823 Years

A passitaround has recently occurred on Facebook that runs, with variations, something like this:

March 2013 will include five Fridays, five Saturdays, and five Sundays, a phenomenon that occurs only once every 823 years.

Is this true or not?

The formula of my last post can help with this. First of all, what does a month look like if it has five of each of those three weekdays? It then will have four of the others, for a total of 31 days. Hence such a month must be a 31-day month. A look at calendar patterns will reveal that such a month must have the 29th of the month on a Friday. The above version of the post sets the month as being March, which has value 0. Let us substitute this in the formula and see what happens:

W = Y + M + D (mod 7)

6 + 28 = Y + 0 + 29 (mod 7) (can add a multiple of 7 to either side)

5 = Y (mod 7)

This says that the Doomsday of the year must be 5, or Thursday. Let's make a table of the first 28 years of this millennium, backing up to include 1999. This table shows for each year in the table the Doomsday number of that year above it.

	1	2	3	4	5	6	7
1	1999	-	2000	2001	2002	2003	-
2	2004	2005	2006	2007	-	2008	2009
3	2010	2011	-	2012	2013	2014	2015
4		2016	2017	2018	2019	-	2020
5	2021	2022	2023		2025	2026	2027

Note that each column has 4 years in it. This means in a period of 28 years, each Doomsday occurs 4 times, so that each Doomsday occurs 4/28 = 1/7 of the time. Further, this pattern repeats indefinitely. The years 2028 - 2055 will show exactly the same pattern of Doomsdays, as well as 2056-2083 and so forth.

This implies that on the average, years with a March with 5 Fridays, Saturdays, and Sundays in it occur every 7 years on the average, not 823 years. It does not occur every 7th year, but in a more complicated pattern, but it is still once every 7 years on the average. This is slightly off because of the Gregorian rule, which alters this sequence of years every time a century year that is not a leap year is encountered.

The questions I have at this point are: Who started this 823 thing to begin with, and why did so many people believe it?