Doomsday method

The Doomsday method is a simple method for determining the weekday of a given date , which with mental arithmetic operations can be performed. It was developed around 1970 by the British mathematician John Horton Conway .

The day of the week on which the date falls according to the Gregorian calendar is determined. This method is initially only suitable for calculating dates from October 15, 1582. For the other dates according to the Julian calendar , the calculation must be modified accordingly (see application in the Julian calendar ).

Calculation of the day of the week

The algorithm is based on the so-called Doomsday (actual meaning of the word: " Judgment Day "), which in this context is the weekday of the last day of February (i.e. the 28th or, in a leap year , February 29th) of a year .

If you know the Doomsday, you can calculate forward and backward the days of the week from the last day of February as a fixed point for all other dates of the year.

In practice, calculations of days of the week in the current year or in the near past or future are the most common. These calculations are quite easy to do in your head and are therefore presented here first.

For these calculations, the doomsday for the current year should simply be memorized. Doomsday moves one day of the week every year, and two days of the week in leap years. So the Doomsday for years in the near past or future can be determined quite easily by calculating forward and backward.

Doomsday of current years
Sunday	Monday	Tuesday	Wednesday	Thursday	Friday	Saturday
1897	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 *

* = Leap year

Remember rules for the doomsday

There are also a number of donkey bridges that make the calculation easier:

In January , the 4.1 is in leap years. a doomsday, in the other years it is the 3.1. (Eselsbrücke: Holy 3 Kings; better: "Three years it is the 3 , and leap years are divisible by 4. " or " 3 times the 3 , the 4th time the 4 ")

In February it is the last day, i.e. the 28.2. or the 29.2. in leap year.

In March it is all days divisible by 7, i.e. the 7th, 14th, 21st and 28th.

From April , in even months, the day with the month number falls on Doomsday ( April 4th, June 6th, August 8th, October 10th and December 12th)

From May in the uneven months are the 9.5., 5.9., 11.7. and 7.11. Doomsdays. There is also an English motto: "I work from 9 to 5 at the 7-11." ( 7-Eleven is an international retail chain.)

The following days of a year always fall on Doomsday:

7.3.
4.4.
9.5.
6.6.
11.7.
8.8.
5.9.
10.10.
7.11.
12.12.

The following months have the same sequence of days of the week:

January (only in leap years), April and July
January and October (except in leap years)
February (except in leap years), March and November
February and August (only in leap years)
September and December

You can also remember other fixed dates, e.g. B. December 24th always falls on the weekday two days before Doomsday. Your own birthday, name day, wedding day etc. can also be used as fixed dates.

As an alternative to the above-mentioned English motto, you can also use the season rule to determine the Doomsday for the odd months from March in the German-speaking area : The year has four seasons. By July it will be warmer - add 4 [to the number of the month]. After that it will get cooler - subtract 4. You get the 7.3., 9.5., 11.7. as well as the 5.9. and the 7.11.

Memo list

By memorizing the following list of memos, one can use the doomsday method.

Memo list
month	Doomsday	in leap year	Memory aid
January	3.1.	4.1.	three years 3 , in the fourth 4
February	28.2.	29.2.	last
March	"0." 3.		zeroth, or divisible by 7
April	4.4.
May	9.5.		nine to five
June	6.6.
July	11.7.		seven-eleven
August	8.8.
September	5.9.		nine to five
October	10.10.
November	7.11.		seven-eleven
December	12.12.

Calculation of the Doomsday

For years that are further in the past or future, the Doomsday can be determined mathematically, for which mental arithmetic is also sufficient.

The days of the week are interpreted as numbers as follows:

Calculation of the doomsday of the century

The starting point is the Doomsday of the first year in a century. Doomsday should be learned for the full centuries from 1800 to 2100 (see table). Since the days of the week are repeated every 400 years, it is possible to calculate forward or backward for other centuries. You can also simply orientate yourself on the Doomsday of the century in which you were born (currently Wednesday or Tuesday).

Century Doomsday of 1800, 1900, 2000 and 2100
Sunday	Monday	Tuesday	Wednesday	Thursday	Friday	Saturday
2100		2000	1900		1800

Using the 400-year cycle described above, you can alternatively make a calculation for the “doomsday of the century”: If you use the method described above to calculate the doomsday back to a fictional year zero , you get the Tuesday as the result Basis, so to speak, the "original doomsday", for which further calculation is.

You still have to calculate

how often the year to be calculated by 100 share (ie the first two digits of the four-digit year) and can
from this result the number before the decimal point modulo 4.

The result multiplied by 2 is then the number of days that you have to calculate back from Tuesday to get the Doomsday of the corresponding century.

This results in the following formula:

${\ displaystyle \ mathrm {Tuesday} - \ left (\ left \ lfloor {\ frac {yyyy} {100}} \ right \ rfloor {\ bmod {4}} \ right) * 2 = {\ text {Doomsday of the century }}}$ ,

where stands for the year for which you want to calculate the doomsday of the century. ${\ displaystyle yyyy}$

Since this formula usually results in negative numbers, you can use the "original Doomsday" with 9 and then subtract 7 from the result:

${\ displaystyle \ mathrm {\ left (9- \ left (\ left \ lfloor {\ frac {yyyy} {100}} \ right \ rfloor {\ bmod {4}} \ right) * 2 \ right)} {\ bmod {7}} = {\ text {Century Doomsday}}}$ .

Alternatively , those who do not want to or cannot remember the above table of the Doomsdays of the century can,

begin with the above-mentioned "original doomsday" of the fictitious year 0 - a Tuesday,
then jump forward in 400-year steps (and stay with Tuesday),
then jump in 100-year steps, each time counting five days of the week (Eselsbrücke: "Another century done! Yeah, give me five!").

Example: For years in the twentieth century you start in the year 0 with Tuesday, then in four 400 jumps in the year 1600 you land again on Tuesday, and then continue three times every 100 years (1700: Tuesday + 5 = Sunday, 1800: Sunday + 5 = Friday, 1900: Friday + 5 = Wednesday).

Calculation of the Doomsday of a year

The calculation of the Doomsday of a certain year is done in four steps:

Determine how often the number 12 fits in the last two digits of the year.
Determine the remainder of Step 1.
Determine how many times the number 4 fits into the remainder from step 1.
Have the doomsday of the century ready.

The results of the four steps are added, subtracting a multiple of 7, resulting in a number from 0 to 6. This is the sought-after doomsday of the year.

As a formula, the procedure can be represented as follows, with the last two digits of the year for which the doomsday is to be determined and denotes the "doomsday of the century" according to the table or calculation above: ${\ displaystyle yy}$ ${\ displaystyle JhD}$

${\ displaystyle \ left ({\ left \ lfloor {\ frac {yy} {12}} \ right \ rfloor + yy \; {\ bmod {1}} 2+ \ left \ lfloor {\ frac {yy \; { \ bmod {1}} 2} {4}} \ right \ rfloor} + JhD \ right) {\ bmod {7}} = \ mathrm {Doomsday}}$

Once the Doomsday has been determined, you can calculate forward and backward to any date of the year as described above.

Alternatively , to calculate the Doomsday of a certain year, the last two digits can be added to the integer result of dividing by 4 of the same two-digit number plus the Doomsday of the century. This sum is then divided modulo 7. ${\ displaystyle yy}$

${\ displaystyle \ left (yy + {\ left \ lfloor {\ frac {yy} {4}} \ right \ rfloor} + JhD \ right) {\ bmod {7}} = \ mathrm {Doomsday}}$

For fast mental arithmetic, however, the detour via the dozen is easier, because smaller numbers have to be divided by 7 for the calculation.

Examples

Summary of the most important data

Day of the week of October 26, 2005

The doomsday of 2005 is calculated as follows:

The last two digits of the year are 05; the 12 fits 0 times into the 5th result: 0
The rest of step 1 is 5. Result: 5
The 4 fits once into the 5. Result: 1
Century Doomsday for 2000 is Tuesday. Result: 2

The sum of the results of the four steps gives 0 + 5 + 1 + 2 = 8. 7 is deducted from this, which then equals 1, i.e. Monday.

So October 10th is a Monday (see rule of thumb ). Then the 24th is a Monday and the October 26th, 2005 you are looking for is a Wednesday.

Day of the week of February 26, 1960

60/12 = 5
Remainder = 0
0/4 = 0
1900: Wednesday = 3

Sum = 8. 7 is deducted from this, which results in 1, i.e. Monday. Since 1960 was a leap year, February 29th is a Monday and therefore the February 26th you are looking for is a Friday.

Application in the Julian calendar

Basically, this method can also be used for dates according to the Julian calendar, as this only differs from the Gregorian calendar in the leap year rule on "smooth" centuries (1800, 1900 etc.).

For this, however, the calculation of the "Doomsday of the century" (see above) must be adjusted as follows:

The "original doomsday" is Sunday.
How often is to be calculated
1. the year can be divided by 100 and
2. this remainder is divisible by 7.

The result is then to be calculated back from Sunday:

${\ displaystyle \ mathrm {Sunday} - \ left \ lfloor {\ frac {a} {100}} \ right \ rfloor {\ bmod {7}} = \ mathrm {Century Doomsday}}$

You can then proceed as described above .

Result control

The perpetual calendars shown are a simple and reliable method of checking the results .

Permanent calendar from 1848 to 2151

Perpetual calendar from Oct. 15, 1582

Julian. Permanent calendar from January 1st of year 1 ( the correction from Augustus up to year 8 is not taken into account )

Others

Many weekday calculation methods were published towards the end of the 19th century. The first publication is likely that of Lewis Carroll in the journal Nature (Volume 35, March 31, 1887, page 517). It is basically very similar to the Doomsday method. In it Carroll writes: “I'm not a high-speed calculator myself and on average I need about 20 seconds to answer a question asked; But I have no doubt that a real high-speed calculator would not even need 15 seconds to answer. "

literature

John Conway, Elwyn Berlekamp, Richard Guy: Winning Ways For Your Mathematical Plays. Vol. 2: Games in Particular. Academic Press, London 1982, 795-797. ISBN 0-120-91102-7
Hans-Christian Solka: Encyclopedia of the weekday calculation . Self-published: Magdeburg 2nd edition 20113. (???)

Web links

The calendar in your head - what day of the week was ...? (A guide that is particularly well described for younger readers)

Individual evidence

↑ Martin Gardner: Mathematical Carnival , Chapter: Tricks of the fast calculator

[1] Martin Gardner: Mathematical Carnival , Chapter: Tricks of the fast calculator