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 Math Help > Counting > Friday the 13th Frequency

My son, Matthew, commented to me this afternoon (August 20, 2005) that there aren't very many Friday-the-13ths.  He said that for some reason it is unlikely for the 13th to fall on a Friday.  He flipped through the calendar and noted that there are no more months this year in which the 13th falls on a Friday.

## The Frequency of Friday the 13th

I said that, in fact, Friday is the most frequent day on which the 13th falls.  He was very doubtful.  That launched me into a lengthy discussion of the calendar, and how wonderful it is that there is an exact multiple of 7 days in 400 years.  For your edification, here's a summary of it.

Every fourth year is a leap year, so in 400 years there are about 300 years that have 365 days, and 100 years that have 366 days.

Except...

Every 100th year is not a leap year, so that makes 304 regular years and 96 leap years.

Except...

Every 400th year is a leap year (that's why 2000 was a leap year), so we have 303 regular years, and 97 leap years.  We're done with the exceptions.  Now here's the beauty part: 303 times 365 plus 97 times 366 is 146,097, which is an exact multiple of 7.  That means the calendar repeats itself exactly, leap years and all, every 400 years.

It's not hard to write an Excel spreadsheet that calculates the frequency of the days of the week on which the 13th falls over the next 400 years.  Here's how to do it.  Start with the 13th of this month.  Put that date in cell A1.  (I put 8/13/2005 in that cell.)  Then in cell A2, put the following formula:

=DATE(YEAR(A1+30),MONTH(A1+30),13)

Copy the contents of cell A2 down to all the cells from A3 through A4800.  Now you have a table of the dates of all the 13ths of every month for the next 400 years.

Next, in cell B1, put this formula:

=MOD(A1,7)

Copy that formula from B1 down to all the cells from B2 through B4800.  Here, numbers from 0 to 6 represent the days of the week, as follows:

 Number Day 0 Sat 1 Sun 2 Mon 3 Tue 4 Wed 5 Thu 6 Fri

If you like, you can have Excel format these cells to show the day of the week instead of just the number.  Click Format, Cells, Custom, and type ddd as the custom format -- it's up to you.

Now, in cells C1:C7, enter the numbers 0, 1, 2, 3, 4, 5, 6 in a vertical column.

In cell D1, enter the following formula, but don't hit enter yet!

=SUM(IF(B\$1:B\$4800=C1,1,0))

After you type the formula in cell D1, hold down the Shift and Ctrl keys, and press enter.  This makes the formula into an "Array Formula", so it counts the number of days in column B that match cell C1.  Then copy this formula from D1 to D2 through D7.

If you format column C to make it show day-of-week (Click Format, Cells, Custom, and type ddd), then you will have a table that looks like this:

 Day Frequency Sat 684 Sun 687 Mon 685 Tue 685 Wed 687 Thu 684 Fri 688

So, as you can see, over any 400-year period, the 13th falls on Friday 688 times, which is more often than any other day of the week!

### Selecting the starting point of the 400-year cycle

 Jay U. writes, So, is this analysis really true for ANY 400 year period, or just the 400 year period that begins with day 1 and ends with day 146,097? It seems to me if you started with an arbitrary day somewhere in the cycle, or even with a different year you would get different frequencies of days of the week being the 13th.

A very good question, Jay.  The short answer is: it doesn't matter when you start, you'll get the same frequencies.  Here's the explanation:

First, let's consider the value of month, day-of-month, day-of-week (where Sat=0), considered as an ordered triple. For example, Friday the 13th of March is (3,13,6). If you line up an infinite number of these triples, one for each day since the beginning of time, through today, and up to the end of time, you would see that the pattern of these triples repeats every 146,097 days. Now, I'm proposing that you begin on an arbitrary day, and go for 146,097 days, to pick up one full cycle. I say it doesn't really matter which day you start, so I just pick a day and start there. To see why this is true, consider a simpler cycle that repeats every 11 days:

Viewing a sequence whose period is 11 through a window with the same width.

If you start at the "21", and check eleven values, this is a "window" of eleven values, and then when you count them, you will find you get 1 2, 3 5's, 1 8, 1 15, 2 18's, 1 21, 1 23, and 1 27. If you start at the 2 instead, then you'll miss the 21 from the first cycle, but you'll pick up the first 21 of the next cycle, so you will *still* count 1 2, 3 5's, 1 8, 1 15, 2 18's, 1 21, 1 23, and 1 27. Each time you shift your window to the right, a value will shift off the left side of the window, but this same value will pop into the right side of the window.

### Internet references

Trivia Magpies -- Friday the 13th

### Related pages in this website

Counting things, such as objects in boxes

The webmaster and author of this Math Help site is Graeme McRae.