Solution Birthday

Be careful of the sequence of dialogues which lead to correct sequence of eliminations.

Dialogue 1: (by red color)

Ben says: I do not  know which month (M), also sure that  Mark does not know which date (N)

=> they are both confused by duplicate M & N

=> eliminate unique dates N: 7/6, 1/12

Ben infers that not in M = 6, 12

 => eliminate 4/6 , 1/12, 8/12 (duplicate Month M)

Dialogue 2: (by Blue color)

Mark says: Initially I don't know, now I know (N)

=> eliminate duplicate dates N : 5/3, 5/9

Dialogue 3: (by yellow)

Ben says: now I also know (which month M)

=> eliminate duplicate month M: 4/3, 8/3

Remain the answer: 1/9 

[QED]