# The Doomsday Rule for Fortnights

The Doomsday rule was invented by John Horton Conway in the 1970's and is given in a book by him and Richard K. Guy entitled "Winning Ways for your Mathematical Plays, Volume II, Games in Particular", p. 795-800. What Conway shows is that remembering what day of the week a certain date is does not have to be that difficult, if you think of it as a game. What's more, Mr. Conway also throws in rules for Easter, the moon's age, and the Jewish Calendar.

This rule is well described in several pages on the World Wide Web, including the Doomsday Algorithm for Day of Week of Rudy Limeback. Go to this page and learn the Doomsday Rule for finding the day of the week. After some practice you will amaze your friends.

This rule does not solve all calendar-related problems. In many instances, days are organized into two-week fortnights. For instance, the US Government pays its workers according to a two-week schedule. Some companies operate according to a two-week plan. There is a recycling company near where I live that picks up every two weeks. Since it may be important to know when you're paid, a rule is needed for fortnights. The Doomsday rule will tell you the day of the week but will not tell you which week of the fortnight it is.

Before describing a Doomsday rule for fortnights, one must agree to a standard fortnight. The one I will select is this:

### 1998 February

 Left Sunday LU 1 Left Monday LM 2 Left Tuesday LT 3 Left Wednesday LW 4 Left Thursday LR 5 Left Friday LF 6 Left Saturday LA 7 Right Sunday RU 8 Right Monday RM 9 Right Tuesday RT 10 Right Wednesday RW 11 Right Thursday RR 12 Right Friday RF 13 Right Saturday RU 14

In other words, the fortnight starts out with a left week and ends with a right week. I have indicated my abbreviations for the days of the fortnight in the middle column, and an ordinal number for each day in the third column. These also by coincidence are also the days of the month for 1998 February!

A full Calendar for 2003 in Fortnights can be obtained by clicking these words, or you can get a 2004 Calendar or a 2005 Calendar.

The rule for Doomsday for fortnights is as follows

1. Octocentennium and Century

An octocentennium is a period of 800 years. A period of 800 years contains 292,194 days, a number evenly divisible by 14, the number of days in a fortnight. Therefore, the pattern of days in a fortnight will repeat each octocentennium. The Conway rule starts out with a quadricentennium (400 years), but the number of days in a quadricentennium is 146,097 which is clearly not divisible by 14.

The rule is (starting at 1600, the start of the current octocentennium):

 1600 Right Tuesday 1700 Right Sunday 1800 Left Friday 1900 Left Wednesday 2000 Left Tuesday 2100 Left Sunday 2200 Right Friday 2300 Right Wednesday

In other words, 1600 is Tues., and you go back by twos, as the poem by Conway says. Note the glitch at 2000. This is not the 2000 bug in action. This results because 2000 is a leap year. The rule is to go back two if the century year is common and to go back one if it is leap. Century years are common years unless they are divisible by 400.

Let's take an example, 1998 February 14, Valentine's day. 1998 is in the 1900's, so take the Doomsday for 1900, which is Left Wednesday.

2. Year.

Now that you've found the Doomsday for the century, it now remains to find the Doomsday for the year. Rudy Limeback observes that the Doomsday advances by one day every twelve years, for Days of the Week. The reason is given below:

 LU LM LT LW LR LF LA RU RM RT RW RT RF RA - - - 1900 1901 1902 1903 - 1904 1905 1906 1907 - 1908 1909 1910 1911 - 1912 1913 1914 1915 - 1916 1917 1918 1919 -
Note that when we advance from 1900 to 1912, the Doomsday for the Week advances by 1. Note even further that we are lucky enough so that the Doomsday for the Fortnight also advances by 1 each 12 years. In fact, the Fortnight Calendar advances by 15 days every 12 years.

Therefore the Doomsday rule for fortnights is exactly the same as the one for weeks! However you can't throw out sevens. Clearly for this rule to work one must throw out fourteens. Therefore the Year rule is:

1. Take the last two digits of the year,
2. Take the Dozens (divide by 12 and take the quotient),
3. The remainder (add it to the remainder),
4. and the Fours in that (divide the remainder by 4 and take the quotient and add it to the total)
5. And throw out the 14's (divide the total by 14 and take the remainder)
6. Take that many days past the Century Doomsday

Take Valentines' Day this year as an example, 1998 February 14.

1. The last two digits are 98.
2. There are 8 dozens in 98.
3. The remainder is 2.
4. There are 0 fours in 2. 8 + 2 + 0 = 10.
5. Throw out the 14's in 10; there are none, get 10
6. 10 days past Left Wednesday is Right Saturday.

Therefore the Doomsday for 1998 is Right Saturday.

3. Months.

The cornerstone of Conway's method is the coincidence that April 4, June 6, August 8, and so forth where the month is even and the day equals the month all occur on the same day. This is Conway's Doomsday. Why does this work? Each of the pairs April-May, June-July, and so forth up to December consist of one 30-day month and one 31-day month. The month number advances by 2 every two months. Add these together, and get 31 + 30 + 2 = 63, which is divisible by 7 but unfortunately not by 14. So Conway's even rule works for weeks, but not for fortnights. However, it works for weeks, so that using his rule will at least get you within one week of the Doomsday.

The rule for Fortnights will start out with the rule for months, and then will modify it in some cases by a week. So here is is:

1. February. Take the last day of the month, either the 28th or the 29th.
2. Even months other than February. The day is the same as the month.
3. Odd months that are long (31 days). Take 4 more than the month.
4. Odd months that are short (30 days). Take 4 less than the month.
5. January. Take January 31 in a common year, and January 32 in a leap year.

Huhhh?? What's January 32? This is what I call an extrapolated calendar date. It simply is the date after January 31, which is usually called February 1. Similarly, March 0 (which appears on Rudy Limeback's page) is the same as February 28.

6. The Snowy Flower Pumpkin rule. If the month is Jan, Feb, May, Jun, Oct, or Nov, then the date we have is the Doomsday. If not, subtract or add 7 days.

It helps to think of a Snowy Flower Pumpkin when applying this rule, a pumpkin with snow on it and a flower sprouting out of it. Snow is January, flower is May as in the Mayflower, and October is Pumpkin for Halloween. A snowy flower pumpkin month is either one of these months or a month AFTER one of these (Feb, Jun, Nov).

Example, 1998 February 14. Doomsday is February 28, since it is February in a common year. February is a Snowy Flower Pumpkin month, since it is the month after the Snow Month of January. So February 28 is the Doomsday, and so is a Right Saturday.

4. Day Rule.

Now go back or forward from the Doomsday in the month to find out what day of the fortnight your date is on.

Example, 1998 February 14. Doomsday is February 28. If you go back a fortnight, you get February 14 exactly. So February 14 is also a Doomsday, and so is a Right Saturday.

That is great, because that means it is at the end of the pay week, so you get paid and can entertain your sweetheart somewhere on that romantic night. It's important to remember Valentine's Day!

Now you know the Doomsday Rule for Fortnights. So you can now amaze your friends by telling them whether their date is in a left or a right week, as well as the day! Happy Calendaring!

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