====== Friday XIII ======

Is there a Friday XIII every year, or are there years without? ((There is always at least one.)) Are there years with lots and lots?((No, most years have either one or two; only one year in seven has three)) How many is the most in one year? ((At most three)) The following simple table answers all these questions for you: I've already done the work. Just find the row that corresponds to the day of the week the first of January falls; the months with a Friday XIII are listed according to whether it is a leap year or not.

^  ^ Months with a Friday XIII ^^
^ 1 Jan ^ Leap Year ^ Non-Leap Year ^
^ Sun | January, April, July | January, October |
^ Mon | September, December | April, July |
^ Tue | June | September, December |
^ Wed | March, November | June |
^ Thu | February, August | February, March, November |
^ Fri | May | August |
^ Sat | October | May |

===== The Reasoning =====

These days, with the availability of computing power, it is quite possible to "brute force" the problem; that is, write a little program, or even a simple spreadsheet, to calculate the fourteen possible calendars ((day of the week January 1 falls has seven possibilities, leap or not has two, 7 x 2 = 14)).  However, I'm old enough to prefer finding the solution on the back of an envelope, and that is exactly what I did; I just recopied the result to this wiki.

  * January 8 and 15 fall on the same day of the week as January 1, so January 13 falls two weekdays earlier, or five later, than January 1. January will have a Friday XIII if and only if the 1st falls on Sunday.
  * When a month has 28 days, its 13th and the 13th of the following month will fall on the same weekday. When February has a Friday XIII in a non-leap year, March will have one, too.
  * When a month has 29 days, as February does in non-leap years, the 13th of the following month will fall one weekday later.
  * Similarly, when a month has 30 or 31 days, the 13th of the following month will fall respectively 2 and 3 weekdays later.
  * The weekdays cycle, i.e. +8 is equivalent to +1 : a cycle of 7, returning to the same weekday, then advancing one.

With these rules, one can construct a table of the off-set to the weekday on which the 13th falls in each month relative to the weekday of January 1.

==== Off-sets versus January 1 ====
^ 13th of... ^ Non-Leap ^ Leap ^
^ January | +5 | +5 |
^ February | +1 | +1 |
^ March | +1 | +2 |
^ April | +4 | +5 |
^ May | +6 | 0 |
^ June | +2 | +3 |
^ July | +4 | +5 |
^ August | 0 | +1 |
^ September | +3 | +4 |
^ October | +5 | +6 |
^ November | +1 | +2 |
^ December | +3 | +4 |

Reading this table, one sees, for instance that in leap years, May 13th falls on the same weekday as January 1, and in non-leap years August 13th does (instead). So if January 1st falls on a Friday, there will be a Friday XIII in May (leap years) or August (non-leap years). The other offsets are converted in the same way : 
  * "+1" will correspond to a Friday XIII if January 1 fell on Thursday.
  * "+2" will correspond to a Friday XIII if January 1 fell on Wednesday.
  * etc.

===== Further Reading =====

An earlier treatment of this topic is [[http://uwwoonp.blogspot.com/2005/10/friday-xiii.html| Friday XIII]] on one of my blogs; I hope this one is an improvement.  I've also written [[http://uwwoonp.blogspot.com/2006/01/delocalization-and-globalization-of.html |Delocalization and Globalization of Holidays]] about "when does a year start."  Then there is a treatment much like to one above applied to [[public holidays in France]].

Another question came up: [[what_year|what year might that have been?]] Given an invitation for an event on "Saturday, April 26th," for instance, might it (if correct, of course) have been from last year?