Friday XIII

Is there a Friday XIII every year, or are there years without? 1) Are there years with lots and lots?2) How many is the most in one year? 3) 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 4). 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 Friday XIII on one of my blogs; I hope this one is an improvement. I've also written 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 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?

1) There is always at least one.
2) No, most years have either one or two; only one year in seven has three
3) At most three
4) day of the week January 1 falls has seven possibilities, leap or not has two, 7 x 2 = 14
friday_xiii.txt · Dernière modification: 2018/01/13 18:13 par suitable
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