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?

 

 

 

Find the Right Date

Many times people ask questions about dates, since dates are not readily obvious. Some of these questions might be:

In what years is the Fourth of July on a Wednesday?
In what months this year is the 13th of the month on a Friday?
When is the third Thursday of August, 2013?
On what day of the week was 1969 July 20?
On what day of the month is the next Sunday after 2013 March 27?

Most people when confronted with these questions will consult a calendar. However there are several ways of finding it without using a calendar or notes. One of these is based on this formula:

W = Y + M + D (mod 7)

where W is the number of the day of the week, Y is the Doomsday Number of the year, M is the number of the month,D is the day of the month (e.g., on the 4th of the month, D = 4), and "mod 7" means we throw out all 7s in the calculation, so for example, 19 = 5, because their difference, 14 = two 7s, which we throw out. In this formula we always take remainders on division by 7.

Let's explain each of these in detail. Y is the Doomsday number of the year, and is based on the Doomsday Rule, which shows how to compute the Doomsday of a year CA, where C is the century, and A is the year of the century; e.g., for 2013, C = 20 and A = 13.

1. Divide A by 12, getting a quotient and remainder. Divide the remainder by 4 and take the quotient. Add both quotients together and take the remainder upon division by 7. Example, 2018. 19 divided by 12 yields 1 with a remainder of 6. 6 divided by 4 gives a quotient of 2. 1 + 6 + 2 = 9, which leaves a remainder of 2 when divided by 7.
2. Divide C by 4 giving a remainder of r. Then the Doomsday number is 3 - 2 x r. If this is negative, add 7 over and over again until it is positive. For 2013, 20 divided by 4 yields a remainder of 0, so compute 3 - 2 x 0 = 3. Or as John Horton Conway puts it: for Quadrennial centuries take Tues., and for every century after that, go back by twos.

The numbers correspond to the days of the week according to this table:

Sunday	1
Monday	2
Tuesday	3
Wednesday	4
Thursday	5
Friday	6
Saturday	7

For 2013, this yields Thursday, so the year number Y = 5.

M is the month number. That is given by this table:

0	February(common year),March,November
1	June
2	September,December
3	April,July,(leap year)January
4	January(common year),October
5	May
6	August,(leapyear)February

It is best to remember the 12-number string (we do something like this all the time with telephone numbers and credit card numbers) 400-351-362-402, and change 400 to 360 in leap years. Or you can remember the 7 categories of months:

Here is an example date. 1969 July 20. 69 divided by 12 yields 5 with a remainder of 9; 9 divided by 4 gives a quotient of 2. 5 + 9 + 2 = 16; remaider after dividing by 7: 2. The 1900s are 3 centuries past the Quadrennium (1600s), so take 2 x 3 = 6 days before Tuesday and get Wednesday. 2 days past Wednesday is Friday, so the Doomsday of 1969 is Friday, and the Doomsday Number Y = 6. The month number M of common year July is 3, and the day number is 20. So we compute the weekday number as:

W = Y + M + D (mod 7)

= 6 + 3 + 20 = 29 = 1 after throwing out the 7s

So 1969 July 20 was a Sunday. That's the day Neil Armstrong landed on the Moon, so the Moon is a Sunday World.

The equation W = Y + M + D (mod 7) has four variables. If any three are known, the fourth can be computed. This leads to four different types of date questions.

Type 1. Y is unknown. We know the month and days of the month and week and want to find the year. A typical question might be "In which year is April 4 on a Tuesday?" The equation is Y = W - M - D (mod 7). A real world example would be if someone found an old document marked "Tuesday, April 4" but the year doesn't appear anywhere. Then this formula gives the Doomsday number of the year. It then remains to search years to find which one has this Doomsday. One can count through the years, adding 1 for each common year and 2 for each leap year.

Type 2. M is unknown. We know the year, day and day of week but want to find the month. An amusing example is to find out when all the Friday the 13ths of the year occur. The equation is M = W - Y - D (mod 7).

Type 3. D is unknown. We know the year and month, and the weekday, but not the day of the month. This appears commonly in finding out when meetings and certain holidays occur.The formula is D = W - Y - M (mod 7). For example, in finding out when the third Thursday of August 2013 is, 2013's Doomsday is Thursday, which has value 5, and August's value is 6. So compute D = W - Y - M = 5 - 5 - 6 = -6 = 1, when we add 7. This just tells us that August 1 is Thursday, and so is the first Thursday of the month. The third Thursday is obtained by adding 14 to this: August 15.

Type 4.W is unknown. We know the date but not the day of the week. The formula is W = Y + M + D (mod 7).

Most date questions fall in these four types. One that doesn't is "What day of the month is the next Sunday after 2013 March 27?". One could ask what day of the week is 2013 March 27. That's a Type 4 question. Then one can figure from that when the next Sunday is.

The things to remember with this system are the Doomsday Rule and the month numbers. Then you can tell what day of the week any date is, as well as answer many other date questions.