Determination of the day of the week

The determination of the day of the week for any date may be performed with a variety of algorithms. In addition, perpetual calendars require no calculation by the user, and are essentially lookup tables. A typical application is to calculate the day of the week on which someone was born or a specific event occurred.


In numerical calculation, the days of the week are represented as weekday numbers. If Monday is the first day of the week, the days may be coded 1 to 7, for Monday through Sunday, as is practiced in ISO 8601. The day designated with 7 may also be counted as 0, by applying the arithmetic modulo 7, which calculates the remainder of a number after division by 7. Thus, the number 7 is treated as 0, 8 as 1, 9 as 2, 18 as 4 and so on. If Sunday is counted as day 1, then 7 days later (i.e. day 8) is also a Sunday, and day 18 is the same as day 4, which is a Wednesday since this falls three days after Sunday.

Standard Monday Tuesday Wednesday Thursday Friday Saturday Sunday Usage examples
ISO 8601 1 2 3 4 5 6 7 %_ISODOWI%, %@ISODOWI[]% (4DOS);[1] DAYOFWEEK() (HP Prime)[2]
0 1 2 3 4 5 6
2 3 4 5 6 7 1 %NDAY OF WEEK% (NetWare, DR-DOS[3]); %_DOWI%, %@DOWI[]% (4DOS)[1]
1 2 3 4 5 6 0 HP financial calculators

The basic approach of nearly all of the methods to calculate the day of the week begins by starting from an ‘anchor date’: a known pair (such as January 1, 1800 as a Wednesday), determining the number of days between the known day and the day that you are trying to determine, and using arithmetic modulo 7 to find a new numerical day of the week.

One standard approach is to look up (or calculate, using a known rule) the value of the first day of the week of a given century, look up (or calculate, using a method of congruence) an adjustment for the month, calculate the number of leap years since the start of the century, and then add these together along with the number of years since the start of the century, and the day number of the month. Eventually, one ends up with a day-count to which one applies modulo 7 to determine the day of the week of the date.[4]

Some methods do all the additions first and then cast out sevens, whereas others cast them out at each step, as in Lewis Carroll's method. Either way is quite viable: the former is easier for calculators and computer programs; the latter for mental calculation (it is quite possible to do all the calculations in one's head with a little practice). None of the methods given here perform range checks, so unreasonable dates will produce erroneous results.

Corresponding days

Every seventh day in a month has the same name as the previous:

Day of
the month
00 07 14 21 28 0
01 08 15 22 29 1
02 09 16 23 30 2
03 10 17 24 31 3
04 11 18 25 4
05 12 19 26 5
06 13 20 27 6

Corresponding months

"Corresponding months" are those months within the calendar year that start on the same day of the week. For example, September and December correspond, because September 1 falls on the same day as December 1. Months can only correspond if the number of days between their first days is divisible by 7, or in other words, if their first days are a whole number of weeks apart. For example, February of a common year corresponds to March because February has 28 days, a number divisible by 7, 28 days being exactly four weeks. In a leap year, January and February correspond to different months than in a common year, since adding February 29 means each subsequent month starts a day later.

The months correspond thus:
For common years:

  • January and October.
  • February, March and November.
  • April and July.
  • No month corresponds to August.

For leap years:

  • January, April and July.
  • February and August.
  • March and November.
  • No month corresponds to October.

For all years:

  • September and December.
  • No month corresponds to May or June.

In the months table below, corresponding months have the same number, a fact which follows directly from the definition.

Common years Leap years m
Jan Oct Oct 0
May 1
Aug Feb Aug 2
Feb Mar Nov Mar Nov 3
Jun 4
Sept Dec 5
Apr July Jan Apr July 6

Corresponding years

There are seven possible days that a year can start on, and leap years will alter the day of the week after February 29. This means that there are 14 configurations that a year can have. All the configurations can be referenced by a dominical letter, but as February 29 has no letter allocated to it a leap year has two dominical letters, one for January and February and the other (one step back in the alphabetical sequence) for March to December. For example, 2019 is a common year starting on Tuesday, meaning that 2019 corresponds to the 2013 calendar year. On the other hand, 2020 is a leap year starting on Wednesday, meaning that 2020 corresponds to the 1992 calendar year, meaning that the first two months of the year begin on the same day as they do in 2014 (i.e. January 1 is a Wednesday and February 1 is a Saturday) but because of a leap day the last ten months correspond to the last ten months in 2015 (i.e. March 1 is a Sunday to December 31 is a Thursday.). 2021 is a common year starting on Friday, meaning that 2021 corresponds to the 2010 calendar year and with the first 2 months corresponds to the 2016 calendar year. 2022 is a common year starting on Saturday, meaning that 2022 corresponds to the 2011 calendar year and with the last 10 months corresponds to the 2016 calendar year. 2023 is a common year starting on Sunday, meaning that 2023 corresponds to the 2017 calendar year. For details see the table below.

Year of the
century mod 28
00 06 12 17 23 0
01 07 12 18 24 1
02 08 13 19 24 2
03 08 14 20 25 3
04 09 15 20 26 4
04 10 16 21 27 5
05 11 16 22 00 6


  • Black means the all months of Common Year
  • Red means the first 2 months of Leap Year
  • Blue means the last 10 months of Leap Year

Corresponding centuries

Julian century
mod 700
Gregorian century
mod 400
400: 1100 1800 ... 300: 1500 1900 ... Sun
300: 1000 1700 ... Mon
200: 0900 1600 ... 200: 1800 2200 ... Tue
100: 0800 1500 ... Wed
000: 1400 2100 ... 100: 1700 2100 ... Thu
600: 1300 2000 ... Fri
500: 1200 1900 ... 000: 1600 2000 ... Sat

The Julian starts on Thursday and the Gregorian on Saturday.

Tabular methods to calculate the day of the week

Complete table: Julian and Gregorian calendars

For Julian dates before 1300 and after 1999 the year in the table which differs by an exact multiple of 700 years should be used. For Gregorian dates after 2299, the year in the table which differs by an exact multiple of 400 years should be used. The values "r0" through "r6" indicate the remainder when the Hundreds value is divided by 7 and 4 respectively, indicating how the series extend in either direction. Both Julian and Gregorian values are shown 1500–1999 for convenience. Bold figures (e.g., 04) denote leap year. If a year ends in 00 and its hundreds are in bold it is a leap year. Thus 19 indicates that 1900 is not a Gregorian leap year, (but 19 in the Julian column indicates that it is a Julian leap year, as are all Julian x00 years). 20 indicates that 2000 is a leap year. Use Jan and Feb only in leap years.

100s of Years D
Remaining Year Digits Month #
(r ÷ 7)
(r ÷ 4)
r5 19 16 20 r0 Sa A Tu 00 06   17 23 28 34   45 51 56 62   73 79 84 90 Jan Oct 0
r4 18 15 19 r3 Su G W 01 07 12 18 29 35 40 46 57 63 68 74 85 91 96 May 1
r3 17 N/A M F Th 02   13 19 24 30   41 47 52 58   69 75 80 86   97 Feb Aug 2
r2 16 18 22 r2 Tu E F 03 08 14   25 31 36 42   53 59 64 70   81 87 92 98 Feb Mar Nov 3
r1 15 N/A W D Sa   09 15 20 26   37 43 48 54   65 71 76 82   93 99 Jun 4
r0 14 17 21 r1 Th C Su 04 10   21 27 32 38   49 55 60 66   77 83 88 94 Sep Dec 5
r6 13 N/A F B M 05 11 16 22 33 39 44 50 61 67 72 78 89 95 Jan Apr Jul 6

For determination of the day of the week (1 January 2000, Saturday)

  • the day of the month: 1 ~ 31 (1)
  • the month: (6)
  • the year: (0)
  • the century mod 4 for the Gregorian calendar and mod 7 for the Julian calendar DW: (Sa)
  • adding Sa + 1 + 6 + 0 = Sa + 7. Dividing by 7 leaves a remainder of 0, so the day of the week is Saturday.

For determination of the dominical letter (2000, BA)

  • the century DL: (A)
  • the year: (0)
  • subtracting A - 0 = A.

For determination of the doomsday (2000, Tuesday)

  • the century DD: (Tu)
  • the year: (0)
  • adding Tu + 0 = Tuesday.

Revised Julian calendar

Note that the date (and hence the day of the week) in the Revised Julian and Gregorian calendars is the same from 14 October 1923 to 28 February AD 2800 inclusive and that for large years it may be possible to subtract 6300 or a multiple thereof before starting so as to reach a year which is within or closer to the table.

To look up the weekday of any date for any year using the table, subtract 100 from the year, divide the difference by 100, multiply the resulting quotient (omitting fractions) by seven and divide the product by nine. Note the quotient (omitting fractions). Enter the table with the Julian year, and just before the final division add 50 and subtract the quotient noted above.

Example: What is the day of the week of 27 January 8315?

8315-6300=2015, 2015-100=1915, 1915/100=19 remainder 15, 19x7=133, 133/9=14 remainder 7. 2015 is 700 years ahead of 1315, so 1315 is used. From table: for hundreds (13): 6. For remaining digits (15): 4. For month (January): 0. For date (27): 27. 6+4+0+27+50-14=73. 73/7=10 remainder 3. Day of week = Tuesday.

Dominical Letter

To find the Dominical Letter, calculate the day of the week for either 1 January or 1 October. If it is Sunday, the Sunday Letter is A, if Saturday B, and similarly backwards through the week and forwards through the alphabet to Monday, which is G.

Leap years have two Sunday Letters, so for January and February calculate the day of the week for 1 January and for March to December calculate the day of the week for 1 October.

Leap years are all years which divide exactly by four with the following exceptions:

In the Gregorian calendar – all years which divide exactly by 100 (other than those which divide exactly by 400).

In the Revised Julian calendar – all years which divide exactly by 100 (other than those which give remainder 200 or 600 when divided by 900).

Check the result

Use this table for finding the day of the week without any calculations.

Index Mon
Perpetual Gregorian and Julian calendar
Use Jan and Feb for leap years
Date letter in year row for the letter in century row

All the C days are doomsdays

Date 01




12 19 16 20 Apr Jul Jan G A B C D E F 01 07 12 18 29 35 40 46 57 63 68 74 85 91 96
13 20 Sep Dec F G A B C D E 02 13 19 24 30 41 47 52 58 69 75 80 86 97
14 21 17 21 Jun E F G A B C D 03 08 14 25 31 36 42 53 59 64 70 81 87 92 98
15 22 Feb Mar Nov D E F G A B C 09 15 20 26 37 43 48 54 65 71 76 82 93 99
16 23 18 22 Aug Feb C D E F G A B 04 10 21 27 32 38 49 55 60 66 77 83 88 94
17 24 May B C D E F G A 05 11 16 22 33 39 44 50 61 67 72 78 89 95
18 25 19 23 Jan Oct A B C D E F G 06 17 23 28 34 45 51 56 62 73 79 84 90 00
[Year/100] Gregorian
Year mod 100


  • For common method
December 26, 1893 (GD)

December is in row F and 26 is in column E, so the letter for the date is C located in row F and column E. 93 (year mod 100) is in row D (year row) and the letter C in the year row is located in column G. 18 ([year/100] in the Gregorian century column) is in row C (century row) and the letter in the century row and column G is B, so the day of the week is Tuesday.

October 13, 1307 (JD)

October 13 is a F day. The letter F in the year row (07) is located in column G. The letter in the century row (13) and column G is E, so the day of the week is Friday.

January 1, 2000 (GD)

January 1 corresponds to G, G in the year row (00) corresponds to F in the century row (20), and F corresponds to Saturday.

A pithy formula for the method: "Date letter (G), letter (G) is in year row (00) for the letter (F) in century row (20), and for the day, the letter (F) become weekday (Saturday)".

1783, September 18 (GD)

Use 17 (in the Gregorian century row, column C) and 83 (in row C) to find the dominical letter that is E. The letter for September 18 is B, so the day of the week is Thursday.

1676, February 23 (JD, non-OS)

Use 16 (in the Julian century row, column E) and 76 (in row D) to find the dominical letter that is A. February 23 is a "D" day, so the day of the week is Wednesday.

Mathematical algorithms

Gauss's algorithm

In a handwritten note in a collection of astronomical tables, Carl Friedrich Gauss described a method for calculating the day of the week for 1 January in any given year.[5] He never published it. It was finally included in his collected works in 1927.[6]

Gauss' method was applicable to the Gregorian calendar. He numbered the weekdays from 0 to 6 starting with Sunday. He defined the following operation: The weekday of 1 January in year number A is[5]


from which a method for the Julian calendar can be derived


where is the remainder after division of y by m,[6] or y modulo m, and Y + 100C = A.

For year number 2000, A - 1 = 1999, Y - 1 = 99 and C = 19, the weekday of 1 January is

The weekday of the last day in year number A - 1 or 0 January in year number A is

The weekday of 0 (a common year) or 1 (a leap year) January in year number A is

In order to determine the week day of an arbitrary date, we will use the following lookup table.

Months 11
Common years 0 3 3 6 1 4 6 2 5 0 3 5 m
Leap years 4 0 2 5 0 3 6 1 4 6

Note: minus 1 if M is 11 or 12 and plus 1 if M less than 11 in a leap year.

The day of the week for any day in year number A is


where D is the day of the month and A - 1 for Jan or Feb.

The weekdays for 30 April 1777 and 23 February 1855 are


This formula was also converted into graphical and tabular methods for calculating any day of the week by Kraitchik and Schwerdtfeger.[6][7]

Disparate variation

Another variation of the above algorithm likewise works with no lookup tables. A slight disadvantage is the unusual month and year counting convention. The formula is


  • Y is the year minus 1 for January or February, and the year for any other month
  • y is the last 2 digits of Y
  • c is the first 2 digits of Y
  • d is the day of the month (1 to 31)
  • m is the shifted month (March=1,...,February=12)
  • w is the day of week (0=Sunday,...,6=Saturday). If w is negative you have to add 7 to it.

For example, January 1, 2000. (year − 1 for January)

Note: The first is only for a 00 leap year and the second is for any 00 years.

The term ⌊2.6m − 0.2⌋ mod 7 gives the values of months: m

Months m
January 0
February 3
March 2
April 5
May 0
June 3
July 5
August 1
September 4
October 6
November 2
December 4

The term y + ⌊y/4⌋ mod 7 gives the values of years: y

y mod 28 y
01 07 12 18 -- 1
02–13 19 24 2
03 08 14–25 3
-- 09 15 20 26 4
04 10–21 27 5
05 11 16 22 -- 6
06–17 23 00 0

The term c/4⌋ − 2c mod 7 gives the values of centuries: c

c mod 4 c
1 5
2 3
3 1
0 0

Now from the general formula: ; January 1, 2000 can be recalculated as follows:

Zeller’s algorithm

In Zeller’s algorithm, the months are numbered from 3 for March to 14 for February. The year is assumed to begin in March; this means, for example, that January 1995 is to be treated as month 13 of 1994.[8] The formula for the Gregorian calendar is


  • Y is the year minus 1 for January or February, and the year for any other month
  • y is the last 2 digits of Y
  • c is the first 2 digits of Y
  • d is the day of the month (1 to 31)
  • m is the shifted month (March=3,...January = 13, February=14)
  • w is the day of week (1=Sunday,..0=Saturday)

The only difference is one between Zeller’s algorithm (Z) and the Gaussian algorithm (G), that is ZG = 1 = Sunday.

(March = 3 in Z but March = 1 in G)

So we can get the values of months from those for the Gaussian algorithm by adding one:

Months m
January 1
February 4
March 3
April 6
May 1
June 4
July 6
August 2
September 5
October 0
November 3
December 5

Other algorithms

Schwerdtfeger's method

In a partly tabular method by Schwerdtfeger, the year is split into the century and the two digit year within the century. The approach depends on the month. For m ≥ 3,

so g is between 0 and 99. For m = 1,2,

The formula for the day of the week is[6]

where the positive modulus is chosen.[6]

The value of e is obtained from the following table:

m 1 2 3 4 5 6 7 8 9 10 11 12
e 0 3 2 5 0 3 5 1 4 6 2 4

The value of f is obtained from the following table, which depends on the calendar. For the Gregorian calendar,[6]

c mod 4 0 1 2 3
f 0 5 3 1

For the Julian calendar,[6]

c mod 7 0 1 2 3 4 5 6
f 5 4 3 2 1 0 6

Lewis Carroll's method

Charles Lutwidge Dodgson (Lewis Carroll) devised a method resembling a puzzle, yet partly tabular in using the same index numbers for the months as in the "Complete table: Julian and Gregorian calendars" above. He lists the same three adjustments for the first three months of non-leap years, one 7 higher for the last, and gives cryptic instructions for finding the rest; his adjustments for centuries are to be determined using formulas similar to those for the centuries table. Although explicit in asserting that his method also works for Old Style dates, his example reproduced below to determine that "1676, February 23" is a Wednesday only works on a Julian calendar which starts the year on January 1, instead of March 25 as on the "Old Style" Julian calendar.


Take the given date in 4 portions, viz. the number of centuries, the number of years over, the month, the day of the month. Compute the following 4 items, adding each, when found, to the total of the previous items. When an item or total exceeds 7, divide by 7, and keep the remainder only.

Century-item: For `Old Style' (which ended 2 September 1752) subtract from 18. For `New Style' (which began 14 September 1752) divide by 4, take overplus from 3, multiply remainder by 2.

Year-item: Add together the number of dozens, the overplus, and the number of 4s in the overplus.

Month-item: If it begins or ends with a vowel, subtract the number, denoting its place in the year, from 10. This, plus its number of days, gives the item for the following month. The item for January is "0"; for February or March, "3"; for December, "12".

Day-item: The total, thus reached, must be corrected, by deducting "1" (first adding 7, if the total be "0"), if the date be January or February in a leap year, remembering that every year, divisible by 4, is a Leap Year, excepting only the century-years, in `New Style', when the number of centuries is not so divisible (e.g. 1800).

The final result gives the day of the week, "0" meaning Sunday, "1" Monday, and so on.


1783, September 18

17, divided by 4, leaves "1" over; 1 from 3 gives "2"; twice 2 is "4". 83 is 6 dozen and 11, giving 17; plus 2 gives 19, i.e. (dividing by 7) "5". Total 9, i.e. "2" The item for August is "8 from 10", i.e. "2"; so, for September, it is "2 plus 31", i.e. "5" Total 7, i.e. "0", which goes out. 18 gives "4". Answer, "Thursday".

1676, February 23

16 from 18 gives "2" 76 is 6 dozen and 4, giving 10; plus 1 gives 11, i.e. "4". Total "6" The item for February is "3". Total 9, i.e. "2" 23 gives "2". Total "4" Correction for Leap Year gives "3". Answer, "Wednesday".

Since 23 February 1676 (counting February as the second month) is, for Carroll, the same day as Gregorian 4 March 1676, he fails to arrive at the correct answer, namely "Friday," for an Old Style date that on the Gregorian calendar is the same day as 5 March 1677. Had he correctly assumed the year to begin on the 25th of March, his method would have accounted for differing year numbers - just like George Washington's birthday differs - between the two calendars.

It is noteworthy that those who have republished Carroll's method have failed to point out his error, most notably Martin Gardner.[10]

In 1752, the British Empire abandoned its use of the Old Style Julian calendar upon adopting the Gregorian calendar, which has become today's standard in most countries of the world. For more background, see Old Style and New Style dates.

Implementation-dependent methods

In the C language expressions below, y, m and d are, respectively, integer variables representing the year (e.g., 1988), month (1-12) and day of the month (1-31).


In 1990, Michael Keith and Tom Craver published the foregoing expression that seeks to minimise the number of keystrokes needed to enter a self-contained function for converting a Gregorian date into a numerical day of the week.[11] It preserves neither y nor d, and returns 0 = Sunday, 1 = Monday, etc.

Shortly afterwards, Hans Lachman streamlined their algorithm for ease of use on low-end devices. As designed originally for four-function calculators, his method needs fewer keypad entries by limiting its range either to A.D. 1905-2099, or to historical Julian dates. It was later modified to convert any Gregorian date, even on an abacus. On Motorola 68000-based devices, there is similarly less need of either processor registers or opcodes, depending on the intended design objective.[12]

Sakamoto's methods

The tabular forerunner to Tøndering's algorithm is embodied in the following K&R C function.[13] With minor changes, it was adapted for other high level programming languages such as APL2.[14] Posted by Tomohiko Sakamoto on the comp.lang.c Usenet newsgroup in 1992, it is accurate for any Gregorian date.[15][16]

    dayofweek(y, m, d)	/* 1 <= m <= 12,  y > 1752 (in the U.K.) */
        static int t[] = {0, 3, 2, 5, 0, 3, 5, 1, 4, 6, 2, 4};
        y -= m < 3;
        return (y + y/4 - y/100 + y/400 + t[m-1] + d) % 7;

Rata Die

IBM's Rata Die method requires that one knows the "key day" of the proleptic Gregorian calendar i.e. the day of the week of January 1, AD 1 (its first date). This has to be done to establish the remainder number based on which the day of the week is determined for the latter part of the analysis. By using a given day August 13, 2009 which was a Thursday as a reference, with Base and n being the number of days and weeks it has been since 01/01/0001 to the given day, respectively and k the day into the given week which must be less than 7, Base is expressed as

                      Base = 7n + k       (i)

Knowing that a year divisible by 4 or 400 is a leap year while a year divisible by 100 and not 400 is not a leap year, a software program can be written to find the number of days. The following is a translation into C of IBM's method for its REXX programming language.

int daystotal (int y, int m, int d)
	static char daytab[2][13] =
		{0, 31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31},
		{0, 31, 29, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31}
	int daystotal = d;
	for (int year = 1 ; year <= y ; year++)
		int max_month = ( year < y ? 12 : m-1 );
		int leap = (year%4 == 0);
		if (year%100 == 0 && year%400 != 0)
			leap = 0;
		for (int month = 1 ; month <= max_month ; month++)
			daystotal += daytab[leap][month];
	return daystotal;

It is found that daystotal is 733632 from the base date January 1, AD 1. This total number of days can be verified with a simple calculation: There are already 2008 full years since 01/01/0001. The total number of days in 2008 years not counting the leap days is 365 *2008 = 732920 days. Assume that all years divisible by 4 are leap years. Add 2008/4 = 502 to the total; then subtract the 15 leap days because the years which are exactly divisible by 100 but not 400 are not leap. Continue by adding to the new total the number of days in the first seven months of 2009 that have already passed which are 31 + 28 + 31 + 30 + 31 + 30 + 31 = 212 days and the 13 days of August to get Base = 732920 + 502 - 20 + 5 + 212 + 13 = 733632.

What this means is that it has been 733632 days since the base date. Substitute the value of Base into the above equation (i) to get 733632 = 7 *104804 + 4, n = 104804 and k = 4 which implies that August 13, 2009 is the fourth day into the 104805th week since 01/01/0001. 13 August 2009 is Thursday; therefore, the first day of the week must be Monday, and it is concluded that the first day 01/01/0001 of the calendar is Monday. Based on this, the remainder of the ratio Base/7, defined above as k, decides what day of the week it is. If k = 0, it's Monday, k = 1, it's Tuesday, etc.[17]

See also


  1. ^ a b Brothers, Hardin; Rawson, Tom; Conn, Rex C.; Paul, Matthias; Dye, Charles E.; Georgiev, Luchezar I. (2002-02-27). 4DOS 8.00 online help.
  2. ^ "HP Prime – Portal: Firmware update" (in German). Moravia Education. 2015-05-15. Archived from the original on 2016-11-05. Retrieved 2015-08-28.
  3. ^ Paul, Matthias (1997-07-30). NWDOS-TIPs — Tips & Tricks rund um Novell DOS 7, mit Blick auf undokumentierte Details, Bugs und Workarounds. MPDOSTIP. Release 157 (in German) (3 ed.). Archived from the original on 2016-11-04. Retrieved 2014-08-06. (NB. NWDOSTIP.TXT is a comprehensive work on Novell DOS 7 and OpenDOS 7.01, including the description of many undocumented features and internals. It is part of the author's yet larger MPDOSTIP.ZIP collection maintained up to 2001 and distributed on many sites at the time. The provided link points to a HTML-converted older version of the NWDOSTIP.TXT file.)
  4. ^ Richards, E. G. (1999). Mapping Time: The Calendar and Its History. Oxford University Press.
  5. ^ a b Gauss, Carl F. (1981). "Den Wochentag des 1. Januar eines Jahres zu finden. Güldene Zahl. Epakte. Ostergrenze.". Werke. herausgegeben von der Königlichen Gesellschaft der Wissenschaften zu Göttingen (2. reprint ed.). Hildesheim: Georg Olms Verlag. pp. 206–207. ISBN 978-3-48704643-3.
  6. ^ a b c d e f g Schwerdtfeger, Berndt E. (2010-05-07). "Gauss' calendar formula for the day of the week" (PDF) (1.4.26 ed.). Retrieved 2012-12-23.
  7. ^ Kraitchik, Maurice (1942). "Chapter 5: The calendar". Mathematical recreations (2nd revised [Dover] ed.). Mineola: Dover Publications. pp. 109–116. ISBN 978-0-48645358-3.
  8. ^ Stockton, J. R. (2010-03-19). "The Calendrical Works of Rektor Chr. Zeller: The Day-of-Week and Easter Formulae". Merlyn. Retrieved 2012-12-19.
  9. ^ a b Dodgson, C.L. (Lewis Carroll). (1887). "To find the day of the week for any given date". Nature, 31 March 1887. Reprinted in Mapping Time, pp. 299-301.
  10. ^ Martin Gardner. (1996). The Universe in a Handkerchief: Lewis Carroll's Mathematical Recreations, Games, Puzzles, and Word Plays, pages 24-26. Springer-Verlag.
  11. ^ Michael Keith; Tom Craver. (1990). The ultimate perpetual calendar? Journal of Recreational Mathematics, 22:4, pp.280-282.
  12. ^ The 4-function Calculator; The Assembly of Motorola 68000 Orphans; The Abacus. gopher://
  13. ^ "Day-of-week algorithm NEEDED!"
  14. ^ APL2 IDIOMS workspace: Date and Time Algorithms, line 15. (2002)
  15. ^ Date -> Day of week conversion. Newsgroups: comp.lang.c.
  16. ^ DOW algorithm. Newsgroups: comp.lang.c. (1994)
  17. ^ REXX/400 Reference manual page 87 (1997).
  • Gauss, Carl F. (1981). "Den Wochentag des 1. Januar eines Jahres zu finden. Güldene Zahl. Epakte. Ostergrenze.". Werke. herausgegeben von der Königlichen Gesellschaft der Wissenschaften zu Göttingen (2. Nachdruckaufl. ed.). Hildesheim: Georg Olms Verlag. pp. 206–207. ISBN 9783487046433.
  • Hale-Evans, Ron (2006). "Hack #43: Calculate any weekday". Mind performance hacks (1st ed.). Beijing: O'Reilly. pp. 164–169. ISBN 9780596101534.
  • Thioux, Marc; Stark, David E.; Klaiman, Cheryl; Schultz, Robert T. (2006). "The day of the week when you were born in 700 ms: Calendar computation in an autistic savant". Journal of Experimental Psychology: Human Perception and Performance. 32 (5): 1155–1168. doi:10.1037/0096-1523.32.5.1155.
  • Treffert, Darold A. "Why calendar calculating?". Islands of genius : the bountiful mind of the autistic, acquired, and sudden savant (1. publ., [repr.]. ed.). London: Jessica Kingsley. pp. 63–66. ISBN 9781849058735.

External links


A day is approximately the period of time during which the Earth completes one rotation around its axis. A solar day is the length of time which elapses between the Sun reaching its highest point in the sky two consecutive times.In 1960, the second was redefined in terms of the orbital motion of the Earth in year 1900, and was designated the SI base unit of time. The unit of measurement "day", was redefined as 86,400 SI seconds and symbolized d. In 1967, the second and so the day were redefined by atomic electron transition. A civil day is usually 86,400 seconds, plus or minus a possible leap second in Coordinated Universal Time (UTC), and occasionally plus or minus an hour in those locations that change from or to daylight saving time.Day can be defined as each of the twenty-four-hour periods, reckoned from one midnight to the next, into which a week, month, or year is divided, and corresponding to a rotation of the earth on its axis. However its use depends on its context, for example when people say 'day and night', 'day' will have a different meaning. It will mean the interval of light between two successive nights; the time between sunrise and sunset, in this instance 'day' will mean time of light between one night and the next. However, in order to be clear when using 'day' in that sense, "daytime" should be used to distinguish it from "day" referring to a 24-hour period; this is since daytime usually always means 'the time of the day between sunrise and sunset. The word day may also refer to a day of the week or to a calendar date, as in answer to the question, "On which day?" The life patterns (circadian rhythms) of humans and many other species are related to Earth's solar day and the day-night cycle.

Dominical letter

Dominical letters or Sunday letters are a method used to determine the day of the week for particular dates. When using this method, each year is assigned a letter (or pair of letters for leap years) depending on which day of the week the year starts on.

Dominical letters are derived from the Roman practice of marking the repeating sequence of eight letters A–H (commencing with A on 1 January) on stone calendars to indicate each day's position in the eight-day market week (nundinae). The word is derived from the number nine due to their practice of inclusive counting. After the introduction of Christianity a similar sequence of seven letters A–G was added alongside, again commencing with 1 January. The dominical letter marks the Sundays. Nowadays they are used primarily as part of the computus, which is the method of calculating the date of Easter.

A common year is assigned a single dominical letter, indicating which lettered days are Sundays in that particular year (hence the name, from Latin dominica for Sunday). Thus, 2017 is A, indicating that all A days are Sunday, and by inference, 1 January 2017 is a Sunday. Leap years are given two letters, the first valid for January 1 – February 28 (or February 24, see below), the second for the remainder of the year.

In leap years, the leap day may or may not have a dominical letter. In the Catholic version it does, but in the 1662 and subsequent Anglican versions it does not. The Catholic version causes February to have 29 days by doubling the sixth day before 1 March, inclusive, because 24 February in a common year is marked "duplex", thus both halves of the doubled day have a dominical letter of F. The Anglican version adds a day to February that did not exist in common years, 29 February, thus it does not have a dominical letter of its own.In either case, all other dates have the same dominical letter every year, but the days of the dominical letters change within a leap year before and after the intercalary day, 24 February or 29 February.

Doomsday rule

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 Conway in 1973 after drawing inspiration from Lewis Carroll's perpetual calendar algorithm. 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 (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 A.D. and the Julian calendar, although their doomsdays are usually different days of the week.

Since this algorithm involves treating days of the week like numbers modulo 7, 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". There are some languages, such as Greek, Portuguese and Galician, that base some of the names of the week days in their positional order.

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.

Gaussian algorithm

Gaussian algorithm may refer to:

Gaussian elimination for solving systems of linear equations

Gauss's algorithm for Determination of the day of the week

Gauss's method for preliminary orbit determination

Gauss's Easter algorithm

List of things named after Carl Friedrich Gauss

Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.

There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy.

Pascal Etcheber

Pascal Etcheber (born August 2, 1963) is a French author and philosopher. He co-edited “Managing Sensitive Projects” published by Macmillan in the UK and Routledge in the US in 1998.

In "The Strength of Becoming: a philosophical treatise on the new art & science of living", Pascal Etcheber argues that there is a positive outlook in Existentialism through creating. Since only individuals can bring meaning into the world “Creativism” allows individuals to both find themselves and fulfillment.

His Philosophy is further outlined in “Vagabond Earth” (published by Substance Publishing in 2004). He purports that Human beings are driven by three essential needs: Freedom, Actions and Ethics, that the world is organized with Religions, Economics and Politics to apparently fulfill those needs. However, human beings can never find fulfillment because Freedom is lost by the obedience to laws; Actions are made useless by the belief that this world is meaningless compared to a higher, spiritual world or afterlife; Ethics is never attained because participation in economics does not lead into abundance for all but the success of a few to the detriment of the majority. “Vagabond Earth” proposes a blue print for a moneyless society of one world with no governments with people actively engaged in making this world a good place to live. In 2012, "7 principles for an Economic Revolution" marks a less utopian outlook and proposes new hopes for full employment and higher standards of living, even in mature economies, despite globalization.

The British Philosophical Association listed Pascal Etcheber in their success stories as one of 38 people who graduated in Philosophy and achieved success in their subsequent careers.

He graduated in Philosophy with a Dean's commendation from Exeter University in the UK. He also holds a diploma in English Literature from Exeter University.

Pascal Etcheber is a Management consultant who has worked in 18 countries. He has been an active speaker notably in the nuclear world to encourage the successful development of difficult projects that are necessary. In 1999, he spoke at the Uranium Institute on “Unlocking local knowledge to benefit public acceptance work”. In 2003, he spoke at the World Nuclear Association on “Controversial projects can succeed in today’s world…even from the brink of defeat”.In 2007, Pascal Etcheber wrote “Freeing the Organization”,a business book that exposes the myths of management consultants, uncovers the root causes of the inefficiencies crippling large organizations and explains how to overcome these issues quickly without large teams of consultants settling in.

Pascal Etcheber, a member of Mensa in France and in the UK, described a simplified method for calculating any day of the Gregorian calendar explained in Determination of the day of the week and made available for free on scribd a knowledge acquisition method that is both fast, fun and effective in the practical guide entitled " The Hard Way or the Easy Way? Fast Fun Learning" On 18 March 2009, Pascal Etcheber was charged with making a false statement to the FBI 18 months earlier. The Post and Courier reported the case on the front page over the 15 months court case. Officially, Pascal Etcheber was prosecuted for his link to Thomas Ravenel. Unofficially, Pascal Etcheber was being investigated for being an agent of the French Secret Service. Eventually, all charges against Pascal Etcheber were dismissed and he pleaded guilty to the fact that a chef who was cooking for a party of 12 people in his house smoked a joint in the kitchen without his consent in the summer of 2004. After passing a polygraph test with the FBI, the prosecutor recognized in court that all National security issues regarding Pascal Etcheber had been cleared.

Perpetual calendar

A perpetual calendar is a calendar valid for many years, usually designed to allow the calculation of the day of the week for a given date in the future.

For the Gregorian and Julian calendars, a perpetual calendar typically consists of one of two general variations:

14 one-year calendars, plus a table to show which one-year calendar is to be used for any given year. These one-year calendars divide evenly into two sets of seven calendars: seven for each common year (year that does not have a February 29) with each of the seven starting on a different day of the week, and seven for each leap year, again with each one starting on a different day of the week, totaling fourteen. (See Dominical letter for one common naming scheme for the 14 calendars.)

Seven (31-day) one-month calendars (or seven each of 28–31 day month lengths, for a total of 28) and one or more tables to show which calendar is used for any given month. Some perpetual calendars' tables slide against each other, so that aligning two scales with one another reveals the specific month calendar via a pointer or window mechanism.The seven calendars may be combined into one, either with 13 columns of which only seven are revealed, or with movable day-of-week names (as shown in the pocket perpetual calendar picture).

Note that such a perpetual calendar fails to indicate the dates of moveable feasts such as Easter, which are calculated based on a combination of events in the Tropical year and lunar cycles. These issues are dealt with in great detail in Computus.

An early example of a perpetual calendar for practical use is found in the manuscript GNM 3227a.

The calendar covers the period of 1390–1495 (on which grounds the manuscript is dated to c. 1389).

For each year of this period, it lists the number of weeks between Christmas day and Quinquagesima. This is the first known instance of a tabular form of perpetual calendar allowing the calculation of the moveable feasts that became popular during the 15th century.

Zeller's congruence

Zeller's congruence is an algorithm devised by Christian Zeller to calculate the day of the week for any Julian or Gregorian calendar date. It can be considered to be based on the conversion between Julian day and the calendar date.

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