the sound of silence

Now that we all have solved the puzzle from Singapore (or at least read the solution), it is time to think about another one; fortunately it is much easier.

Your boss wants to fire two of three employees, unfortunately you are one of them, and only keep the smartest. This is her challenge: There are 2 black hats and 3 white ones. You three are blindfolded and one hat placed on each of your heads (the two unused hats discarded). After the blindfolds are removed you cannot see your own hat, only the other two. The first employee to figure out the color of his own hat keeps the job.
After the blindfolds are removed you see that the other two guys have white hats. There is an awkward moment of silence and that silence in fact tells you the color of your hat.
What is it?

13 comments:

Anonymous said...

White, because I know that they know what we all know.

wolfgang said...

Anonymous, I guess you will keep your job.

Lee said...

Thanks Wolfgang. That was fun.

PS. The problem you linked to wasn't very hard (I'm of course assuming I got the right answer), but if I remember correctly I was learning how to multiply, divide, add, and subtract fractions in 5th grade along with some story problems. I'm quite certain at age 10 I wouldn't have been able to figure that problem out in a test setting where you're supposed to be quick. I guess that's why I'm not a physicist.

wolfgang said...

>> I'm of course assuming I got the right answer

July 16 (for the puzzle from Singapore)

Lee said...

Yeah, since that was the only possibility I was fairly certain I got the Singapore problem right. Your problem was more nuanced and more fun, but didn't take too long to figure out. I am really impressed though by how succinctly anonymous stated the answer.

wolfgang said...

>> Your problem was more nuanced
Btw it also contains an ambiguity imho:
On the one hand we assume that one of the three is faster thinking than the other two.

On the other hand, if the other two are really slow thinkers then I cannot be sure that my conclusion is correct.

So the puzzle (implicitly) assumes some difference in the speed of thinking but not too much...

Lee said...

In the Singapore problem all a third party has to do is assume all of the statements made are truthful in order to figure out the answer. The statements could come a year apart, and the third party would still be able to figure out the answer.

I agree there is much more ambiguity in your problem because the winner has to make an estimate of how quickly her coworkers need in order to correctly assess the problem and then answer the problem before the coworkers realize they've correctly assessed the problem. So it is fairly easy to come up with what the correct answer should be looking at it from the outside, but would be really hard to do if one were actually a participant in the game.

Lee said...

Now that I think about the ambiguity a little more, I think a person who had thought about the problem previously and then became a participant, would immediately shout out white.

The way the problem is set up, regardless of what combination of black and white hats are passed out, a white hat wearer should always be the winner. So why not just shout out white at the beginning of the game? If you have a black hat on and lose what difference does it make? You would have lost to a white hat wearer anyway.

wolfgang said...

Btw there is one more possibility.
I could wear a black hat but the two conspire, making me think it is white with their silence...

If the game would be set up so that the first wrong answer means the other two can stay it would actually make sense...


Lee said...

I think it would be hard to conspire in this game without advance knowledge.

But I have a question. Let's assume all three get to keep their job if they get their hat color right. With the setup you described as soon as the first person says white, the remaining two have no way of determining their hat color because they won't know if she said white because she saw two white hats or she saw a black hat and a white hat. So should the remaining two assume there is a probability of 1/2 that they are wearing a black hat or should they assume a 2/5 probability they are wearing a black hat when trying to guess hat color?

wolfgang said...

This is a very good question!

A priori one would think 2/3 , because there were initially 5 hats and 2 whites are seen on the others (this leaves 2 black hats and only 1 white).

However, there was that awkward moment of silence at the beginning and this should inform the 2nd fastest guy.
On the other hand, if he used your strategy...

Lee said...

I think you have to make the assumption that all the participants are capable of assessing the situation correctly and winning the game, or the problem becomes unsolvable (meaningless?). In that case 2/3 can't be the right answer because they all end up realizing that the initial state was either one black hat and two white hats or three white hats. They can see the initial state wasn't two black hats and one white hat.

So my first guess was the remaining two participants must only have a 1/2 probability of guessing correctly. I still think that's right but then I got to thinking about 2/5.

Damn, I have more problem with the reCAPTCHA than the riddle.

Lee said...

Now that I think about it a little more, since the two remaining know that at most one black hat was out in the initial state, and if they want to assume that the hats were randomly distributed initially, the second player to guess should use 1/5 as the probability that she is wearing a black hat.