Doomsday rule

Doomsday's anchor day for the current year in the Gregorian calendar (2020) is Saturday. 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 (like 1900 which is not a leap year) 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.

As mentioned above, the last day of February defines the 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 the pseudo-date "March 0", which refers 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.[7]

Several common holidays are also on doomsday. The chart below includes only dates covered by the mnemonics 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 suggests thinking of the days of the week as "Noneday"; or as "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.

There are some languages, such as Slavic languages, Greek, Portuguese, Galician, Hebrew and Chinese, that base some of the names of the week days in their positional order.

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 1800–1899, 1900–1999, 2000–2099 and 2100–2199.

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: c = ⌊year/100⌋.

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

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 the sum of the last two digits of the year taken collectively plus the floor of those collective digits 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:

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

Why it works

Doomsday rule

The doomsday's anchor day calculation is effectively calculating the number of days between any given date in the base year and the same date in the current year, then taking the remainder modulo 7. When both dates come after the leap day (if any), the difference is just 365y + y/4 (rounded down). But 365 equals 52 × 7 + 1, so after taking the remainder we get just

${\displaystyle \left(y+\left\lfloor {\frac {y}{4}}\right\rfloor \right){\bmod {7}}.}$

This gives a simpler formula if one is comfortable dividing large values of y by both 4 and 7. For example, we can compute

${\displaystyle \left(66+\left\lfloor {\frac {66}{4}}\right\rfloor \right){\bmod {7}}=(66+16){\bmod {7}}=82{\bmod {7}}=5}$

which gives the same answer as in the example above.

Where 12 comes in is that the pattern of (y + ⌊y/4⌋) mod 7 almost repeats every 12 years. After 12 years, we get (12 + 12/4) mod 7 = 15 mod 7 = 1. If we replace y by y mod 12, we are throwing this extra day away; but adding back in ⌊y/12⌋ compensates for this error, giving the final formula.

The "odd + 11" method

A simple flowchart showing the Odd+11 method.

A simpler method for finding the year's anchor day was discovered in 2010 by Chamberlain Fong and Michael K. Walters,[9] and described in their paper submitted to the 7th International Congress on Industrial and Applied Mathematics (2011). Called the "odd + 11" method, it is equivalent[9] to computing

${\displaystyle \left(y+\left\lfloor {\frac {y}{4}}\right\rfloor \right){\bmod {7}}}$.

It is well suited to mental calculation, because it requires no division by 4 (or 12), and the procedure is easy to remember because of its repeated use of the "odd + 11" rule.

Extending this to get the anchor day, the procedure is often described as accumulating a running total T in six steps, as follows:

1. Let T be the year's last two digits.
2. If T is odd, add 11.
3. Now let T = T/2.
4. If T is odd, add 11.
5. Now let T = 7 − (T mod 7).
6. Count forward T days from the century's anchor day to get the year's anchor day.

Applying this method to the year 2005, for example, the steps as outlined would be:

1. T = 5
2. T = 5 + 11 = 16 (adding 11 because T is odd)
3. T = 16/2 = 8
4. T = 8 (do nothing since T is even)
5. T = 7 − (8 mod 7) = 7 − 1 = 6
6. Doomsday for 2005 = 6 + Tuesday = Monday

The explicit formula for the odd+11 method is:

${\displaystyle 7-\left[{\frac {y+11(y\,{\bmod {2}})}{2}}+11\left({\frac {y+11(y\,{\bmod {2}})}{2}}{\bmod {2}}\right)\right]{\bmod {7}}}$.

Although this expression looks daunting and complicated, it is actually simple[9] because of a common subexpression y + 11(y mod 2)/2 that only needs to be calculated once.

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 2017, the dominical letter is A - 0 = A.

* 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 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, 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

Wikimedia Commons has media related to Doomsday rule.

• Encyclopedia of Weekday Calculation by Hans-Christian Solka, 2010
• Doomsday calculator that also "shows all work"
• World records for mentally calculating the day of the week in the Gregorian Calendar
• National records for finding Calendar Dates
• World Ranking of Memoriad Mental Calendar Dates (all competitions combined)
• What is the day of the week, given any date?
• Doomsday Algorithm
• Finding the Day of the Week
• Poem explaining the Doomsday rule at the Wayback Machine (archived October 18, 2006)