Newsgroups: rec.puzzles
From: hoey@AIC.NRL.Navy.Mil (Dan Hoey)
Date: Thu, 11 Mar 1993 21:57:28 GMT
Subject: Computer dating (was Re: Computer logic)

emc...@bnr.ca (hume smith) asks on what day of the week the 13th is
most likely to fall.

ab...@Freenet.carleton.ca (Guy Cousineau) answers:

>In the last 400 years, which should be a reasonably full cycle,
>Christmas has fallen on a particular day of the week:

I should note that Guy's figures for the days of Christmas are correct
for the Gregorian calendar, and he then calculates the probable day of
December 13th correctly.  Unfortunately, December 13th is not typical
of thirteenths, so he answers the question wrong.

But while the last 400 years were a full cycle in countries that were
controlled by Pope Gregory in the 16th century, England and her
colonies did not adopt the Gregorian calendar until 1752.  My figures,
said to reflect England and colonies, indicate that the Christmases of
1593 through 1992 fell on 58 Sundays,
                          56 Mondays,
                          60 Tuesdays,
                          58 Wednesdays,
                          57 Thursdays,
                          57 Fridays, and
                          54 Saturdays.

But knowing the days of Christmas does not help us count all the
thirteenths during a time of calendar reform.  In 1593 through 1992 I
calculate there were 685 Sunday the thirteenths,
                     684 Monday the thirteenths,
                     685 Tuesday the thirteenths,
                     688 Wednesday the thirteenths,
                     684 Thursday the thirteenths,
                     691 Friday the thirteenths, and
                     682 Saturday the thirteenths,

for a total of 4799 thirteenths in all.  The reason there are only
4799 thirteenths in those 400 years is that September 13, 1752 did not
occur.

We see that Friday was the most common thirteenth in the last 400
years, but to answer the question question as posed, it would make
sense to answer it for the Gregorian calendar, since that will
continue for quite a few centuries to come (and no one has any good
plans for its successor.)  In any four hundred consecutive years of
the Gregorian calendar there will be 687 Sunday the thirteenths,
                                     685 Monday the thirteenths,
                                     685 Tuesday the thirteenths,
                                     687 Wednesday the thirteenths,
                                     684 Thursday the thirteenths,
                                     688 Friday the thirteenths, and
                                     684 Saturday the thirteenths.
So Friday will continue to be the most popular thirteenth, but not to
such an extreme extent as it was recently.

To summarize these results, if the date is the thirteenth, there will
be a 14 1/4% chance of it being Thursday, and likewise for Saturday, a
14 13/48% chance of it being Monday, and likewise for Tuesday, and a
14 5/16% chance of it being Wednesday, and likewise for Sunday.  But
the chance of it being Friday is 14 1/3%, an entire twelfth of a
percent more likely than the least probable day.

Dan Hoey
Hoey@AIC.NRL.Navy.Mil