Editing Doomsday rule (section)

==Finding the day of the week from a year's doomsday==
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&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 4th year". For March, one can remember either [[Pi Day]] or "[[March 0]]", the latter referring 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#Workweek structure|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 (this is true for both the Day/Month and Month/Day conventions).<ref name="Torrence">{{cite web |last1=Torrence |first1=Bruce |last2=Torrence |first2=Eve |author2-link=Eve Torrence|title=John H. Conway - Doomsday, part 1 |url=https://www.youtube.com/watch?v=T_nQG-Bzxsg  |archive-url=https://ghostarchive.org/varchive/youtube/20211221/T_nQG-Bzxsg |archive-date=December 21, 2021 |url-status=live|website=YouTube |publisher=Mathematical Association of America |access-date=April 14, 2020}}{{cbignore}}</ref>

Several well-known dates, such as [[Independence Day (United States)|Independence Day in United States]], [[Boxing Day]], [[Halloween]] and [[Valentine's Day]] in common years, also fall on doomsdays every year.

{| class="wikitable"
|-
! Month !! Memorable date !! Month/Day !! Mnemonic<ref>{{cite web |title=The Doomsday Algorithm - Numberphile | website=[[YouTube]] |url=https://www.youtube.com/watch?v=z2x3SSBVGJU |access-date=July 9, 2023}}</ref><ref name="Torrence" /><ref name="Rudy">{{cite web |url=http://rudy.ca/doomsday.html |last=Limeback |first=Rudy |title=Doomsday Algorithm |date=January 3, 2017 |access-date=May 27, 2017}}</ref>
!Complete list of days
|-
! January
| January 3 (common years), <br/>January 4 (leap years) || 1/3 '''OR''' 1/4 (1/31 '''OR''' 1/32) || the 3rd '''3''' years in 4 and the 4th in the '''4'''th<ref name="Rudy" /> (or: last day of January, pretending leap years have a January 32nd<ref name="Torrence" />) <!-- Do NOT add mnemonics not in the cited source -->
|'''3''', 10, 17, 24, 31 '''OR''' {{nowrap|'''4''', 11, 18, 25, 32<ref name="Torrence" />}}
|-
! February
| February 28 (common years), February 29 (leap years) || 2/0 '''OR''' 2/1 (2/28 '''OR''' 2/29) || last day of January, pretending leap years have a January 32nd<ref name="Torrence" /> (or: last day of February) <!-- Do NOT add mnemonics not in the cited source -->
|0, 7, 14, 21, '''28''' '''OR''' {{nowrap|1, 8, 15, 22, '''29'''}}
|-
! March
| "[[March 0]]," March 14 || 3/0 '''AND''' 3/14 || last day of February, [[Pi Day]] <!-- Do NOT add mnemonics not in the cited source -->
|'''0''', 7, '''14''', 21, 28
|-
! April
| April 4 || 4/4 || '''4/4''', 6/6, 8/8, 10/10, 12/12 <!-- Do NOT add mnemonics not in the cited source -->
|'''4''', 11, 18, 25
|-
! May
| May 9 || 5/9 || '''[[Working time|9-to-5]]''' at [[7-Eleven|7-11]]<!-- Do NOT add mnemonics not in the cited source -->
|2, '''9''', 16, 23, 30
|-
! June
| June 6 || 6/6 || 4/4, '''6/6''', 8/8, 10/10, 12/12 <!-- Do NOT add mnemonics not in the cited source -->
|'''6''', 13, 20, 27
|-
! July
| July 11 || 7/11 || 9-to-5 at '''7-11''' <!-- Do NOT add mnemonics not in the cited source -->
|4, '''11''', 18, 25
|-
! August
| August 8 || 8/8 || 4/4, 6/6, '''8/8''', 10/10, 12/12 <!-- Do NOT add mnemonics not in the cited source -->
|1, '''8''', 15, 22, 29
|-
! September
| September 5 || 9/5 || '''9-to-5''' at 7-11 <!-- Do NOT add mnemonics not in the cited source -->
|'''5''', 12, 19, 26
|-
! October
| October 10 || 10/10 || 4/4, 6/6, 8/8, '''10/10''', 12/12 <!-- Do NOT add mnemonics not in the cited source -->
|3, '''10''', 17, 24, 31
|-
! November
| November 7 || 11/7 || 9-to-5 at '''7-11''' <!-- Do NOT add mnemonics not in the cited source -->
|'''7''', 14, 21, 28
|-
! December
| December 12 || 12/12 || 4/4, 6/6, 8/8, 10/10, '''12/12''' <!-- Do NOT add mnemonics not in the cited source -->
|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.

{| class="wikitable sortable mw-collapsible mw-collapsed nowrap"
|+ List of all doomsdays in a year
!Month!!Dates!!Week numbers<ref>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]] (such as 2010, 2021, & 2027).</ref>
|-
| 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
|}

<!--
For the Gregorian calendar, the formula is
:<math> w = \left(d - dd + 5 \left(y \bmod 4 \right) + 3y + 5 \left(c \bmod 4 \right) + 2 \right) \bmod 7,</math>
and for the Julian
:<math> w = \left(d - dd + 5 \left(y \bmod 4 \right) + 3y - c \right) \bmod 7,</math>
where
* <math>d</math> is the day of the month (1 to 31)
* <math>dd</math> is a doomsday
* <math>y</math> is the last two digits of the year (00 to 99)
* <math>c</math> is the first 2 digits of the year
* <math>w</math> is the day of the week (0=Sunday,...6=Saturday)
''For the date 2000-01-01,''
:<math> w = \left(1 - 2 + 5 \left(99 \bmod 4 \right) + 3 \times 99 + 5 \left(19 \bmod 4 \right) + 2 \right) \bmod 7</math>
:<math> w = \left(1 - 2 + 15 + 3 + 15 + 2 \right) \bmod 7</math>
:<math> w = 6 = Saturday </math>
''For the date 1582-10-04,''
:<math> w = \left(4 - 10 + 5 \left(82 \bmod 4 \right) + 3 \times 82 - 15 \right) \bmod 7</math>
:<math> w = \left(4 - 10 + 10 + 15 - 15 \right) \bmod 7</math>
:<math> w = 4 = Thursday </math>
-->
===Example===
To find which day of the week [[Christmas Day]] of 2021 is, proceed as follows: in the year 2021, doomsday is on Sunday. Since December&nbsp;12 is a doomsday, December&nbsp;25, being thirteen days afterwards (two weeks less a day), fell on a Saturday. Christmas Day is always the day of the week before doomsday. In addition, July&nbsp;4 ([[Independence Day (United States)|U.S. Independence Day]]) is always on the same day of the week as a doomsday, as are [[Halloween]] (October&nbsp;31), [[Pi Day]] (March 14), and December&nbsp;26 ([[Boxing Day]]).