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For any year, Doomsday is the day of the week on which the last day of February falls. It is also the day of the week of 4/4, 6/6, 8/8, 10/10 and 12/12, as well as 5/9, 9/5, 7/11, and 11/7.

It is a convenient way of characterizing each of the 7 possible calendars for the months March - December. Only for the calendar of January and February a further distinction between common year and leap year is needed. Alternatively the fixed connection with the Doomsday of the previous year is used for these two months.

The Doomsday rule

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 and 12/12 always occur on the same day of week (also that of the last day of February). 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 is 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. Remember, for the purposes of the Doomsday rule, a century starts with a "00" year and ends with a "99" year.

Next we find the year's Doomsday. To accomplish that according to Conway, begin by dividing the year's last two digits (call this y) by 12 and taking the integral value of the quotient (a). Then take the remainder of the same quotient (b). After that, divide that remainder by 4 and take the integral value (c). Finally, determine the sum of the three numbers (add a, b, and c to get d). (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 plus the integral value those digits divided by four.)
Now count forward the specified number of days (d or the remainder of d/7) from the anchor day to get the year's Doomsday.
${\displaystyle \left({\left\lfloor {\frac {y}{12}}\right\rfloor +y{\bmod {1}}2+\left\lfloor {\frac {y{\bmod {1}}2}{4}}\right\rfloor }\right){\bmod {7}}+{\rm {{anchor}={\rm {Doomsday}}}}}$

For the twentieth-century year 1966, for example:
${\displaystyle {\begin{matrix}\left({\left\lfloor {\frac {66}{12}}\right\rfloor +66{\bmod {1}}2+\left\lfloor {\frac {66{\bmod {1}}2}{4}}\right\rfloor }\right){\bmod {7}}+{\rm {Wednesday}}&=&\left(5+6+1\right){\bmod {7}}+{\rm {Wednesday}}\\\ &=&{\rm {Monday}}\end{matrix}}}$
So Doomsday in 1966 fell on Monday.

Similarly, Doomsday in 2005 is on a Monday:
${\displaystyle \left({\left\lfloor {\frac {5}{12}}\right\rfloor +5{\bmod {1}}2+\left\lfloor {\frac {5{\bmod {1}}2}{4}}\right\rfloor }\right){\bmod {7}}+{\rm {{Tuesday}={\rm {Monday}}}}}$

Finding the day of the week of a given calendar date

One can easily find the day of the week of a given calendar date from a nearby Doomsday.

The following days all occur on Doomsday for any given Gregorian or Julian year:

• January 3rd 3 years in 4, ie in common years; January 4th in the 4th year, leap year
• The last day of February - the 28th in common years, the 29th if it's a leap year
• "March 0" (a backwards extension of the calendar, equivalent to the last day of February.)
• April 4th
• May 9th
• June 6th
• July 11th
• August 8th
• September 5th
• October 10th
• November 7th
• December 12th

The dates listed above were chosen to be easy to remember; the ones for even months are simply doubles, 4/4, 6/6, 8/8, 10/10, and 12/12. Four of the odd month dates (5/9, 9/5, 7/11, and 11/7) are recalled using the mnemonic "I work from 9 to 5 at the 7-11."

For dates in March, March 7 falls on Doomsday, but the pseudodate "March 0" is easier to remember, as it is necessarily the same as the last day of February.

Doomsday is directly related to weekdays of dates in the period from March through February of the next year. For January and February of the same year, common years and leap years have to be distinguished.

Overview of all Doomsdays

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.

Formula for the Doomsday of a year

For computer use the following formulas for the Doomsday of a year are convenient.

For the Gregorian calendar:

${\displaystyle {\mbox{Doomsday}}={\mbox{Tuesday}}+y+\left\lfloor {\frac {y}{4}}\right\rfloor -\left\lfloor {\frac {y}{100}}\right\rfloor +\left\lfloor {\frac {y}{400}}\right\rfloor }$

For the Julian calendar:

${\displaystyle {\mbox{Doomsday}}={\mbox{Sunday}}+y+\left\lfloor {\frac {y}{4}}\right\rfloor }$

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.

Compare Julian day#Calculation.

Cycle

The full 400-year cycle of Doomsdays is given in the following table. The centuries are for the Gregorian and proleptic Gregorian calendar, unless marked with a J for Julian (for the latter not all centuries are shown, for the missing ones it is easy to interpolate). The Gregorian leap years are widened.

            ,-----,-----,-----,-----,
            |-200J|  00J| 200J| 400J|
            | 500J| 700J| 900J|1100J|
            |1200J|1400J|1600J|1800J|
            |1900J|2100J|2300J|2500J|
            |     |     |     |     |
            |-400 |-300 |-200 |-100 |
            |  00 | 100 | 200 | 300 |
            | 400 | 500 | 600 | 700 |
            | 800 | 900 |1000 |1100 |
            |1200 |1300 |1400 |1500 |
            |1600 |1700 |1800 |1900 |
            |2000 |2100 |2200 |2300 |
,-----------+-----+-----+-----+-----|
|         00| T U | SU  | FR  | WE  |
|-----------+-----+-----+-----+-----|
|85 57 29 01| WE  | MO  | SA  | TH  |
|86 58 30 02| TH  | TU  | SU  | FR  |
|87 59 31 03| FR  | WE  | MO  | SA  |
|88 60 32 04| S U | F R | W E | M O |
|-----------+-----+-----+-----+-----|
|89 61 33 05| MO  | SA  | TH  | TU  |
|90 62 34 06| TU  | SU  | FR  | WE  |
|91 63 35 07| WE  | MO  | SA  | TH  |
|92 64 36 08| F R | W E | M O | S A |
|-----------+-----+-----+-----+-----|
|93 65 37 09| SA  | TH  | TU  | SU  |
|94 66 38 10| SU  | FR  | WE  | MO  |
|95 67 39 11| MO  | SA  | TH  | TU  |
|96 68 40 12| W E | M O | S A | T H |
|-----------+-----+-----+-----+-----|
|97 69 41 13| TH  | TU  | SU  | FR  |
|98 70 42 14| FR  | WE  | MO  | SA  |
|99 71 43 15| SA  | TH  | TU  | SU  |
|   72 44 16| M O | S A | T H | T U |
|-----------+-----+-----+-----+-----|
|   73 45 17| TU  | SU  | FR  | WE  |
|   74 46 18| WE  | MO  | SA  | TH  |
|   75 47 19| TH  | TU  | SU  | FR  |
|   76 48 20| S A | T H | T U | S U |
|-----------+-----+-----+-----+-----|
|   77 49 21| SU  | FR  | WE  | MO  |
|   78 50 22| MO  | SA  | TH  | TU  |
|   79 51 23| TU  | SU  | FR  | WE  |
|   80 52 24| T H | T U | S U | F R |
|-----------+-----+-----+-----+-----|
|   81 53 25| FR  | WE  | MO  | SA  |
|   82 54 26| SA  | TH  | TU  | SU  |
|   83 55 27| SU  | FR  | WE  | MO  |
|   84 56 28| T U | S U | F R | W E |
'-----------+-----+-----+-----+-----|
            |1600 |1700 |1800 |1900 |
            |2000 |2100 |2200 |2300 |
            '-----'-----'-----'-----'

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.

Frequency in the 400-year cycle (leap years are widened again):

• 44 × TH, SA
• 43 × MO, TU, WE, FR, SU
• 15 × M O, W E
• 14 × F R, S A
• 13 × T U, T H, S U

Adding common and leap years:

• 58 × Mo, Wo, Sa
• 57 × Th, Fr
• 56 × Tu, Su

A leap year with Monday as Doomsday means that Sunday is one of 97 days skipped in the 497-day 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.

28-year cycle

Regarding the frequency of Doomsdays in a Julian 28-year cycle, there are 1 leap year and 3 common years for every weekday, the latter 6, 17 and 23 years after the former (so with intervals of 6, 11, 6, and 5 years; not evenly distributed because after 12 years the day is skipped in the sequence of Doomsdays). The same cycle applies for any given date from 1 March falling on a particular weekday.

For any given date up to 28 February falling on a particular weekday, the 3 common years are 5, 11, and 22 years after the leap year, so with intervals of 5, 6, 11, and 6 years. Thus the cycle is the same, but with the 5-year interval after instead of before the leap year.

Thus, for any date except 29 February, the intervals between common years falling on a particular weekday are 6, 11, 11. See e.g. at the bottom of the page Common year starting on Monday the years in the range 1906 - 2091.

For 29 February falling on a particular weekday, there is just one in every 28 years, and it is of course a leap year.

Doomsdays for some contemporary years

Doomsday for the current year (2025) is , and for some other contemporary years:

Correspondence with dominical letter

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

Examples

Example 1 (this year)

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

Example 2 (other years of this 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 is Tuesday, and one day beyond is Wednesday. September 5 is a Doomsday, and September 11, six days later, falls 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 is 99 days after Thursday, or Friday. The digits 61 give a displacement of six days, so Doomsday was Thursday. Therefore, April 4 was Thursday, so April 12, eight days later, is 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 (is 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 was not 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.

See also

Calendars:

• Mon - Tue - Wed - Thu - Fri - Sat - Sun - common years by Doomsday
• Mon - Tue - Wed - Thu - Fri - Sat - Sun - leap years by Doomsday

• Ordinal date

External links

• What is the day of the week, given any date?
• Doomsday Algorithm
• Poem explaining the Doomsday rule

• Weeks
• Algorithms
• Calendars
• Unusual days