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Every 823 Years?!?

Published on Sunday, June 19, 2011 in , , ,

Dafne Cholet's Calendar* photoYou've probably received the e-mail that says something like, “July 2011 will have 5 Fridays, 5 Saturdays, and 5 Sundays all in one month. This happens only once in every 823 years.”

A quick double-check of the calendar will verify that July 2011 does indeed have 5 Fridays, 5 Saturdays, and 5 Sundays. But is it really true that this happens only once in every 823 years? Since I teach calendar math on this site, I thought I'd take a closer look at this.

For those who don't already know, over at the Mental Gym section of this site, I teach a method for determining the day of the week for any date in your head. Using the math from this article, I'm going to examine the 823-year claim.

Don't worry, you don't have to have learned the method to follow this explanation. The whole method boils down to adding a key number for the year, month, and date given, and then reducing that to a number of 7 or less to get the day. If you can understand that, you should be able to understand what follows.

Since we're spanning so many centuries with this 823-year claim, let's take a look at the patterns over the centuries. In particular, notice this part:

  • 2300 to 2399 = add 1
  • 2200 to 2299 = add 3
  • 2100 to 2199 = add 5
  • 2000 to 2099 = add 0
  • 1900 to 1999 = add 1
  • 1800 to 1899 = add 3
  • 1700 to 1799 = add 5
  • 1600 to 1699 = add 0
The instructions to add a number simply indicate a way to compensate for the same date in a given century. The main thing to notice here is the 1/3/5/0 pattern that repeats. In short, this means that every 400 years, the calendar repeats exactly.

Since July 2011 has 5 Fridays, 5 Saturdays, and 5 Sundays, that also means that July 2411, 2811, 3211, and so on, will also have that same quality. We've already destroyed the 823-year claim!

Still, it does seem to be an unusual quality, and obviously doesn't happen too frequently. Just how often does a month have 5 Fridays, 5 Saturdays, and 5 Sundays?

Let's start by taking a look at a month that couldn't possibly have 5 of any day of the week. If you imagine a non-leap year February that begins on Sunday (such as February 2009), you can see that there are exactly 4 of each day of the week, because it only has 28 days. It starts on a Sunday, and ends on a Saturday.

For any 28-day February, whatever day it starts on, the last day will fall on the day of the week before it. If it started on Monday, it would end on Sunday. If it started on Tuesday, it would end on Monday, and so on.

What happens if we add a 29th day, as in a leap year? Then that February will have 5 of whatever day of the week it begins. If it starts on a Sunday, then it will have 5 Sundays: 1, 8, 15, 22, and 29. This is just as true for any other day of the week.

If we add in a 30th day, then such a month would have 5 of two days of the week. If that month started on a Sunday, then it would have 5 Sundays and 5 Mondays. By the same logic, a 31-day month beginning on a Sunday would have 5 Sundays, 5 Mondays and 5 Tuesdays. Check the calendar for May 2011 to see this for yourself.

The original claim is looking less and less unusual, isn't it? To have 5 of any three consecutive days simply requires a 31-day month, and 7 of the 12 months in any year fit that description.

Let's take the specific case mentioned in the original claim of 5 Fridays, 5 Saturdays, and 5 Sundays. After all, getting a full extra weekend in a month is quite nice!

What we're looking for, then, is a 31-day month that begins on a Friday. Any month can, of course, begin on any one of the 7 days in a week. It would seem, then, that with 7 months of the year having 31 days and a month beginning on any one of 7 days that this should happen once every year!

Let's take a closer look, and see if that's true.

See the code numbers for months on this page (scroll down towards the bottom)? The same key number means that the month will start on the same day of the week as any other month with the same key number.

Since we're just looking for 31-day months, let's examine just those months a little more carefully.

Examining the calendar patterns closely, we see that, in any given year, March, May, August, and December will always begin on different days of the week (Remember, we're only considering 31-day months). In a non-leap year, July will begin on a 5th day of week (different from the previous 4), and that January and October will share the same starting day of the week, different from the previous 5.

What about leap years? In leap years, October and July switch places. October begins on a day different from all other 31-day months, and January and July begin on the 6th remaining day of the week.

So, in any given year, the 31-day months will only cover 6 different days of the week! There will always be one day of the week on which no 31-day month begins. If that day happens to be a Friday, you won't get your 5 Fridays, 5 Saturdays, and 5 Sundays in that year.

If the Friday doesn't show up as the starting day of any 31-day months in a given year (which last happened in 2007), it's more likely to happen in the following year. This is because the same date in each year happens 1 day later in the following year (or 2 days later after a leap year).

In short, finding a month with 5 Fridays, 5 Saturdays, and 5 Sundays happens a little over once a year. If a January happens to fall on a Friday, then it will happen twice in the same year, with other occurrence being in either October (in a non-leap year) or July (in a leap-year).

Taking a look at the calendar beyond July 2011, we see that it happens again in March 2013, August 2014, May 2015, and January and July of 2016, so that seems about right.

If you decide you do want to preserve the magic of an 823-year occurence, you can always offer a 2012 calendar as proof, since 5 Fridays, 5 Saturdays, and 5 Sundays don't happen that year.

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