The Doomsday rule, Doomsday algorithm or Doomsday method is an algorithm of determination of the day of the week for a given date. It provides a perpetual calendar because the Gregorian calendar moves in cycles of 400 years. The algorithm for mental calculation was devised by John Conway in 1973,^[1][2] drawing inspiration from Lewis Carroll's perpetual calendar algorithm.^[3][4][5] It takes advantage of each year having a certain day of the week upon which certain easy-to-remember dates, called the doomsdays, fall; for example, the last day of February, 4/4, 6/6, 8/8, 10/10, and 12/12 all occur on the same day of the week in any year.

Applying the Doomsday algorithm involves three steps: determination of the anchor day for the century, calculation of the anchor day for the year from the one for the century, and selection of the closest date out of those that always fall on the doomsday, e.g., 4/4 and 6/6, and count of the number of days (modulo 7) between that date and the date in question to arrive at the day of the week. The technique applies to both the Gregorian calendar and the Julian calendar, although their doomsdays are usually different days of the week.

The algorithm is simple enough that it can be computed mentally. Conway could usually give the correct answer in under two seconds. To improve his speed, he practiced his calendrical calculations on his computer, which was programmed to quiz him with random dates every time he logged on.^[6]

Doomsday's anchor day for the current year in the Gregorian calendar (2024) is Thursday. For some other contemporary years:

The table is filled in horizontally, skipping one column for each leap year. This table cycles every 28 years, except in the Gregorian calendar on years that are a multiple of 100 (such as 1900 and 2100 which are not leap years) that are not also a multiple of 400 (like 2000 which is still a leap year). The full cycle is 28 years (1,461 weeks) in the Julian calendar, 400 years (20,871 weeks) in the Gregorian calendar.

One can find the day of the week of a given calendar date by using a nearby doomsday as a reference point. To help with this, the following is a list of easy-to-remember dates for each month that always land on the doomsday.

The last day of February is always a doomsday. For January, January 3 is a doomsday during common years and January 4 a doomsday during leap years, which can be remembered as "the 3rd during 3 years in 4, and the 4th in the 4th year". For March, one can remember either Pi Day or "March 0", the latter referring to the day before March 1, i.e. the last day of February.

For the months April through December, the even numbered months are covered by the double dates 4/4, 6/6, 8/8, 10/10, and 12/12, all of which fall on the doomsday. The odd numbered months can be remembered with the mnemonic "I work from 9 to 5 at the 7-11", i.e., 9/5, 7/11, and also 5/9 and 11/7, are all doomsdays (this is true for both the Day/Month and Month/Day conventions).^[7]

Several well-known dates, such as Independence Day in United States, Boxing Day, and Valentine's Day in common years, also fall on doomsdays every year. The chart below includes only the mnemonics covered in the sources listed.

Since the doomsday for a particular year is directly related to weekdays of dates in the period from March through February of the next year, common years and leap years have to be distinguished for January and February of the same year.

January and February can be treated as the last two months of the previous year.

Since this algorithm involves treating days of the week like numbers modulo 7, John Conway suggested thinking of the days of the week as "Noneday" or "Sansday" (for Sunday), "Oneday", "Twosday", "Treblesday", "Foursday", "Fiveday", and "Six-a-day" in order to recall the number-weekday relation without needing to count them out in one's head.^[10]

There are some languages, such as Slavic languages, Chinese, Greek, Portuguese, Galician and Hebrew, that base some of the names of the week days in their positional order. The Slavic and Chinese agree with the table above; the other languages mentioned count from Sunday as day one.

First take the anchor day for the century. For the purposes of the doomsday rule, a century starts with '00 and ends with '99. The following table shows the anchor day of centuries 1600–1699, 1700–1799, 1800–1899, 1900–1999, 2000–2099, 2100–2199 and 2200–2299.

For the Gregorian calendar:

Mathematical formula

5 × (c mod 4) mod 7 + Tuesday = anchor.

Algorithmic

Let r = c mod 4

if r = 0 then anchor = Tuesday

if r = 1 then anchor = Sunday

if r = 2 then anchor = Friday

if r = 3 then anchor = Wednesday

For the Julian calendar:

6c mod 7 + Sunday = anchor.

Note: ${\displaystyle c={\biggl \lfloor }{{\text{year}} \over 100}{\biggr \rfloor }}$.

Next, find the year's anchor day. To accomplish that according to Conway:^[11]

1. Divide the year's last two digits (call this y) by 12 and let a be the floor of the quotient.
2. Let b be the remainder of the same quotient.
3. Divide that remainder by 4 and let c be the floor of the quotient.
4. 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 y plus the floor of y divided by four.)
5. Count forward the specified number of days (d or the remainder of d/7) from the anchor day to get the year's one.

${\displaystyle {\begin{matrix}\left({\left\lfloor {\frac {y}{12}}\right\rfloor +y{\bmod {1}}2+\left\lfloor {\frac {y{\bmod {1}}2}{4}}\right\rfloor }\right){\bmod {7}}+{\rm {{anchor}={\rm {Doomsday}}}}\end{matrix}}}$

For the twentieth-century year 1966, for example:

${\displaystyle {\begin{matrix}\left({\left\lfloor {\frac {66}{12}}\right\rfloor +66{\bmod {1}}2+\left\lfloor {\frac {66{\bmod {1}}2}{4}}\right\rfloor }\right){\bmod {7}}+{\rm {Wednesday}}&=&\left(5+6+1\right){\bmod {7}}+{\rm {Wednesday}}\\\ &=&{\rm {Monday}}\end{matrix}}}$

As described in bullet 4, above, this is equivalent to:

${\displaystyle {\begin{matrix}\left({66+\left\lfloor {\frac {66}{4}}\right\rfloor }\right){\bmod {7}}+{\rm {Wednesday}}&=&\left(66+16\right){\bmod {7}}+{\rm {Wednesday}}\\\ &=&{\rm {Monday}}\end{matrix}}}$

So doomsday in 1966 fell on Monday.

Similarly, doomsday in 2005 is on a Monday:

${\displaystyle \left({\left\lfloor {\frac {5}{12}}\right\rfloor +5{\bmod {1}}2+\left\lfloor {\frac {5{\bmod {1}}2}{4}}\right\rfloor }\right){\bmod {7}}+{\rm {{Tuesday}={\rm {Monday}}}}}$

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

Look up the table below for the dominical letter (DL).

For the year 2024, the dominical letter is BA + 2 = GF.

* In leap years the nth doomsday is in ISO week n. In common years the day after the nth doomsday is in week n. Thus in a common year the week number on the doomsday itself is one less if it is a Sunday, i.e. in a common year starting on Friday.

For computer use, the following formulas for the anchor day of a year are convenient.

For the Gregorian calendar:

${\displaystyle {\mbox{anchor day}}={\mbox{Tuesday}}+y+\left\lfloor {\frac {y}{4}}\right\rfloor -\left\lfloor {\frac {y}{100}}\right\rfloor +\left\lfloor {\frac {y}{400}}\right\rfloor ={\mbox{Tuesday}}+5\times (y{\bmod {4}})+4\times (y{\bmod {1}}00)+6\times (y{\bmod {4}}00)}$

For example, the doomsday 2009 is Saturday under the Gregorian calendar (the currently accepted calendar), since

${\displaystyle {\mbox{Saturday (6)}}{\bmod {7}}={\mbox{Tuesday (2)}}+2009+\left\lfloor {\frac {2009}{4}}\right\rfloor -\left\lfloor {\frac {2009}{100}}\right\rfloor +\left\lfloor {\frac {2009}{400}}\right\rfloor }$

As another example, the doomsday 1946 is Thursday, since

${\displaystyle {\mbox{Thursday (4)}}{\bmod {7}}={\mbox{Tuesday (2)}}+1946+\left\lfloor {\frac {1946}{4}}\right\rfloor -\left\lfloor {\frac {1946}{100}}\right\rfloor +\left\lfloor {\frac {1946}{400}}\right\rfloor }$

For the Julian calendar:

${\displaystyle {\mbox{anchor day}}={\mbox{Sunday}}+y+\left\lfloor {\frac {y}{4}}\right\rfloor ={\mbox{Sunday}}+5\times (y{\bmod {4}})+3\times (y{\bmod {7}})}$

The formulas apply also for the proleptic Gregorian calendar and the proleptic Julian calendar. They use the floor function and astronomical year numbering for years BC.

For comparison, see the calculation of a Julian day number.

Since in the Gregorian calendar there are 146,097 days, or exactly 20,871 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, i.e. Sunday.

The full 400-year cycle of doomsdays is given in the adjacent table. The centuries are for the Gregorian and proleptic Gregorian calendar, unless marked with a J for Julian. The Gregorian leap years are highlighted.

Negative years use astronomical year numbering. Year 25BC is −24, shown in the column of −100J (proleptic Julian) or −100 (proleptic Gregorian), at the row 76.

A leap year with Monday as doomsday means that Sunday is one of 97 days skipped in the 400-year sequence. Thus the total number of years with Sunday as doomsday is 71 minus the number of leap years with Monday as doomsday, etc. Since Monday as doomsday is skipped across 29 February 2000 and the pattern of leap days is symmetric about that leap day, the frequencies of doomsdays per weekday (adding common and leap years) are symmetric about Monday. The frequencies of doomsdays of leap years per weekday are symmetric about the doomsday of 2000, Tuesday.

The frequency of a particular date being on a particular weekday can easily be derived from the above (for a date from 1 January – 28 February, relate it to the doomsday of the previous year).

For example, 28 February is one day after doomsday of the previous year, so it is 58 times each on Tuesday, Thursday and Sunday, etc. 29 February is doomsday of a leap year, so it is 15 times each on Monday and Wednesday, etc.

• Ordinal date
• Computus – Gauss algorithm for Easter date calculation
• Zeller's congruence – An algorithm (1882) to calculate the day of the week for any Julian or Gregorian calendar date.
• Mental calculation