Guess your hat color, or abstain

[plasmid]

I'm fairly sure I've seen this puzzle or something similar here before. In fact, I was trying to search for it to list it as one of my favorites in the "Best Brain Teasers" topic because if its sheer simplicity but unintuitive answer, and being the simplest case of what turns out to be an entire class of problems that can get quite complex. But my search turned up no hits If anyone's able to find the original post of this problem then please give a link so I can credit it as a favorite brain teaser. Otherwise, have a go at solving it.

You and two buddies are about to play a game where the host randomly places either a white or black hat on everyone's head. You can each see the hats of the other two players, but not your own. You each get a slip of paper and may write down your guess for your hat color, which must be either "white", "black", or "abstain".

If at least one person guesses their color correctly and everyone who does not abstain answers correctly, the entire team wins. If anyone guesses incorrectly or if everyone abstains, the team loses. Peeking at what the other people are writing, or any other form of communication, is cheating that will be punished by forcing the entire team to watch an episode of Sarah Palin's Alaska.

What's your team's strategy?

[bonanova]

You see two Like-color hats or two Opposite-color hats.

Your strategy has to include a case to Guess a color and a case to Abstain.

If seeing Like colors is the abstain signal, the team loses [all abstain]

whenever all three have the same color, which happens 25% of the time.

So Let's say seeing Opposite colors is the Abstain signal.

Then, the cases are [letting a,b = W,B in whatever order]

1. [2 cases] aaa - Guess Guess Guess [win if guess Same]
   
2. [6 cases] baa - Guess Abstain Abstain [win if guess Opposite]
   

So a strategy of Guess Opposite [where Opposite means White or Black, depending on what you see]

if you see two like-colored hats, else Abstain, gives a 75% win percentage.

A strategy that guarantees 100% would be great, and it seems there is such a thing.

If my memory serves me.

[CaptainEd]

for 100% chance, don't we need to know how many of each color are to be placed? Otherwise, how can anybody infer their color? Am I missing something?

[Guest]

The least likely combination of hat colours is all three the same, so if you see 2 the same colour guess the other colours, otherwise abstain. It is impossible to be 100% certain without more information. I am assuming that there is absolutely no communication or anything else similar, i.e. you have 2 peices of data and have to make a binary decision.

Edited August 7, 2011 by James22

[Guest]

If both colors are use.

I see 1 of each color, I abstain.

If I see 2 of the same color, I write the opposite color.

Same for the other players.

I'll watch Sarah Palins Alaska anyway. I like most Alaska shows.

[plasmid]

Right you are. Sorry for the omission from the OP; this is asking for an optimal strategy, not a 100% successful strategy (which I don't think exists).

The uninitiated would likely say "The more people who answer, the lower the probability that everyone will be right, so you should just designate one person to answer. Since everyone could be wearing a black or white hat regardless of what the others are wearing, the person who answers might as well answer randomly." It all seems perfectly logical on the face of it, but leads to a strategy with a 50% chance of winning instead of the 75% chance of winning that could be had otherwise. (In fact, spotting the fallacy in that logic is part of why I liked this one so much.)

Now I remember bushindo posting a much more complex variation on this theme, but have yet to re-discover the original post of this puzzle.

[CaptainEd]

Sorry to be a broken record. What is the probability distribution? Both Bonanova and James22 seem to know the distribution, so I recognize I'm being dense.

[plasmid]

Indeed, the probability distribution could change things quite a bit. For example, if the host were picking the players' hats out of a bag of N white and M black hats...

The intended probability distribution is a 50% chance of receiving either a white hat or a black hat for each of the players, independent of the hat colors of the others.

[Guest]

I would say everyone guess white. BBB is the only situation out of Eight possible situations that would be a loser. This gives a winning percentage of 87.5.

Edited August 8, 2011 by thomalikona

[k-man]

Here is the more interesting (and much more difficult) version of the same puzzle. Same rules, but 15 people.

[Guest]

Missed a part of the problem. Please disregard.

[Guest]

since black and white absorbe and reflect heat differently, see how hot your head gets. i know its as stupid answer, but ill think of something cclever later

Edited August 8, 2011 by mudroze

[plasmid]

Here is the more interesting (and much more difficult) version of the same puzzle. Same rules, but 15 people.

Aha, I knew I wasn't hallucinating it (although perhaps I was hallucinating about seeing this simple case before). Thanks for finding the hard version, k-man.

since black and white absorbe and reflect heat differently, see how hot your head gets. i know its as stupid answer, but ill think of something cclever later

lawl, I would have accepted that as a valid answer if this were one of my "How would you cross puzzle land" riddles

[Guest]

<p>

Team of 3 appoints one of them (say you) as the voter. Only the voter writes the answer the other 2 write &quot;abstain&quot;.</p>

<p>The secret code between the 3 of you is as follows:</p>

<p>1. Assign sequence of writing between the 3 of you<br />

2. Member 1, Member 2, and then You<br />

3. If you are wearing a white hat : Member 1 writes &quot;abstain&quot; first and then Member2 writes &quot;abstain&quot;.<br />

If you are wearing a black hat : Member 2 writes &quot;abstain&quot; first and then Member2 writes &quot;abstain&quot;.<br />

4. You write the correct answer after noticing this sequence.<br />

</p>