Perpetual Calendar

Date: 10/21/98 at 17:57:08
From: chris
Subject: Perpetual calendar

How do you figure the perpetual calendar?

Date: 10/22/98 at 10:34:30
From: Doctor Rob
Subject: Re: Perpetual calendar

The perpetual calendar is rather complicated, I'm afraid, so this 
answer is quite long. For that, I apologize, but I hope that this will 
answer your question fully.

A monthly calendar tells you on what day of the week each date falls.
There are only seven different monthly calendars, since the first of 
the month can occur on any of the seven days of the week. You can 
label them 1 through 7 if the first of the month falls on Sunday, 
Monday, ..., Saturday, respectively. 

Example:  Monthly calendar 5 is:

    S  M  T  W  T  F  S
    -  -  -  -  1  2  3
    4  5  6  7  8  9 10
   11 12 13 14 15 16 17
   18 19 20 21 22 23 24
   25 26 27 28 29 30 31

A yearly calendar tells you which monthly calendar to use each month.  
It consists of a list of 12 numbers from 1 to 7, corresponding to the 
12 months, and the monthly calendar to use for each. There are only 14 
yearly calendars, since January 1 can occur on any of the seven days 
of the week, and the year can either be an ordinary year or a leap 
year. All you have to do is tell which of the 14 calendars to use in 
any given year. These can be labeled with the letters from A through 
N, as given below.

Once you know which day January 1 falls upon, you can readily figure 
out which day February 1 falls upon, since it is 3 days later in the 
week. If January 1 falls on Sunday, then February 1 falls on Wednesday.  
March 1 falls on the same day that February 1 does in ordinary years, 
but one day later in leap years. April 1 is 3 days later in the week 
than March 1; May 1 is 2 days later in the week than April 1; and so 
it continues. 

When the current month has 31 days, the next month starts 3 days later. 
When the current month has 30 days, the next month starts 2 days later. 
When the current month has 29 days, the next month starts 1 day later. 
When the current month has 28 days, the next month starts 0 days later. 

These things happen because 31 = 4*7 + 3, 30 = 4*7 + 2, 29 = 4*7 + 1, 
and 28 = 4*7 + 0.

Then the yearly calendars look like this:

Ordinary years:

       J  F  M  A  M  J  J  A  S  O  N  D
   A:  1  4  4  7  2  5  7  3  6  1  4  6
   B:  2  5  5  1  3  6  1  4  7  2  5  7
   C:  3  6  6  2  4  7  2  5  1  3  6  1
   D:  4  7  7  3  5  1  3  6  2  4  7  2
   E:  5  1  1  4  6  2  4  7  3  5  1  3
   F:  6  2  2  5  7  3  5  1  4  6  2  4
   G:  7  3  3  6  1  4  6  2  5  7  3  5

Leap years:

   H:  1  4  5  1  3  6  1  4  7  2  5  7
   I:  2  5  6  2  4  7  2  5  1  3  6  1
   J:  3  6  7  3  5  1  3  6  2  4  7  2
   K:  4  7  1  4  6  2  4  7  3  5  1  3
   L:  5  1  2  5  7  3  5  1  4  6  2  4
   M:  6  2  3  6  1  4  6  2  5  7  3  5
   N:  7  3  4  7  2  5  7  3  6  1  4  6

In the Gregorian calendar (which we use today), leap years fall 
whenever the year is divisible by 4, except when the year is divisible 
by 100, but including the years divisible by 400. In the Julian 
calendar (which was formerly used), leap years fall whenever the year 
is divisible by 4.

After an ordinary year, January 1 falls one day later on that year than 
it did in the year preceding. After a leap year, it falls two days 
later. That is because 365 = 52*7 + 1, and 366 = 52*7 + 2. For example, 
in 1995, January 1 fell on a Sunday, so in 1996 it fell on a Monday, 
and in 1997 it fell on a Wednesday (Tuesday was skipped because 1996 
was a leap year).

That leads to a 28-year cycle for the use of the yearly calendars. In 
the year 1753, the yearly calendar was B. Starting with that year, the 
28-year cycle is:

   B C D L G A B J E F G H C D E M A B C K F G A I D E F N

That cycle corresponds to the years 1753-1780, and then the first part 
of the cycle also corresponds to the years 1781-1799. According to this 
cycle, the next year, 1800, should use K, but 1800 is not a leap year, 
so the actual calendar to use is D. The special leap year rule causes a 
disruption of the cycle at this point. Then the cycling is resumed with 
E F G H C D E M ... N, which covers the years 1801-1820. Two more 
cycles cover 1821-1848 and 1849-1876. Continuing the cycle works 
through 1899; then 1900, which the cycle said should use letter I, is 
not a leap year, so it must use letter B instead. Then the cycling is 
resumed with C D E M A B C K ... N, through 1916. Now six more cycles 
will cover the years to 2084, and beginning another cycle will take us 
to 2099. 2100 is not a leap year, so instead of using M, use F, then 
continue on the cycle from G A B J ... N, covering years 2101-2124.

When there is a centurial ordinary year, substitute for the leap-year
letter the ordinary-year letter 7 back in the alphabet, back up 12 
letters (or go forward 16 letters) on the cycle of 28, and continue.

I started with 1753 because the English-speaking world converted from 
the Julian to the Gregorian calendar in 1752. To do that, the day after
September 2, 1752 was decreed to be September 14, 1752. After that 
date, the calendar for 1752 was G or N, and before that date it was K.  
The previous beginning of the cycle was in 1733, the one before that 
in 1705, then 1677, 1649, 1621, 1593, 1565, 1537, 1509, 1481, 1453, 
1425, 1397, 1369, 1341, 1313, 1285, 1257, 1229, 1201, and so on.  
There are no centurial ordinary years in the Julian calendar, so the 
cycle is never interrupted before 1752.

Confusing the matter is the fact that during much of the life of the 
Julian calendar, the first day of the year was decreed to be Lady Day, 
March 25th, not January 1st. That is another matter, however. It does 
not apply for the Gregorian calendar, so we can ignore it for dates 
after September 14, 1752.

I think you get the idea. 1998 uses yearly calendar E, and October 1998 
uses monthly calendar 5, so October 1, 1998 was a Thursday, and so is 
today, October 22, 1998.

If you think this is complicated, you are right. Don't ask about how 
to figure the date of Easter, because it is worse!

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/   

Date: 10/22/98 at 10:37:35
From: Doctor Sarah
Subject: Re: Perpetual calendar

Hi Chris -

Wow - that's a complicated question.

A perpetual calendar can be used to answer some interesting questions, 
like what day of the week July 4 will fall on in whatever year you 
want to know about, or what months will have a Friday the 13th, or in 
what years your birthday will fall on a Saturday.  

There's a Web page by Ron Knott for you to look at that shows a 
perpetual calendar and explains how to use it:

 http://www.mcs.surrey.ac.uk/Personal/R.Knott/PerpetualCalendar.html   

Here's the calendar:

                          M  O  N  T  H
                  Jan  Apr  Sep  Jun  Feb  Aug  May  /
                  Oct  July Dec       Mar  Feb      /
   DAY OF MONTH        Jan            Nov          /     DAY OF WEEK
   1  8 15 22 29   A    B    C    D    E    F    G      Monday
   2  9 16 23 30   G    A    B    C    D    E    F      Tuesday
   3 10 17 24 31   F    G    A    B    C    D    E      Wednesday
   4 11 18 25      E    F    G    A    B    C    D      Thursday
   5 12 19 26      D    E    F    G    A    B    C      Friday
   6 13 20 27      C    D    E    F    G    A    B      Saturday
   7 14 21 28      B    C    D    E    F    G    A      Sunday
                 /      68   69   70   71        72  
                /  73   74   75        76   77   78 
               /   79        80   81   82   83           1900s
                   84   85   86   87        88   89
                   90   91        92   93   94   95
                        96   97   98   99        00      ......
                   01   02   03        04   05   06
                   07        08   09   10   11           2000s
                   12   13   14   15        16   17 
                   18   19        20   21   22   23
                     Y E A R    o f     C E N T U R Y

Have fun with it!

- Doctor Sarah, The Math Forum
  http://mathforum.org/dr.math/