I am stuck of this question, for longer that I should be and also convinced that there is a typo in the question.
I am going to take a screen shot of the question as to make sure I don’t miss type the information.
Now my problem lie with the last part of the question, what I cannot figure is how the middle knows for sure that he has a red how the reason for how is below.
So the possible permutations of hat are
${RRR , RRB , RBR , BRR , BBB , BBR , BRB , RBB}$
The question says at least 1 blue hat which to me mean that they could all have blue hats.
Now if I work backwards and think how would the back person know what colour hat they have then that would be $RRB$ but the back person doesn’t know, so that leaves
$RBR , BRR , BBB , BBR , BRB , RBB$
and this is where I am stuck because how can M know his hat is red. One conclusion I thought off is if the middle man has said "if, back don’t know his hat, then he must have then he must not have a blue hat", so then that would remove $BBB,BRB,RBB$ and then left with $RBR,BRB$ so then M must have a red hat.
But this logic breaks down because M cannot assume that B has a blue hat from he just saying he dose not know the colour of his hat because, the B would not know the colour of his hat if there was a $BBB$.
So I honestly cannot see how M knows for sure he has a red hat, from the information given.
So is there a typo in this question.
Any advice would be much appreciated.