Friday the 13ths in a Year

Date: 08/10/99 at 04:06:28
From: Stephanie Cole
Subject: Max number of Friday the 13ths in a year

How can I figure out the maximum number of Friday the 13ths there are 
in one year? I know that 1998 had 3 because I looked at my mom's old 
calendar. But why do some years have 2 or 3, and why do all years have 
at least one?

Thank you. (I think your web site is great).

Date: 08/20/99 at 10:55:39
From: Doctor TWE
Subject: Re: Max number of Friday the 13ths in a year

Hi Stephanie -

The trick here is to figure out which months start on the same day of 
the week. In a standard (non-leap) year, if January starts on a 
Monday, then February starts on a Thursday, March also starts on a 
Thursday, etc.

Since January has 31 days, it has 4 full weeks (28 days) plus 3 days. 
Hence, February must start 3 weekdays later. Similarly, since February 
has 28 days (4 weeks + 0 days,) March starts on the same weekday as 
February. Continuing this pattern, you can see which months start on 
the same days, and you can count how many months start on each day. 
Remember to do the counts for both a standard year and a leap year!

If months (like February and March) start on the same weekday, 
logically the 13th of those months will also fall on the same weekday 
- which will, in certain years be a Friday. So whatever was the 
highest count of months starting on the same weekday, that number is 
the most Friday the 13th's there could possibly be.

Try counting them out and see what you come up with!

As a related aside, the average number of Friday the 13th's is 
really quick to compute: There are 12 months (therefore 12 "13th's") 
and 7 weekdays for them to fall on, so the average number of Friday 
the 13th's in a year is 12/7, or about 1.714.

- Doctor TWE, The Math Forum
  http://mathforum.org/dr.math/