10 people in a circle and they are each given a red or blue hat. They can see each other's hats, but not their own. They are told that at least one red hat was assigned (we know that 3 red and 7 blue were assigned). They can't talk to each other. They are asked, "put your hand up if you know you are wearing a red hat" repeatedly and they have time to respond before being asked again. How many times do they need to be asked before those wearing a red hat have figured it out?
The answer to the question is as follows… If only one person was wearing a red hat, then they would see only blue hats on the others and on the first time of asking, would raise their hand as at least one red hat was assigned. If there were two people with red hats, consider one of them. They will see that the person wearing the other red hat didn't put up his hand on the first time of asking, can see the eight others are wearing blue, so deduces he is wearing red. Then both red hat people raise their hand on the second time of asking. With three red hats, consider one of them. They notice that both of the red hat people they can see don't raise their hand on the second time of asking, can see the other seven are wearing blue, so deduces he is wearing red. Then all three of them raise their hand on the third time of asking.
He's my question! The line in the question "they are told that at least one red hat was assigned" seems redundant because everyone in the circle can see a red hat since three were assigned. Yet without this line, the basis of the solution for the case "if only one person was wearing a red hat" is removed and the solution breaks down. Can someone clear this up for me please? Thank you!
Best Answer
It's called meta information. Basically, it boils down to “I know that you know, what I know. But do you know that?”.
So if there is only one red hat and the intel about red hat was not given. I am looking at the person with red hat and think myself “I know that there is a red hat, but I don't know if the person in red hat knows about it”. With two hats, the chain of thought is a step longer: “I don't know if
person in red hat #1knows thatperson in red hat #2knows that there is a red hat”.With intel that not only everyone hears, but also everyone hears that everyone hears and so on (so called common knowledge), this ladder of meta-information can never end, so eventually everyone will guess their red hat. However, if everyone is given this intel secretly from everyone else, this no longer works.
P.S. There is an interesting speculation about meta-information and flirting/euphemisms. If man asks a woman to go home and listen to his collection of medieval music, they both understand that man suggests a night together. However, if a man asks woman directly, woman can be dis-pleasured and likely to say no. The idea is that meta-informational decay “I know that he suggests sex, but I am not sure if he knows that I know that” allows her to keep face and bail out if necessary (“I didn't know!”), whereas common knowledge (“Come with me for sexy time!”) doesn't leave room for that.