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. This article details various mathematical algorithms to calculate the day of the week for any particular date in the past or future. ... A perpetual calendar is a calendar which is good for a span of many years, such as the Runic calendar. ... The Gregorian calendar is the most widely used calendar in the world. ... A year is the time between two recurrences of an event related to the orbit of the Earth around the Sun. ...

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. In mathematics, computing, linguistics, and related disciplines, an algorithm is a procedure (a finite set of well-defined instructions) for accomplishing some task which, given an initial state, will terminate in a defined end-state. ... Mental calculation is the practice of doing mathematical calculations using only the human brain, with no help from any computing devices. ... John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. ... The Gregorian calendar is the most widely used calendar in the world. ... Dionysius Exiguus invented Anno Domini years to date Easter. ... The Julian calendar was introduced in 46 BC by Julius Caesar and came into force in 45 BC (709 ab urbe condita). ...

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. The following table shows the anchor day of centuries 1800-1899, 1900-1999, 2000-2099 and 2100-2199.

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 year 2000 problem (also known as the Y2K problem and the millennium bug) was a flaw in computer program design that caused some date-related processing to operate incorrectly for dates and times on and after January 1, 2000. ... 2100 can refer to either: The year at the end of the 21st century. ... A week is a [ PER [unit]] of time longer than a day and shorter than a month. ...

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

1. Divide the year's last two digits (call this y) by 12 and taking the integral value of the quotient (a)
2. Take the remainder of the same quotient (b).
3. Divide that remainder by 4 and take the integral value (c).
4. 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.)
5. Count forward the specified number of days (d or the remainder of d/7) from the anchor day to get the year's Doomsday.

For the twentieth-century year 1966, for example:

So Doomsday in 1966 fell on Monday.

Similarly, Doomsday in 2005 is on a 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 3 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 4
• May 9
• June 6
• July 11
• August 8
• September 5
• October 10
• November 7
• December 12

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." January 3 is the 3rd day of the year in the Gregorian calendar. ... January 4 is the 4th day of the year in the Gregorian calendar. ... A leap year (or intercalary year) is a year containing an extra day, week or month in order to keep the calendar year synchronised with the astronomical or seasonal year. ... February 28 is the 59th day of the year in the Gregorian calendar. ... February 29th, or bissextile day, is the 60th day of a leap year in the Gregorian Calendar, with 306 days remaining. ... April 4 is the 94th day of the year in the Gregorian calendar (95th in leap years). ... May 9 is the 129th day of the year in the Gregorian Calendar (130th in leap years). ... June 6 is the 157th day of the year in the Gregorian calendar (158th in leap years), with 208 days remaining // 1508 - Maximilian I, Holy Roman Emperor, is defeated in Friulia by Venetian forces; he is forced to sign a three-year truce and cede several territories to Venice 1513... July 11 is the 192nd day (193rd in leap years) of the year in the Gregorian Calendar, with 173 days remaining. ... August 8 is the 220th day of the year in the Gregorian Calendar (221st in leap years), with 145 days remaining. ... September 5 is the 248th day of the year (249th in leap years). ... October 10 is the 283rd day of the year (284th in Leap years). ... November 7 is the 311th day of the year (312th in leap years) in the Gregorian Calendar, with 54 days remaining. ... December 12 is the 346th day (347th in leap years) of the year in the Gregorian calendar, with 19 days remaining. ... 7-Eleven is an international franchisor, licensor, and operator of the largest chain of convenience stores in eighteen countries including: Canada, United States, Mexico and Puerto Rico in the Americas; Norway, Sweden and Denmark in Europe; Taiwan, Peoples Republic of China (China), Hong Kong, South Korea and Japan in...

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. March 7 is the 66th day of the year in the Gregorian Calendar (67th in leap years). ...

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. January 3 is the 3rd day of the year in the Gregorian calendar. ... January 10 is the 10th day of the year in the Gregorian calendar. ... January 17 is the 17th day of the year in the Gregorian calendar. ... January 24 is the 24th day of the year in the Gregorian calendar. ... January 31 is the 31st day of the year in the Gregorian calendar. ... February 7 is the 38th day of the year in the Gregorian Calendar. ... February 14 is the 45th day of the year in the Gregorian calendar. ... February 21 is the 52nd day of the year in the Gregorian Calendar. ... February 28 is the 59th day of the year in the Gregorian calendar. ... February 1 is the 32nd day of the year in the Gregorian Calendar. ... February 8 is the 39th day of the year in the Gregorian Calendar. ... February 15 is the 46th day of the year in the Gregorian Calendar. ... February 22 is the 53rd day of the year in the Gregorian Calendar. ... February 29th, or bissextile day, is the 60th day of a leap year in the Gregorian Calendar, with 306 days remaining. ... March 7 is the 66th day of the year in the Gregorian Calendar (67th in leap years). ... March 14 is the 73rd day of the year in the Gregorian Calendar (74th in leap years) with 292 days remaining in the year. ... March 21 is the 80th day of the year in the Gregorian Calendar (81st in leap years). ... March 28 is the 87th day of the year in the Gregorian Calendar (88th in leap years). ... April 4 is the 94th day of the year in the Gregorian calendar (95th in leap years). ... April 11 is the 101st day of the year in the Gregorian calendar (102nd in leap years). ... April 18 is the 108th day of the year in the Gregorian calendar (109th in leap years). ... April 25 is the 115th day of the year in the Gregorian Calendar (116th in leap years). ... May 2 is the 122nd day of the year in the Gregorian calendar (123rd in leap years). ... May 9 is the 129th day of the year in the Gregorian Calendar (130th in leap years). ... May 16 is the 136th day of the year in the Gregorian Calendar (137th in leap years). ... May 23 is the 143rd day of the year in the Gregorian calendar (144th in leap years). ... May 30 is the 150th day of the year in the Gregorian calendar (151st in leap years). ... June 6 is the 157th day of the year in the Gregorian calendar (158th in leap years), with 208 days remaining // 1508 - Maximilian I, Holy Roman Emperor, is defeated in Friulia by Venetian forces; he is forced to sign a three-year truce and cede several territories to Venice 1513... June 13 is the 164th day of the year in the Gregorian calendar (165th in leap years), with 201 days remaining. ... June 20 is the 171st day of the year (172nd in leap years) in the Gregorian Calendar, with 194 days remaining. ... June 27 is the 178th day of the year (179th in leap years) in the Gregorian calendar, with 187 days remaining. ... For the United States holiday, the Fourth of July, see Independence Day (United States). ... July 11 is the 192nd day (193rd in leap years) of the year in the Gregorian Calendar, with 173 days remaining. ... July 18 is the 199th day (200th in leap years) of the year in the Gregorian Calendar, with 166 days remaining. ... July 25 is the 206th day (207th in leap years) of the year in the Gregorian calendar, with 159 days remaining. ... August 1 is the 213th day of the year in the Gregorian Calendar (214th in leap years), with 152 days remaining. ... August 8 is the 220th day of the year in the Gregorian Calendar (221st in leap years), with 145 days remaining. ... August 15 is the 227th day of the year in the Gregorian Calendar (228th in leap years), with 138 days remaining. ... August 22 is the 234th day of the year in the Gregorian calendar (235th in leap years), with 131 days remaining. ... August 29 is the 241st day of the year in the Gregorian Calendar (242nd in leap years), with 124 days remaining. ... September 5 is the 248th day of the year (249th in leap years). ... Portal:Currentevents September 12 is the 255th day of the year (256th in leap years). ... September 19 is the 262nd day of the year (263rd in leap years). ... September 26 is the 269th day of the year (270th in leap years) in the Gregorian calendar. ... October 3 is the 276th day of the year (277th in leap years) in the Gregorian Calendar. ... October 10 is the 283rd day of the year (284th in Leap years). ... October 17 is the 290th day of the year (291st in leap years) in the Gregorian calendar. ... October 24 is the 297th day of the year (298th in leap years) in the Gregorian Calendar, with 68 days remaining. ... October 31 is the 304th day of the year (305th in leap years) in the Gregorian Calendar, with 61 days remaining. ... November 7 is the 311th day of the year (312th in leap years) in the Gregorian Calendar, with 54 days remaining. ... November 14 is the 318th day of the year (319th in leap years) in the Gregorian Calendar, with 47 days remaining until the end of the year. ... November 21 is the 325th day of the year (326th in leap years) in the Gregorian Calendar. ... November 28 is the 332nd day (333rd on leap years) of the year in the Gregorian Calendar. ... December 5 is the 339th day (340th in leap years) of the year in the Gregorian calendar. ... December 12 is the 346th day (347th in leap years) of the year in the Gregorian calendar, with 19 days remaining. ... December 19 is the 353rd day of the year (354th in leap years) in the Gregorian calendar. ... December 26 is the 360th day of the year in the Gregorian Calendar, 361st in leap years. ... January 2 is the second day of the year in the Gregorian calendar. ... January 9 is the 9th day of the year in the Gregorian calendar. ... January 16 is the 16th day of the year in the Gregorian calendar. ... January is the first month of the year and one of seven Gregorian months with the length of 31 days. ... January 30 is the 30th day of the year in the Gregorian calendar. ... February 6 is the 37th day of the year in the Gregorian Calendar. ... February 13 is the 44th day of the year in the Gregorian calendar. ... February 20 is the 51st day of the year in the Gregorian Calendar. ... February 27 is the 58th day of the year in the Gregorian Calendar. ... The ISO week date system is a leap week calendar system that is part of the ISO 8601 date and time standard. ... This is the calendar for a common year starting on Friday (dominical letter C), e. ...

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:

For the Julian calendar:

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. The proleptic Gregorian calendar is produced by extending the Gregorian Calendar to dates preceding its official introduction in 1582. ... The proleptic Julian calendar is produced by extending the Julian calendar to dates preceding its official introduction in 45 BC. Historians since Bede have traditionally represented the years preceding AD 1 as 1 BC, 2 BC, etc. ... The floor and fractional part functions In mathematics, the floor function of a real number x, denoted or floor(x), is the largest integer less than or equal to x (formally, ). For example, floor(2. ... Astronomical year numbering is based on BCE/CE (or BC/AD) year numbering, but follows normal decimal integer numbering more strictly. ...

Compare Julian day#Calculation. The Julian day or Julian day number (JDN) is the (integer) number of days that have elapsed since Monday, January 1, 4713 BC in the proleptic Julian calendar [1]. That day is counted as Julian day zero. ...

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 and 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. Astronomical year numbering is based on BCE/CE (or BC/AD) year numbering, but follows normal decimal integer numbering more strictly. ...

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. This is the calendar for a common year starting on Monday (dominical letter G), e. ...

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 (2006) is Tuesday, and for some other contemporary years:

Correspondence with dominical letter

Doomsday is related to the dominical letter of the year as follows. The days of the year are sometimes designated letters A, B, C, D, E, F and G in a cycle of 7 as an aid for finding the day of week of a given calendar date and in calculating Easter. ...

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. Joseph and Mary with baby Jesus, at the first Christmas Christmas (literally, the Mass of Christ) is a holiday in the Christian calendar, usually observed on December 25, which celebrates the birth of Jesus. ... December 12 is the 346th day (347th in leap years) of the year in the Gregorian calendar, with 19 days remaining. ... December 25 is the 359th day of the year (360th in leap years) in the Gregorian Calendar, with 6 days remaining for the year. ...

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. This article is about the date September 11 in general. ... This article is about the year 2001. ... A sequential look at United Flight 175 crashing into the south tower of the World Trade Center The September 11, 2001 attacks (often referred to as 9/11—pronounced nine eleven or nine one one) consisted of a series of coordinated terrorist[1] suicide attacks upon the United States, predominantly... 1 World Trade Center redirects here. ... September 5 is the 248th day of the year (249th in leap years). ... This article is about the date September 11 in general. ...

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. Combatants United States of America (Union) Confederate States of America (Confederacy) Commanders Lincoln, President Ulysses S. Grant, General Jefferson Davis, President Robert E. Lee, General Strength 2,200,000 1,064,000 Casualties 110,000 killed in action, 360,000 total dead, 275,200 wounded 93,000 killed in action... Fort Sumter, located in Charleston, South Carolina, harbor, was named after General Thomas Sumter. ... April 12 is the 102nd day of the year in the Gregorian calendar (103rd in leap years). ... 1861 is a common year starting on Tuesday. ... April 4 is the 94th day of the year in the Gregorian calendar (95th in leap years). ... April 12 is the 102nd day of the year in the Gregorian calendar (103rd in leap years). ...

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. Solstice is an astronomical term regarding the position of the Sun in relation to the celestial equator. ... Events January 15 - Russia cedes Livonia and Estonia to Poland February 24 - Pope Gregory XIII implements the Gregorian Calendar. ... The Julian calendar was introduced in 46 BC by Julius Caesar and came into force in 45 BC (709 ab urbe condita). ...

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. The Gregorian calendar is the most widely used calendar in the world. ...

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

This is the calendar for any common year starting on Saturday (dominical letter B) e. ... This is the calendar for any common year starting on Sunday (dominical letter A), in other words, a common year where Doomsday is Tuesday. ... This is the calendar for a common year starting on Monday (dominical letter G), e. ... This is the calendar for a common year starting on Tuesday (dominical letter F), e. ... This is the calendar for a common year starting on Wednesday (dominical letter E), e. ... This is the calendar for any common year starting on Thursday (dominical letter D). ... This is the calendar for a common year starting on Friday (dominical letter C), e. ... This is the calendar for any leap year starting on a Friday (dominical letter CB). ... This is the calendar for any leap year starting on Saturday (dominical letter BA), e. ... Here is a calendar for any leap year starting on Sunday (dominical letter AG). ... This is a calendar for a leap year starting on Monday (dominical letter GF). ... This is the calendar for a leap year starting on Tuesday (dominical letter FE) January February March Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa 1 2 3 4 5 1 2 1 6 7 8 9... This is the calendar for any leap year starting on Wednesday (dominical letter ED), e. ... This is a calendar for any leap year starting on Thursday (dominical letter DC), e. ... The ordinal date within a year together with the year form the full ISO 8601 ordinal date. ...

External links

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