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{{short description|Way of calculating the day of the week of a given date}}
[[File:John H Conway 2005.jpg|thumb|[[John Horton Conway|John Conway]], inventor of the Doomsday algorithm.]]

The '''Doomsday rule''' 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 '''Doomsday algorithm''' for [[mental calculation]] was devised by [[John Horton Conway|John Conway]]<ref>John Horton Conway, "Tomorrow is the Day After Doomsday", ''Eureka'', volume 36, pages 28–31, October 1973.</ref><ref>Richard Guy, John Horton Conway, Elwyn Berlekamp : "Winning Ways: For Your Mathematical Plays, Volume. 2: Games in Particular", pages 795–797, Academic Press, London, 1982, {{ISBN|0-12-091102-7}}.</ref> in 1973 after drawing inspiration from [[Lewis Carroll|Lewis Carroll's]] [[Determination of the day of the week#Lewis Carroll%27s method|perpetual calendar algorithm]].<ref>Lewis Carroll, "To Find the Day of the Week for Any Given Date", ''Nature'', March 31, 1887. {{doi|10.1038/035517a0}}</ref><ref>Martin Gardner, ''The Universe in a Handkerchief: Lewis Carroll's Mathematical Recreations, Games, Puzzles, and Word Plays'', pages 24–26, Springer-Verlag, 1996.</ref><ref>{{cite web |url=
http://www.mathaware.org/mam/2014/calendar/doomsday.html |title=What Day is Doomsday |date=April 2014  |publisher=Mathematics Awareness Month }}</ref> It takes advantage of each year having a certain day of the week, called the ''doomsday'', upon which certain easy-to-remember dates fall; for example, 4/4, 6/6, 8/8, 10/10, 12/12, and the last day of February 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 doomsday for the year from the anchor day, 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 ([[Modular arithmetic|modulo 7]]) between that date and the date in question to arrive at the day of the week.

This 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 for anyone with basic arithmetic ability to do the calculations mentally. Conway can usually give the correct answer in under two seconds. To improve his speed, he practices his calendrical calculations on his computer, which is programmed to quiz him with random dates every time he logs on.<ref>Alpert, Mark. "Not Just Fun and Games", [[Scientific American]], April, 1999. {{doi|10.1038/scientificamerican0499-40}}</ref>

==Doomsdays for some contemporary years==
Doomsday for the current year in the Gregorian calendar '''({{CURRENTYEAR}})''' is '''{{#switch:{{CURRENTYEAR}}
 |2018=Wednesday
 |2019=Thursday
 |2020=Saturday
 |2021=Sunday
 |2022=Monday
 |2023=Tuesday
 |2024=Thursday
 |2025=Friday
 |2026=Saturday
 |2027=Sunday
 |2028=Tuesday
 |#default=<span class=error>Error. Year not recognized.</span>
}}'''.

For some other contemporary years :

{|class="wikitable" style="line-height:normal;margin:auto;table-layout:fixed;text-align:center"
|+'''Doomsdays for the Gregorian calendar'''
|-
!scope="col" style="width:2.5em"|<abbr title="Monday">Mon.</abbr>
!scope="col" style="width:2.5em"|<abbr title="Tuesday">Tue.</abbr>
!scope="col" style="width:2.5em"|<abbr title="Wednesday">Wed.</abbr>
!scope="col" style="width:2.5em"|<abbr title="Thursday">Thu.</abbr>
!scope="col" style="width:2.5em"|<abbr title="Friday">Fri.</abbr>
!scope="col" style="width:2.5em"|<abbr title="Saturday">Sat.</abbr>
!scope="col" style="width:2.5em"|<abbr title="Sunday">Sun.</abbr>
!scope="col" style="width:2.5em"|<abbr title="Monday">Mon.</abbr>
!scope="col" style="width:2.5em"|<abbr title="Tuesday">Tue.</abbr>
!scope="col" style="width:2.5em"|<abbr title="Wednesday">Wed.</abbr>
!scope="col" style="width:2.5em"|<abbr title="Thursday">Thu.</abbr>
!scope="col" style="width:2.5em"|<abbr title="Friday">Fri.</abbr>
!scope="col" style="width:2.5em"|<abbr title="Saturday">Sat.</abbr>
!scope="col" style="width:2.5em"|<abbr title="Sunday">Sun.</abbr>
|-
|1898||1899||1900||1901||1902||1903||   →||1904||1905||1906||1907||   →||1908||1909
|-
|1910||1911||   →||1912||1913||1914||1915||   →||1916||1917||1918||1919||   →||1920
|-
|1921||1922||1923||   →||1924||1925||1926||1927||   →||1928||1929||1930||1931||   →
|-
|1932||1933||1934||1935||   →||1936||1937||1938||1939||   →||1940||1941||1942||1943
|-
|   →||1944||1945||1946||1947||   →||1948||1949||1950||1951||   →||1952||1953||1954
|-
|1955||   →||1956||1957||1958||1959||   →||1960||1961||1962||1963||   →||1964||1965
|-
|1966||1967||   →||1968||1969||1970||1971||   →||1972||1973||1974||1975||   →||1976
|-
|1977||1978||1979||   →||1980||1981||1982||1983||   →||1984||1985||1986||1987||   →
|-
|1988||1989||1990||1991||   →||1992||1993||1994||1995||   →||1996||1997||1998||1999
|-
|   →||2000||2001||2002||2003||   →||2004||2005||2006||2007||   →||2008||2009||2010
|-
|2011||   →||2012||2013||2014||2015||   →||2016||2017||2018||2019||   →||2020||2021
|-
|2022||2023||   →||2024||2025||2026||2027||   →||2028||2029||2030||2031||   →||2032
|-
|2033||2034||2035||   →||2036||2037||2038||2039||   →||2040||2041||2042||2043||   →
|-
|2044||2045||2046||2047||   →||2048||2049||2050||2051||   →||2052||2053||2054||2055
|-
|   →||2056||2057||2058||2059||   →||2060||2061||2062||2063||   →||2064||2065||2066
|-
|2067||   →||2068||2069||2070||2071||   →||2072||2073||2074||2075||   →||2076||2077
|-
|2078||2079||   →||2080||2081||2082||2083||   →||2084||2085||2086||2087||   →||2088
|-
|2089||2090||2091||   →||2092||2093||2094||2095||   →||2096||2097||2098||2099||2100
|}
'''Notes:''' Fill in the table 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.

==Memorable dates that always land on Doomsday==
One can easily 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&nbsp;3 is a doomsday during common years and January&nbsp;4 a doomsday during leap years, which can be remembered as "the 3rd during '''3''' years in 4, and the 4th in the '''4'''th year". For March, one can remember the pseudo-date "[[March 0]]", which refers to the day before March&nbsp;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 [[Working time#9-to-5|9 to 5]] at the [[7-Eleven|7-11]]", i.e., 9/5, 7/11, and also 5/9 and 11/7, are all doomsdays.

Several common holidays are also on doomsday. The chart below includes only dates covered by the mnemonics in the sources listed. 

{| class="wikitable"
|-
! Month !! Memorable date !! Month/Day !! Mnemonic<ref>{{cite web |url=http://rudy.ca/doomsday.html |last=Limeback |first=Rudy |title=Doomsday Algorithm |date=3 January 2017 |accessdate=27 May 2017}}</ref>
!Complete list of days
|-
! January
| January 3 (common years), <br/>January 4 (leap years) || 1/3 '''OR''' 1/4 || the 3rd '''3''' years in 4 and the 4th in the '''4'''th
|3, 10, 17, 24, 31 '''OR''' {{nowrap|4, 11, 18, 25}}
|-
! February
| February 28 (common years), February 29 (leap years) || 2/28 '''OR''' 2/29 || last day of February
|0, 7, 14, 21, 28 '''OR''' {{nowrap|1, 8, 15, 22, 29}}
|-
! March
| "[[March 0]]" || 3/0 || last day of February
|0, 7, 14, 21, 28
|-
! April
| April 4 || 4/4 || '''4/4''', 6/6, 8/8, 10/10, 12/12
|4, 11, 18, 25
|-
! May
| May 9 || 5/9 || '''9-to-5''' at 7-11
|2, 9, 16, 23, 30
|-
! June
| June 6 || 6/6 || 4/4, '''6/6''', 8/8, 10/10, 12/12
|6, 13, 20, 27
|-
! July
| July 11 || 7/11 || 9-to-5 at '''7-11'''
|4, 11, 18, 25
|-
! August
| August 8 || 8/8 || 4/4, 6/6, '''8/8''', 10/10, 12/12
|1, 8, 15, 22, 29
|-
! September
| September 5 || 9/5 || '''9-to-5''' at 7-11
|5, 12, 19, 26
|-
! October
| October 10 || 10/10 || 4/4, 6/6, 8/8, '''10/10''', 12/12
|3, 10, 17, 24, 31
|-
! November
| November 7 || 11/7 || 9-to-5 at '''7-11'''
|0, 7, 14, 21, 28
|-
! December
| December 12 || 12/12 || 4/4, 6/6, 8/8, 10/10, '''12/12'''
|5, 12, 19, 26
|}

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.

===Example===
To find which day of the week [[Christmas Day]] of 2018 was, proceed as follows: in the year 2018, doomsday was Wednesday. Since December&nbsp;12 is a doomsday, December&nbsp;25, being thirteen days afterwards (two weeks less a day), fell on a Tuesday.

It is useful to note that Christmas Day is always the day before doomsday ("One off doomsday"). In addition, July&nbsp;4 ([[Independence Day (United States)|U.S. Independence Day]]) is always on a doomsday, as are [[Halloween]] (October&nbsp;31), [[Pi Day]] (March 14), and [[Boxing Day]] (December&nbsp;26).

==Mnemonic weekday names==
Since this algorithm involves treating days of the week like numbers modulo 7, [[John Horton Conway|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. 
{| class="wikitable" style="margin: 1em auto 1em auto"
! day of week || Index<br/>number || Mnemonic
|-
| Sunday    || 0 || Noneday or<br/>Sansday
|-
| Monday    || 1 || Oneday
|-
| Tuesday   || 2 || Twosday
|-
| Wednesday || 3 || Treblesday
|-
| Thursday  || 4 || Foursday
|-
| Friday    || 5 || Fiveday
|-
| Saturday  || 6 || Six-a-day
|}
There are some languages, such as [[Greek language|Greek]], [[Portuguese language|Portuguese]] and [[Galician language|Galician]], that base some of the [[Names of the days of the week#Numbered days of the week|names of the week days in their positional order]].

==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 '00 and ends with '99. The following table shows the anchor day of centuries 1800–1899, 1900–1999, 2000–2099 and 2100–2199.

{| class="wikitable" style="margin: 1em auto 1em auto"
! Century || Anchor day || Mnemonic || Index (day of week)
|-
|- 1700–1799 || Sunday || — || 0 (Noneday)
|-
| 1800–1899 || Friday || — || 5 (Fiveday)
|-
| 1900–1999 || Wednesday || We-in-dis-day<br />(most living people were born in that century) || 3 (Treblesday)
|-
| '''2000–2099''' || Tuesday || Y-Tue-K or Twos-day<br />([[Y2K]] was at the head of this century) || '''2 (Twosday)'''
|-
| 2100–2199 || Sunday || Twenty-one-day is Sunday<br />(2100 is the start of the next century) || 0 (Noneday)
|}

Next, we find the year's doomsday. To accomplish that according to Conway:

#Divide the year's last two digits (call this {{math|''y''}}) by 12 and let {{math|''a''}} be the [[floor and ceiling functions|floor]] of the [[quotient]].
#Let {{math|''b''}} be the remainder of the same quotient.
#Divide that remainder by 4 and let {{math|''c''}} be the floor of the quotient.
#Let {{math|''d''}} be the sum of the three numbers ({{math|''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 ({{math|''d''}} or the remainder of {{math|{{sfrac|''d''|7}}}}) from the anchor day to get the year's doomsday.

:<math>\begin{matrix}\left({\left\lfloor{\frac{y}{12}}\right\rfloor+y \bmod 12+\left\lfloor{\frac{y \bmod 12}{4}}\right\rfloor}\right) \bmod 7+\rm{anchor}=\rm{Doomsday}\end{matrix}</math>

For the twentieth-century year 1966, for example:
:<math>\begin{matrix}\left({\left\lfloor{\frac{66}{12}}\right\rfloor+66 \bmod 12+\left\lfloor{\frac{66 \bmod 12}{4}}\right\rfloor}\right) \bmod 7+\rm{Wednesday} & = & \left(5+6+1\right) \bmod 7+\rm{Wednesday} \\
\ & = & \rm{Monday}\end{matrix}</math>

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

:<math>\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}</math>

So doomsday in 19'''66''' fell on Monday.

Similarly, doomsday in 20'''05''' is on a Monday:
:<math>\left({\left\lfloor{\frac{5}{12}}\right\rfloor+5 \bmod 12+\left\lfloor{\frac{5 \bmod 12}{4}}\right\rfloor}\right) \bmod 7+\rm{Tuesday}=\rm{Monday}</math>

===Why it works===
[[File:Doomsday Rule.svg|thumb|Doomsday rule]]
The doomsday 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 {{math|365''y'' + {{sfrac|''y''|4}}}} (rounded down).  But 365 equals 52 × 7 + 1, so after taking the remainder we get just

:<math>\left(y + \left\lfloor \frac{y}{4} \right\rfloor\right) \bmod 7.</math>

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

:<math>\left(66 + \left\lfloor \frac{66}{4} \right\rfloor\right) \bmod 7 = (66 + 16) \bmod 7 = 82 \bmod 7 = 5</math>

which gives the same answer as in the example above.

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

===The "odd + 11" method===
[[File:Odd+11 doomsday flowchart.svg|thumb|right|270px| A simple flowchart showing the Odd+11 method.]]
A simpler method for finding the year's doomsday was discovered in 2010 by Chamberlain Fong and Michael K. Walters,<ref name=Odd11paper>Chamberlain Fong, Michael K. Walters: [https://arxiv.org/abs/1010.0765 "Methods for Accelerating Conway's Doomsday Algorithm (part 2)"], 7th International Congress on Industrial and Applied Mathematics (2011).</ref> 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<ref name=Odd11paper /> to computing

:<math>\left(y + \left\lfloor \frac{y}{4} \right\rfloor\right) \bmod 7</math>.

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 doomsday, the procedure is often described as accumulating a running total {{math|''T''}} in six steps, as follows:
# Let {{math|''T''}} be the year's last two digits.
#If {{math|''T''}} is odd, add 11.
#Now let {{math|''T'' {{=}} {{sfrac|''T''|2}}}}.
#If {{math|''T''}} is odd, add 11.
#Now let {{math|''T'' {{=}} 7 − (''T'' mod 7)}}.
#Count forward {{math|''T''}} days from the century's anchor day to get the year's doomsday.

Applying this method to the year 2005, for example, the steps as outlined would be:
#{{math|''T'' {{=}} 5}}
#{{math|''T'' {{=}} 5 + 11 {{=}} 16}} (adding 11 because {{math|''T''}} is odd)
#{{math|''T'' {{=}} {{sfrac|16|2}} {{=}} 8}}
#{{math|''T'' {{=}} 8}} (do nothing since {{math|''T''}} is even)
#{{math|''T'' {{=}} 7 − (8 mod 7) {{=}} 7 − 1 {{=}} 6}}
#Doomsday for 2005 = 6 + Tuesday = Monday

The explicit formula for the odd+11 method is:
:<math> 7- \left[\frac{y+11(y\,\bmod 2)}{2} + 11 \left(\frac{y+11(y\,\bmod 2)}{2}\bmod 2\right)\right] \bmod 7</math>.

Although this expression looks daunting and complicated, it is actually simple<ref name=Odd11paper /> because of a [[common subexpression]] {{math|{{sfrac|''y'' + 11(''y'' mod 2)|2}}}} that only needs to be calculated once.

==Correspondence with dominical letter==
Doomsday is related to the [[dominical letter]] of the year as follows.

{| class="wikitable" style="text-align:center"
! rowspan=2 valign="bottom" | Doomsday !! colspan=2 | Dominical letter
|-
!<small>Common year</small>!!<small>Leap year</small>
|-
| Sunday    || C || DC
|-
| Monday    || B || CB
|-
| Tuesday   || A || BA
|-
| Wednesday || G || AG
|-
| Thursday  || F || GF
|-
| Friday    || E || FE
|-
| Saturday  || D || ED
|}

Look up the table below for the dominical letter (DL).
{| class=wikitable
!colspan=2 style="line-height:10px; border-bottom:none"|<small>100s of Years</small>||colspan=1 rowspan=2|<small style="line-height:10px">D<br />L</small>||colspan=4 rowspan=2|Remaining Year Digits||colspan=1 rowspan=2|#
|-
! style="border-top:none; line-height:10px"|<small>Julian<br />(r ÷ 7)</small>
! style="border-top:none; line-height:10px"|<small>Gregorian<br />(r ÷ 4)</small>
|-
|<small>'''r5'''</small> '''19'''||'''16''' '''20''' <small>'''r0'''</small>||A||00 06 &emsp; 17 23||'''28''' 34 &emsp; 45 51||'''56''' 62 &emsp; 73 79||'''84''' 90 ||0
|-
|<small>'''r4'''</small> '''18'''||15 19 <small>r3</small>||G||01 07 '''12''' 18 ||29 35 '''40''' 46||57 63 '''68''' 74 ||85 91 '''96'''||1
|-
|<small>'''r3'''</small> '''17'''||<center><small>N/A</small></center>||F||02 &emsp; 13 19 '''24'''||30 &emsp; 41 47 '''52'''||58 &emsp; 69 75 '''80'''||86 &emsp; 97||2
|-
|<small>'''r2'''</small> '''16'''||18 22 <small>r2</small>||E||03 '''08''' 14 &emsp; 25||31 '''36''' 42 &emsp; 53||59 '''64''' 70 &emsp; 81||87 '''92''' 98||3
|-
|<small>'''r1'''</small> '''15'''||<center><small>N/A</small></center>||D|| &emsp; 09 15 '''20''' 26|| &emsp; 37 43 '''48''' 54|| &emsp; 65 71 '''76''' 82|| &emsp; 93 99 ||4
|-
|<small>'''r0'''</small> '''14'''||17 21 <small>r1</small>||C||'''04''' 10 &emsp; 21 27||'''32''' 38 &emsp; 49 55||'''60''' 66 &emsp; 77 83||'''88''' 94 ||5
|-
|<small>'''r6'''</small> '''13'''||<center><small>N/A</small></center>||B||05 11 '''16''' 22||33 39 '''44''' 50||61 67 '''72''' 78||89 95 ||6
|}
For the year 2017, the dominical letter is A - 0 = A.

==Finding a century's anchor day==

For the Gregorian calendar:
:'''Mathematical formula'''
:{{math|5 × (''c'' mod 4) mod 7 +}} Tuesday = anchor.
:'''Algorithmic'''
:Let r = {{math|''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:
:{{math|6 × (''c'' mod 7) mod 7 +}} Sunday = anchor.
Note: {{math|''c'' {{=}} ⌊{{sfrac|year|100}}⌋}}.

==Overview of all Doomsdays==

{| class="wikitable sortable"
!Month!!Dates!!Week numbers *
|-
| January (common years) || 3, 10, 17, 24, 31 || 1–5
|-
| January (leap years) || 4, 11, 18, 25 || 1–4
|-
| February (common years) || 7, 14, 21, 28 || 6–9
|-
| February (leap years) || 1, 8, 15, 22, 29 || 5–9
|-
| March || 7, 14, 21, 28 || 10–13
|-
| April || 4, 11, 18, 25 || 14–17
|-
| May || 2, 9, 16, 23, 30 || 18–22
|-
| June || 6, 13, 20, 27 || 23–26
|-
| July || 4, 11, 18, 25 || 27–30
|-
| August || 1, 8, 15, 22, 29 || 31–35
|-
| September || 5, 12, 19, 26 || 36–39
|-
| October || 3, 10, 17, 24, 31 || 40–44
|-
| November || 7, 14, 21, 28 || 45–48
|-
| December || 5, 12, 19, 26 || 49–52
|}

<nowiki>*</nowiki> In leap years the {{math|''n''}}th doomsday is in [[ISO week date|ISO week]] {{math|''n''}}. In common years the day after the {{math|''n''}}th doomsday is in week {{math|''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]].

==Computer 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:

:<math>\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 = \mbox{Tuesday} + 5\times (y\bmod 4) + 4\times (y\bmod 100) + 6\times (y\bmod 400)</math>
For example, the year 2009 has a doomsday of Saturday under the Gregorian calendar (the currently accepted calendar), since

:<math>\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</math>

As another example, the year 1946 has a doomsday of Thursday, since

:<math>\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</math>

For the Julian calendar:

:<math>\mbox{Doomsday} =  \mbox{Sunday} + y + \left\lfloor\frac{y}{4}\right\rfloor = \mbox{Sunday}+ 5\times (y\bmod 4) + 3\times (y\bmod 7)</math>

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 [[Julian day#Calculation|the calculation of a Julian day number]].


==400-year cycle of Doomsdays==
{|class="wikitable" style="clear:right;float:right;margin:0.5em 1em 0.5em 0;text-align:center"
! colspan=4 | Julian centuries
! style="text-align:right" | -1600J<br/>  -900J<br/>  -200J<br/>  500J<br/> 1200J<br/> 1900J<br/> 2600J<br/> 3300J
! style="text-align:right" | -1500J<br/>  -800J<br/>  -100J<br/>  600J<br/> 1300J<br/> 2000J<br/> 2700J<br/> 3400J
! style="text-align:right" | -1400J<br/>  -700J<br/>     0J<br/>  700J<br/> 1400J<br/> 2100J<br/> 2800J<br/> 3500J
! style="text-align:right" | -1300J<br/>  -600J<br/>   100J<br/>  800J<br/> 1500J<br/> 2200J<br/> 2900J<br/> 3600J
! style="text-align:right" | -1200J<br/>  -500J<br/>   200J<br/>  900J<br/> 1600J<br/> 2300J<br/> 3000J<br/> 3700J
! style="text-align:right" | -1100J<br/>  -400J<br/>   300J<br/> 1000J<br/> 1700J<br/> 2400J<br/> 3100J<br/> 3800J
! style="text-align:right" | -1000J<br/>  -300J<br/>   400J<br/> 1100J<br/> 1800J<br/> 2500J<br/> 3200J<br/> 3900J
|-
! colspan=4 {{Diagonal split header|1={{indent|2}}Years|2=<span style="line-height:1em;{{transform-rotate|90}}">Gregorian<br/>centuries</span>}}
! valign="top" scope="col" style="text-align:right" |-1600<br /> -1200<br />  -800<br />  -400<br />   0<br /> 400<br />  800<br /> 1200<br /> 1600<br /> 2000<br /> 2400<br /> 2800<br /> 3200<br /> 3600
! valign="top" scope="col" style="text-align:right" |
! valign="top" scope="col" style="text-align:right" |-1500<br /> -1100<br />  -700<br />  -300<br /> 100<br /> 500<br />  900<br /> 1300<br /> 1700<br /> 2100<br /> 2500<br /> 2900<br /> 3300<br /> 3700
! valign="top" scope="col" style="text-align:right" |
! valign="top" scope="col" style="text-align:right" |-1400<br /> -1000<br />  -600<br />  -200<br /> 200<br /> 600<br /> 1000<br /> 1400<br /> 1800<br /> 2200<br /> 2600<br /> 3000<br /> 3400<br /> 3800
! valign="top" scope="col" style="text-align:right" |
! valign="top" scope="col" style="text-align:right" |-1300<br /> -900<br />   -500<br />  -100<br /> 300<br /> 700<br /> 1100<br /> 1500<br /> 1900<br /> 2300<br /> 2700<br /> 3100<br /> 3500<br /> 3900
|-style="background:#EEF"
!scope="row" | 00 || 28 || 56 || 84
| Tue. || Mon. || Sun. || Sat. || Fri. || Thu. || Wed.
|-
!scope="row" | 01 || 29 || 57 || 85
| Wed. || Tue. || Mon. || Sun. || Sat. || Fri. || Thu.
|-
!scope="row" | 02 || 30 || 58 || 86
| Thu. || Wed. || Tue. || Mon. || Sun. || Sat. || Fri.
|-
!scope="row" | 03 || 31 || 59 || 87
| Fri. || Thu. || Wed. || Tue. || Mon. || Sun. || Sat.
|-style="background:#EEF"
!scope="row" | 04 || 32 || 60 || 88
| Sun. || Sat. || Fri. || Thu. || Wed. || Tue. || Mon.
|-
!scope="row" | 05 || 33 || 61 || 89
| Mon. || Sun. || Sat. || Fri. || Thu. || Wed. || Tue.
|-
!scope="row" | 06 || 34 || 62 || 90
| Tue. || Mon. || Sun. || Sat. || Fri. || Thu. || Wed.
|-
!scope="row" | 07 || 35 || 63 || 91
| Wed. || Tue. || Mon. || Sun. || Sat. || Fri. || Thu.
|-style="background:#EEF"
!scope="row" | 08 || 36 || 64 || 92
| Fri. || Thu. || Wed. || Tue. || Mon. || Sun. || Sat.
|-
!scope="row" | 09 || 37 || 65 || 93
| Sat. || Fri. || Thu. || Wed. || Tue. || Mon. || Sun.
|-
!scope="row" | 10 || 38 || 66 || 94
| Sun. || Sat. || Fri. || Thu. || Wed. || Tue. || Mon.
|-
!scope="row" | 11 || 39 || 67 || 95
| Mon. || Sun. || Sat. || Fri. || Thu. || Wed. || Tue.
|-style="background:#EEF"
!scope="row" | 12 || 40 || 68 || 96
| Wed. || Tue. || Mon. || Sun. || Sat. || Fri. || Thu.
|-
!scope="row" | 13 || 41 || 69 || 97
| Thu. || Wed. || Tue. || Mon. || Sun. || Sat. || Fri.
|-
!scope="row" | 14 || 42 || 70 || 98
| Fri. || Thu. || Wed. || Tue. || Mon. || Sun. || Sat.
|-
!scope="row" | 15 || 43 || 71 || 99
| Sat. || Fri. || Thu. || Wed. || Tue. || Mon. || Sun.
|-style="background:#EEF"
!scope="row" | 16 || 44 || 72 ||
| Mon. || Sun. || Sat. || Fri. || Thu. || Wed. || Tue.
|-
!scope="row" | 17 || 45 || 73 ||
| Tue. || Mon. || Sun. || Sat. || Fri. || Thu. || Wed.
|-
!scope="row" | 18 || 46 || 74 ||
| Wed. || Tue. || Mon. || Sun. || Sat. || Fri. || Thu.
|-
!scope="row" | 19 || 47 || 75 ||
| Thu. || Wed. || Tue. || Mon. || Sun. || Sat. || Fri.
|-style="background:#EEF"
!scope="row" | 20 || 48 || 76 ||
| Sat. || Fri. || Thu. || Wed. || Tue. || Mon. || Sun.
|-
!scope="row" | 21 || 49 || 77 ||
| Sun. || Sat. || Fri. || Thu. || Wed. || Tue. || Mon.
|-
!scope="row" | 22 || 50 || 78 ||
| Mon. || Sun. || Sat. || Fri. || Thu. || Wed. || Tue.
|-
!scope="row" | 23 || 51 || 79 ||
| Tue. || Mon. || Sun. || Sat. || Fri. || Thu. || Wed.
|-style="background:#EEF"
!scope="row" | 24 || 52 || 80 ||
| Thu. || Wed. || Tue. || Mon. || Sun. || Sat. || Fri.
|-
!scope="row" | 25 || 53 || 81 ||
| Fri. || Thu. || Wed. || Tue. || Mon. || Sun. || Sat.
|-
!scope="row" | 26 || 54 || 82 ||
| Sat. || Fri. || Thu. || Wed. || Tue. || Mon. || Sun.
|-
!scope="row" | 27 || 55 || 83 ||
| Sun. || Sat. || Fri. || Thu. || Wed. || Tue. || Mon.
|}

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.

{|class="wikitable" style="margin:1em auto;text-align:center"
|+ '''Frequency of Gregorian Doomsday in the 400-year cycle per weekday and year type'''
|-
! !!scope="col"|Sunday!!scope="col"|Monday!!scope="col"|Tuesday!!scope="col"|Wednesday!!scope="col"|Thursday!!scope="col"|Friday!!scope="col"|Saturday
!!scope="col"|Total
|-
!scope="row" style="text-align:left"|Non-leap years
| 43 || 43 || 43 || 43 || 44 || 43 || 44 || '''303'''
|-
!scope="row" style="text-align:left"|Leap years
| 13 || 15 || 13 || 15 || 13 || 14 || 14 || '''97'''
|-style="font-weight:bold"
!scope="row" style="text-align:left"|Total
| 56 || 58 || 56 || 58 || 57 || 57 || 58 || '''400'''
|}

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.

===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).{{Citation needed|date=January 2008}} 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.

===Julian calendar===
The [[Gregorian calendar]] is currently accurately lining 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 (i.e. 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.
{{Clear}}

==Full examples==

===Example 1 (1985)===
Suppose you want to know the day of the week of September 18, 1985.  You begin with the century's anchor day, Wednesday.  To this, we'll add three things, called {{math|''a''}}, {{math|''b''}}, and {{math|''c''}} above:

*{{math|''a''}} is the floor of {{math|{{sfrac|85|12}}}}, which is 7.
*{{math|''b''}} is {{math|85 mod 12}}, which is {{math|1}}.
*{{math|''c''}} is the floor of {{math|{{sfrac|''b''|4}}}}, which is 0.

This yields {{math|''a'' + ''b'' + ''c'' {{=}} 8}}. Counting 8 days from Wednesday, we reach Thursday, which is the doomsday in 1985. (Using numbers: In modulo 7 arithmetic, 8 is congruent to 1. Because the century's anchor day is Wednesday (index 3), and 3 + 1 = 4, doomsday in 1985 was Thursday (index 4).) We now compare September 18 to a nearby doomsday, September 5. We see that the 18th is 13 past a doomsday, i.e. one day less than two weeks. Hence, the 18th was a Wednesday (the day preceding Thursday). (Using numbers: In modulo 7 arithmetic, 13 is congruent to 6 or, more succinctly, −1. Thus, we take one away from the doomsday, Thursday, to find that September 18, 1985 was a Wednesday.)

===Example 2 (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 {{math|(18 + 1) × 5 + ⌊{{sfrac|18|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.

==See also==
{{List of calendars}}

*[[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]]

==References==
{{Reflist}}

==External links==
{{commons}}
*[https://web.archive.org/web/20061018103244/http://www.cs.wustl.edu/~ksl2/mathpoem.html Encyclopedia of Weekday Calculation by Hans-Christian Solka, 2010]
* [https://web.archive.org/web/20090614053119/http://homer.freeshell.org/dd.cgi Doomsday calculator that also "shows all work"]
* [http://www.recordholders.org/en/records/dates.html World records for mentally calculating the day of the week in the Gregorian Calendar ]
* [http://www.recordholders.org/en/list/mental-calculation-rankings.html National records for finding Calendar Dates]
* [http://www.memoriad.com/index.asp?s=kategoriler&b=kategori-detay&kategoriid=dcacea11f97125360e50694fd11c2ae4&lang=EN World Ranking of Memoriad Mental Calendar Dates] (all competitions combined)
* [https://web.archive.org/web/20061018103244/http://www.cs.wustl.edu/~ksl2/mathpoem.html What is the day of the week, given any date?]
* [http://rudy.ca/doomsday.html Doomsday Algorithm]
* [https://web.archive.org/web/20070115153606/http://www.theworldofstuff.com/other/day.html Finding the Day of the Week]
* {{webarchive |url=https://web.archive.org/web/20061018103244/http://www.cs.wustl.edu/~ksl2/mathpoem.html |date=October 18, 2006 |title=Poem explaining the Doomsday rule }}

{{DEFAULTSORT:Doomsday Rule}}
[[Category:Gregorian calendar]]
[[Category:Julian calendar]]
[[Category:Calendar algorithms]]
[[Category:1973 introductions]]
[[Category:John Horton Conway]]