Perpetual CalendarDate: 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/ |
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