# Jack's birthday riddle

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?

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Is it Ramadan?? – The Chaz 2.0 Feb 13 '12 at 11:00

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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You were just a bit faster :) – Beni Bogosel Feb 13 '12 at 11:35
The birthday is not Feb 29th :-). Also, just for completion (assuming no leap year and assuming I have got it right), if Jan 7th is Friday, then Jan,Feb,Mar have 4 fridays each and if Feb 7th is friday, Feb,Mar,Apr have four fridays and Jan has 5. – Aryabhata Feb 13 '12 at 12:49
@Aryabhata: There are two more possibilities: February 6th also gives four Fridays each in February, March and April; and December 7 gives four Fridays each in December, January and February; in fact that last constellation will occur this/next year. – joriki Feb 13 '12 at 13:11
@joriki: I was just trying to prove that there are at least two sets of months, with only Feb being common among them. Basically eliminating the case that it will always be jan,feb,mar or something like that (in which case, the answer could be any one of those). – Aryabhata Feb 13 '12 at 13:19
@Aryabhata: I thought I'd covered that by saying that they have at most 90 days iff one of them is February. That implies that there are examples for each combination with February, so it can only be February. Also your examples both include February and March, so it seems they don't serve the intended purpose? – joriki Feb 13 '12 at 13:24

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.

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