Anytime each of three consecutive months has exactly four Fridays, Jack's birthday will fall in one of those three months. Which month is that?
The three months with $4$ Fridays each have $12$ Fridays together, so they can have at most $12\cdot7+6=90$ days. Conversely, if three consecutive months have at most $90$ days, they sometimes contain only $12$ Fridays, and since every month contains at least $4$ Fridays they then contain $4$ Fridays each. In summary, three consecutive months may contain $4$ Fridays each iff they have at most $90$ days. Three consecutive months can have at most $90$ days iff one of them is February. Thus Jack's birthday is in February.
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Three consecutive months which do not contain February have at least $91$ days. Because $91=7 \times 13$, each day of the week will repeat itself at least 13 times, so Jack's birthday cannot fall in any of these months.
But if we consider February (in years in which it has $28$ days) then any three consecutive months containing February have at most 90 days, so there is a day of the week which repeats itself exactly 12 times in these three months. As time goes by, this day will be a friday. So Jack's birthday will fall in a month out of any three which contain february. Therefore Jack's birthday comes in February.