An Adaption Of Logic Hats
3 people were playing a logic game together.
In the game, there were 3 black hats and 2 white hats.
The gamemaster randomly placed a coloured hat onto the head of each of the 3 people, such that none of them could see their own hat, but can see the other 2 hats on the other 2 people.
Let the 3 people be Mr A, Mr B and Mr C. A conversation ensued amongst them:
Mr A: "Mr B does not know his hat colour."
Mr B: "Mr C does not know his hat colour."
Mr C: "Mr A know his hat colour."
By looking at their conversation, can it be deduced that all 3 people have black hats?
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You cannot deduce that all of them are wearing black hats because B will not make his statement if they are.
Let's call the three statements SA, SB, and SC.
Without hearing anything, B can only know his own color if he sees two white hats. A will only make his claim if he is certain that B can see at most one white hat. For A to make his statement, B must be wearing a white hat and/or C must be wearing a black hat. So once A makes his statement, B and C gain knowledge based off of each other's hat colors. If B is wearing black, C now knows he must be wearing black too. If C is wearing white, B now knows he is wearing white as well.
For B to say SB he has to know his own hat color. We can see this because if he doesn't then he always has to consider the possibility that he is wearing a black hat and therefore C knows his own hat color.
In scenario All Black Hats (BBB), A will say SA. Once he does, C knows his hat color. B does not know his own, but he knows it is possible that his own hat tipped C off, so he can't say SB.
If you want to see what scenarios all three statements can be made for, take the original seven hat combinations for ABC: BBB, BBW, BWB, WBB, BWW, WBW, and WWB
SA eliminates any scenario where B is wearing a black hat and C is wearing a white one, leaving: BBB, BWB, WBB, BWW, WWB
Of those five, B can only know his hat color in one: BWW.
For SB to be true, A must be wearing Black, B must be wearing White, and C must be wearing White.
And yes, as there is only one combination that both SA and SB are true for, A will immediately know his own hat color and SC can also be stated truthfully.