# Weekend birthdays

My birthday this year (2011) is on a Friday. In most years, one's birthday the following year is on the subsequent day of the week, and in that pattern, my birthday next year (2012) it is on a Saturday.

However, due to 2012 being a leap year, my birthday in 2013 will be on Monday - I miss out on having a Sunday birthday.

I quite like having weekend birthdays, so that disappoints me a bit. However it leads me to a question:

Over a reasonable-length life-time, does any given person's number of weekend birthdays even out to an average, or are some people significantly more blessed than others?

If some people do have an advantage in this respect, is there any way mathematical way to work out whether being born on any given day of any given year will confer an advantage? And further, what are the probabilities of being 'lucky' in your number of weekend birthdays.

My guess is that for any given year, on each side of Feb 29th, birthdays on any given day of the week throughout the year will have the same score.

To make it a bit easier, lets say for a 'reasonable length life-time', we mean a fixed length of 80 years (though I for one intend to beat that!). That said, I would also be interested in how the maths change as we vary the life-span.

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This isn't really probability, since the day of the week a particular day falls on is not a random event; more combinatorics. Also: you probably want to ignore the real leap year rule, which would necessita complications if the 80 years includes a year that is a multiple of 100 but not of 400... –  Arturo Magidin Feb 12 '11 at 21:05
In fact, once you include the real leap year rule, the calendar repeats on a 400 year cycle. This is because 7 divides 400*365+97. So there are only 146097 starting days to try to find which is best (though the differences will be small.) –  Ross Millikan Feb 12 '11 at 21:11
Well, I can tell you this: people born on the 29th of February lose out on weekend birthdays big time. –  Paul Fisher Feb 13 '11 at 2:13
@PaulFisher. They miss out on birthdays big time. Which can be a good thing if you're trying to stay "younger for longer". –  Samuel Tan Dec 21 '11 at 11:53

The calculations are slightly different depending on whether it is the Julian or Gregorian calendar, and whether the date in question occurs between January 1st and February 28, or March 1st and later (we'll ignore February 29, I think).

Consider first the situation in the Julian calendar, where there is a leap year every $4$ years without fail.

Suppose you have January 1st on a Monday, in the year immediately after a leap year. When will January 1st again fall on a Monday in the year immediately after a leap year? The pattern is:

M, T, W, R, Sa, Su, M, T, R, F, Sa, Su, T, W, R, F, Su, M, T, W, F, Sa, Su, M, W, R, F, Sa,

and then repeats; that is, a 28 year cycle. So, ignoring the leap year rule for Gregorian calendar, you could check for a period of 84 years (three full cycles). The situation is similar if your birthday occurs March 1st or later, except that the leap year affects you in position 4 instead of 5; you would have:

M, T, W, F, Sa, Su, M, W, R, F, Sa, M, T, W, R, Sa, Su, M, T, R, F, Sa, Su, T, W, R, F, Su,

and the pattern repeats.

In our putative example, which began on a Monday, we got each day of the week exactly $4$ times; that is, they all occur equally likely. If you chop it off at 80 years, then you'd be one short on four days. Changing the day of the week, or the time of the most recent leap year, would not affect the outcome: it cycles every 28 years.

However, the Gregorian calendar has a slightly different rule for leap years: in the Gregorian calendar, a year is leap year if and only if it is a multiple of $4$ that is not a multiple of $100$, except for the multiples of $400$ (that's why $1900$ was not a leap year, $2000$ was, but $2100$ will not). For someone born before $2016$, assuming a period of $84$ years, this will not matter because we will have a cycle just as above, with leap years every four years. But if you extend the lifetime so that it includes the year $2100$, then you start running into trouble because you skip a leap year.

You can think of the Julian calendar as repeating the pattern of leap years every four years; the Gregorian calendar, on the other hand, has a cycle of 400 years for repeating leap years. Added: And as Ross pointed out, while the Julian calendar has a cycle of 28 years, the Gregorian calendar has a cycle of 400. So you want to check what happens over a period of 400 years.

Over a cycle of 400 years, the number of times a given date (other than February 29) falls on a particular day does change (it's not uniform, like in the Julian calendar). If you start counting on a Monday, over the next 400 years, that date will fall on Monday, Wednesday, and Saturday 58 times each, Thursday and Friday 57 times each; and Tuesday and Sunday 56 times each. It's a pretty small difference over a period of 400 years, but there you go.

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Aewesome answer. Thank you. –  Spudley Feb 12 '11 at 21:50