Monday, January 10, 2011

The Five Weekend Myth, Debunked

This year, July has 5 Fridays, 5 Saturdays, and 5 Sundays. This happens once every 823 years...
- Too many Facebook statuses to count

If you want to believe that people are stupid, have really short memories, and really suck at math, here's your chance. There's a meme going around that July 2011 will have 5 full weekends, and it'll be the first time this has happened in centuries.

It won't be. In fact, the last time this happened was in October of 2010 (yes, not even three months before this post), and the same meme went around the web then.

If you're thinking, "Omigod! This thing only happens once every few centuries and it's happened twice in less than 12 months! Awesome!" Well, I have bad news for you: this happens all the freaking time. Rather than just pull out a calendar and show you how often this happens*, I'll try to explain using math.

* - SPOILER ALERT: By the end of this post, I'll be pulling out a calendar to show you how often this happens. Yes, I'm a hypocrite.

Let's begin with something easy. Any month that starts on a Friday will have five Fridays: the 1st, 8th, 15th, 22nd, and 29th. If that month happens to have 31 days, then it will also have five full weekends, since Saturday will be the 30th and Sunday the 31st. (It is true that, because of this, only a month with 31 days can have five full weekends -- so if someone tries to claim that a shorter month will also have five full weekends, you can debunk that really easily.)

So, how often do months with 31 days roll around?

Thirty days hath September
April, June, and November
All the rest have thirty-one
Excepting February alone

The old rhyme is right on, and by a process of elimination we can easily see that there are seven months each year with 31 days (January, March, May, July, August, October, and December). So you start with seven chances every year to see a month with five full weekends.

Next, note that just about every month contains a number of days that doesn't work out to an even number of weeks -- only February (in non-leap years) has exactly 28 days. So, since the number of days in each month is causing the start of each month to shift around the week, chances get even better that you'll see some month in any given year start on a Friday. This is not to say that every year gets a month with five full weekends (in fact, 2012 has no such month), but here's the thing:

If a given year doesn't have a month that starts on a Friday, the next year is more likely to.

Why? Because the number of days in the year also don't divide evenly into weeks. If they did, then years would start on the same day every year, and the progression of weeks would be eminently predictable -- July might always start on a Sunday, for instance, as it will in 2012. But because the normal year has 365 days, while an even 52 weeks contains just 364 days, each year progresses by a day on the weekly calendar. So in 2012, January will start on a Sunday, not a Saturday like it did this year.

So if you have a year where all the 31-day months start on some day other than a Friday (like 2012), the shifting of all those starting days by one (or two, in the case of leap years like 2012) means that you get seven new shots at the apple, so to speak.

And lo and behold! The next 'money month' that only happens once every 823 years will actually March of 2013.

"Okay, okay," I hear you thinking, "so it's not as rare as all that. It's still gotta be pretty rare, though, right? I mean, three times in less than three years has to be a fluke?"

No, my good person, it doesn't. If you understand enough math to build a calendar, you can build what's called a perpetual calendar, in which any combination of day, month, year, and day of the week can be determined. And since the math required to calculate calendars has been around for many centuries, the ability to produce perpetual calendars has also been around for centuries -- see that Wikipedia page linked above, which contains an image of a Swedish perpetual calendar used to calculate which day Easter falls on...from 1140 through 1671.

But for our purposes, this perpetual calendar works best. The provider of the calendar gives you a handy guide for using it as intended, but we can also use it to find out how often five-full-weekend months come around:

  • Start at the '1' in the 'days of the month' box in the lower right of the calendar,
  • Go to the right until you reach the 'Friday' on the same row.
  • Go up.

Each month you see, in each year in the same row -- both to the left and right of the month names -- starts on Friday the 1st. And thus each such month with 31 days will have five full weekends. The particular calendar I've linked to covers 1775 to 2025, a span of 250 years. And during that 250 year span, it's easy to see that there are a lot of five-full-weekend months**; heck, there are years that have more than one month with five full weekends, such as 2010, for example. Yes, the very year in which people started sharing online about how rare it was to see a month with five full weekends was a year that had two such months!

**If you don't feel like doing the comparison yourself, the total is 227 five-full-weekend months during this 250 year period, or nearly one every year.

In truth, a year like 2012 that won't have a five-full-weekend month in it is actually somewhat more noteworthy, even if it's not particularly rare, either. According to the same calendar, 36 of the 250 years on the calendar won't see a five-full-weekend month.

And OMG! Every one of those years has June as the month that starts on a Friday! That has to be when the aliens will come!

Or not.

1 comment:

Phil Church said...

I cannot believe no one has commented on this! I told this to 2 work friends who were adament the 823 day fact was true (as they had a text about it), now they tell everyone its not true and word it in a way that makes the other person stupid for believing it. I tried showing them before i read this, but you seemed to have gift of telling it how it is. Thanks