Ben is told the month, but you cannot identify the birthday from
the month, because each month covers at least two birthdays. This
is why Ben cannot identify the birthday initially.
Mark is told the day, and there is a small chance that you can
identify the birthday from the day, because the 7 (for June) and 2
(for December) are unique in the list.
Ben insists however that Mark
cannot identify the birthday, even though Ben doesn't know what
"day" Mark was told! This means that Ben somehow knows that,
whatever Mark was told, it was not 7 or 2. This means that Ben
knows that the birthday is not in June or December.
So he must have been told either
March or September.
Mark sees this and announces that he now knows the birthday. So
Mark must have been told either 1, 4 or 8, because from 1, he can
now deduce September, and from either 4 or 8, he can now deduce
March. (The 5 is ruled out because it is ambiguous between March
and September.)
Ben sees this and says that he now knows the birthday too. So he
must have been told September, because if he was told March, he
wouldn't be able to deduce whether Mark was told 4 or 8, whereas if
he was told September, he can deduce at once that Mark was told
1.
So the answer is 1 September.
