Hat Game

Logic Level 2

Three people are to play a hat game. They are to guess their hat color and say it out loud. If someone speaks up, then the game is over (there's no coordinated effort to speak in codes). If the person guesses correctly, then they win; otherwise, they loose. They also each know that there are 2 blue, 1 red and 1 white. Assuming that they can see everyone else's hat, except their own, is it possible for at least one of them to know their hat color?

Yes No Sometimes

2 solutions

Rein Gomez
Nov 26, 2019

I don’t understand how there are 4 hats for a game of 3

Think about other games. In poker, you could use some but not all of the cards and still play. Or a card game where the remaining cards are not used. I hope that clears things up :)

Jay B - 1 year, 6 months ago

Jay B
Nov 22, 2019

Case 1: Two of them are wearing blue. Suppose that Mary is wearing blue. Then she will see that one of the other two, John, is wearing blue. If she were wearing other than blue, then John would know that Mary and the other person, Peter, have red and white, not necessarily in that order. Then John would know for sure that he's wearing blue and therefore would say something. But this won't happen. Then Mary would know that she's also wearing blue. The same thinking process goes for John and in this case, both would know their hat color.

Case 2. All colors are distinct. Then one wearing blue would know.

Two cases:

First case: all three colors are used. Person wearing blue hat can see other two colors and conclude he is wearing blue hat.

Second case: two blue hats used. WLOG, third hat is white. Let's say person 3 is wearing white.

Person 1 sees blue hat and white hat. Doesn't know whether he's wearing blue or red. Person 2 sees blue hat and white hat. Doesn't know whether he's wearing blue or red. Person 3 sees two blue hats. Doesn't know whether he's wearing white or red.

"The second case is if Peter see’s two different colors. Then he know what he’s wearing." Not if one of those colors is blue.

The way you condition your argument is very confusing. You start with Peter seeing two blue hats and then talk about Mary seeing two colors. You seem to want to argue that Mary can make a conclusion based on information about whether the third person can identify his own hat color. But that's not the same problem you've stated. Your framing doesn't include any statement indicating that any of the people has any knowledge about what conclusions other people have drawn.

Yes, if Mary knows that the third person cannot draw a conclusion, that helps her draw a conclusion. But your question doesn't have any language implying that she has any such knowledge.

Richard Desper - 1 year, 6 months ago

I'm sorry man, I don't get your explanation. Mary will definitely see two colors, one of which would be blue (second case). If she were wearing other than blue, then John, the one who's wearing blue, would have seen read and white, WLOG, Peter wears white and Mary, under her assumption, would have red. This means that John would have known he's wearing blue. He doesn't draw that conlcusion, however. Then Mary's assumption is wrong. Therefore, she is not wearing a different color other than blue.

Jay B - 1 year, 6 months ago

Hat Game

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