I feel like I’ve seen a logic problem like this before…
Also lol CAAT didn’t bother finding new names (check MAT 2011 Q6)
i) The only way for Alice to know her hat’s color must be that Bob and Charlie are wearing blue hats. Since at least one of the (poorly drawn) logicians must be wearing a red hat, Alice then knows she is wearing a red hat.
If any of Bob and Charlie wore a red hat, the information the father gave would be no use for Alice.
ii) Since Alice cannot deduce the color of her hat, one of Bob and Charlie are wearing a new hat.
Bob took the information that Alice does not know her own hat’s color. Since he does not know, Charlie must be wearing a red hat. If Charlie was wearing a blue hat instead, Bob can deduce that he is wearing a red hat, as otherwise Alice would know her hat’s color.
iii) As in part i, Alice sees that Bob and Charlie are wearing hats of the same color, in order to deduce she must have the other color.
Bob then just needs to see which hat color Charlie is wearing to know his own (which is the same color). Therefore, both Bob and Charlie are wearing red hats and Alice is wearing a blue hat.
iv) Since Alice cannot deduce her own hat’s color, Bob and Charlie must be wearing two different hat colors. Bob deduced this, and thus knows his hat is red, so Charlie has a blue hat.
v) Clearly Bob must have a red hat, as otherwise there is a contradiction with the father’s statement.
If Charlie wore a blue hat, then Alice would know she has a red hat, as otherwise we have a contradiction. Since Alice does not know, Charlie wears a blue hat.