Doomsday rule Totally Explained

Everything about The Doomsday Rule totally explained

The Doomsday rule or Doomsday algorithm is a way of calculating the day of the week of a given date. It provides a perpetual calendar since the Gregorian calendar moves in cycles of 400 years.
The algorithm for mental calculation was invented by John Conway. It takes advantage of the fact that within any calendar year, the days of 4/4, 6/6, 8/8, 10/10, 12/12, and the last day of February always occur on the same day of week - the so-called "doomsday" (and furthermore that other months have "doomsday" on the pairs 5/9 and 9/5 as well as 7/11 and 11/7, which can be remembered using a simple mnemonic). This applies to both the Gregorian calendar A.D. and the Julian calendar, but note that for the Julian calendar the Doomsday of a year is a weekday that's usually different from that for the Gregorian calendar.
The algorithm has three steps: namely, finding the anchor day for the century, finding a year's Doomsday, and finding the day of week of the day in question.

Finding a year's Doomsday

We first take the anchor day for the century. For the purposes of the Doomsday rule, a century starts with a 00 year and ends with a 99 year. The following table shows the anchor day of centuries 1800–1899, 1900–1999, 2000–2099 and 2100–2199.

Century	Anchor day	Mnemonic
1800–1899	Friday	-
1900–1999	Wednesday	We-in-dis-day (most living people were born in that century)
2000–2099	Tuesday	Y-Tue-K (Y2K was at the head of this century)
2100–2199	Sunday	20-One-day is Sunday (2100 is the start of the next century)

Since in the Gregorian calendar there are 146097 days, or exactly 20871 seven-day weeks, in 400 years, the anchor day repeats every four centuries. For example, the anchor day of 1700–1799 is the same as the anchor day of 2100–2199, for example Sunday.
Next, we find the year's Doomsday. To accomplish that according to Conway:

Divide the year's last two digits (call this y) by 12 and let a be the floor of the quotient.
Let b be the remainder of the same quotient.
Divide that remainder by 4 and let c be the floor of the quotient.
Let d be the sum of the three numbers (d = a + b + c). (It is again possible here to divide by seven and take the remainder. This number is equivalent, as it must be, to the sum of the last two digits of the year taken collectively plus the floor of those collective digits divided by four.)
Count forward the specified number of days (d or the remainder of d/7) from the anchor day to get the year's Doomsday.

left(and for some other contemporary years:

2004	Sunday
2005	Monday
2006	Tuesday
2007	Wednesday
2008	Friday
2009	Saturday
2010	Sunday
2011	Monday

Correspondence with dominical letter

Doomsday is related to the dominical letter of the year as follows.

Dominical letter	Doomsday
A or BA	Tuesday
B or CB	Monday
C or DC	Sunday
D or ED	Saturday
E or FE	Friday
F or GF	Thursday
G or AG	Wednesday

Examples

Example 1 (2006)

Suppose you want to know which day of the week Christmas Day of 2006 was. In the year 2006, Doomsday was Tuesday. (The century's anchor day was Tuesday, and 2006's Doomsday was seven days beyond and was a Tuesday.) This means that December 12, 2006 was a Tuesday. December 25, being thirteen days afterwards, fell on a Monday.

Example 2 (other years of the 21st century)

Suppose that you want to find the day of week that the September 11, 2001 attacks on the World Trade Center occurred. The anchor was Tuesday, and one day beyond was Wednesday. September 5 was a Doomsday, and September 11, six days later, fell on a Tuesday.

Example 3 (other centuries)

Suppose that you want to find the day of week that the American Civil War broke out at Fort Sumter, which was April 12, 1861. The anchor day for the century was 99 days after Thursday, or, in other words, Friday (calculated as (19+1)*5+floor(19/4); or just look at the chart, above, which lists the century's anchor days). The digits 61 gave a displacement of six days so Doomsday was Thursday. Therefore, April 4 was Thursday so April 12, eight days later, was a Friday.

Julian calendar

The Gregorian calendar accurately lines up with astronomical events such as solstices. In 1582 this modification of the Julian calendar was first instituted. In order to correct for calendar drift, 10 days were skipped, so Doomsday moved back 10 days (for example 3 days): Thursday 4 October (Julian, Doomsday is Wednesday) was followed by Friday 15 October (Gregorian, Doomsday is Sunday). The table includes Julian calendar years, but the algorithm is for the Gregorian and proleptic Gregorian calendar only.
Note that the Gregorian calendar wasn't adopted simultaneously in all countries, so for many centuries, different regions used different dates for the same day. More information can be found in the Gregorian Calendar article.

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