I think it’s time for a math post. Yesterday was Friday the 13th (of May, 2011). I don’t consider myself superstitious, but it got me wondering as to how frequent such a date is. To be definite, let’s ask the question: How many Friday the 13ths have there been in the past 1,000 years, i.e. since May 14 1011?

My initial intuition was: every month has exactly one 13th day (unlike the 31st, for instance) and on average it will be just as likely to be any one of the days of the week, so it should be Friday about 1/7ths of the time. There have been 12,000 months in the past 1,000 years, so there should be about 12,000 x (1/7) ~ 1714 Friday the 13ths in that time span.

Is this close to the exact number? I don’t know, because I don’t know the exact number. Clearly it could be found in a variety of “brute-force” ways — combing through the past 12,000 monthly calendars (um, no), or finding some electronic date database and performing a quick search (unfortunately I don’t know of such a database). In any case, these methods do not appeal to the mathematician in me since they are rather inelegant and unsystematic — can we set up a tidy equation to solve this, without having to make use of historical records? Alas, the calendar that we are stuck with is not very systematic either — months do not all have the same number of days, and there are those pesky leap years.

I am fairly confident, however, that my answer of 1714 above is very close to being correct, even if you (rightfully) find my reasoning fishy. It turns out that the irregularity of the calendar is precisely what makes it work! To see this, imagine the case where every month always had exactly 28 days. Then the 13ths of every month would be exactly 28 days apart, and would thus all fall on the same weekday (since 28 is a multiple of 7). If this day happens to be Friday, then the answer is 12,000; if not, the answer is 0. My reasoning in that case would therefore be way off, either way. If every month had exactly 31 days, then things actually work — turns out that since 7 and 31 are *relatively prime* (meaning they have no common factor other than 1), the 13ths of every month would cycle through all seven days in the same repeating order. The key is that the length of the cycle of the days of the month cannot be a multiple of 7. In reality, however, the variation in month lengths precludes the possibility of having such a tidy argument, but we should be OK since there are no cycle lenghts lurking that are multiples of 7 – not a regular monthly pattern, nor a yearly one, since neither 365 nor 366 is a multiple of 7.

The issue of when this kind of cycling can provide an even distribution over a set of possibilities is explored in a slightly different context in Hermann Weyl’s paper *On the distribution of numbers mod 1*, which I translated from the German during a lazy quarter in grad school where I didn’t want to do any real math.

Also notice that in my reasoning in the second paragraph I used words like “average” and “likely” to justify my answer, but these words belong to the realm of probability — there is no randomness in the assignment of days of the week to days of the month. My justification might make sense if, for example, every day of the month, God rolled a fair 7-sided die to determine what day it is to be. (I realize that neither Einstein nor Plato would be happy with that last sentence.) This is a nice feature of some random systems that are called *ergodic* – even though the state (in this case, day of the week) that they’re in at any one time is random, in the long run, the proportion of time in each state is a predictable quantity. I was in effect using the converse of this property — treating a deterministic but messy system as if it were random made it easier to think about, but as noted above with the 28-day months, you need to be careful when you do this!

P.S. If you’re wondering about the title of the post, it’s derived from *friggatriskaidekaphobia*, a somewhat whimsical term for the “fear of Friday the 13th”; I just replaced the *-phobia* suffix with *-sophy*, which I guess would be something like “science of Friday the 13th”.

on May 16, 2011 at 5:55 am |elenamaryi once owned a car formerly owned by a mathematician. he kept meticulous records on his car, and used his own calendar/dating system because he found our Julian calendar to be, perhaps, inadequate. that man rocked almost as much as you.

on May 17, 2011 at 5:33 pm |mathmongerFalkner!!! Not only is he a mathematician but he is also Canadian.