When Does Your Birthday Fall on the Day You Were Born?

Date: 02/20/2004 at 23:35:16
From: Peggy Lee Gazay
Subject: This is just something I'm curious about!

I'm curious how many years pass before your birthday falls on the day 
of the week you were born?

Date: 02/21/2004 at 12:01:09
From: Doctor Rob
Subject: Re: This is just something I'm curious about!

Thanks for writing to Ask Dr. Math, Peggy Lee!

Your question is made a lot harder by leap years, but let's wait on
that for a moment and just think about a 365 day year.  Since a week
has 7 days, 52 weeks covers 52*7 or 364 days, leaving one day left
out.  That means that ordinarily your birthday day shifts by one day
each year.  If it was on a Tuesday this year, it will be on Wednesday
next year.

Since a leap year adds an extra day, your birthday day will jump two
days during a leap year.  If it was on Tuesday this year, it will jump
to Thursday next year. 

So the answer to your question really depends on how the leap years
fall in relation to the year of your birth.  Since 14 September 1752,
when the Gregorian calendar was adopted by English-speaking countries,
this is how it works:

The answer is either 5, 6, 7, 11, or 12 years.

Let a 29th of February be called a leap day.  If you are born on
a leap day, consider the 1st of March to be your birthday in
nonleap years.

  If there is a leap day before your first birthday and between
     your fourth and fifth birthdays, it will be 5 years.

  If there is a leap day before your first birthday but not
     between your fourth and fifth birthdays, it will be 6 years.

  If there is a leap day between your fourth and fifth birthdays
     but not before your first birthday, it will be 6 years.

  If there is a leap day between your first and second birthdays,
     your fifth and sixth birthdays, and your ninth and tenth
     birthdays, it will be 11 years.

  If there is a leap day between your first and second birthdays
     but not between your ninth and tenth birthdays, it will be 12
     years.

  If there is a leap day between your first and second birthdays
     but not between your fifth and sixth birthdays, it will be 6
     years.

  If there is a leap day between your fifth and sixth birthdays
     but not between your first and second birthdays, it will be 6
     years.

  If there is a leap day between your second and fourth
     birthdays, it will be 6 years.

  If there is no leap day before your seventh birthday, it will be
     7 years.

This list should cover all cases.

Feel free to reply if I can help further with this question.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/