100%... you just need to win 3/4 of the times.

**Patrick** · 05-21-2007 04:25 PM

Originally Posted by Umbrella

I know that problem. The solution is insanely difficult. Figuring it out by yourself is really quite something, congrats.

Thanks. It was a very rare flash of genius.

**blade5x** · 05-21-2007 11:26 PM

Pensive if that's the answer then I guess I got it right - and I thought my logic was stupid to

I pictures red/blue in my mind, then I pictured blue/blue... and OH... hehe (maybe this will confuse people some more)

**The Cusp** · 05-22-2007 11:56 PM

Well in that case, I'll remove my answer so I don't spoil it for the next person,

**sogart** · 05-23-2007 11:48 AM

yeah... exactly

**Skibbaz** · 05-30-2007 02:26 PM

Ever heard of Einsteins 2% riddle?

If anyones interested, i will post it.

**sogart** · 05-30-2007 02:48 PM

go ahead and post it

PS: I saw somewhere a question about a garden and 4 trees being in equal distances has anybody heard of that? i cant seem to find it on google...

**Xaqaria** · 05-30-2007 09:25 PM

My answer in white below. Highlight my post if you would like to see it.

It seems to me that if you just decided that if any of the people saw that the other two were wearing the same color, probability wise they should guess the opposite color.