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in: Week starts Monday, ISO standards, Leap Week Calendars, Date notations

ISO week date

Revision as of 20:32, 23 May 2016 by Crissov (talk | contribs)
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Wikipedia This page uses content from the English Wikipedia. The original article was at ISO week date. The list of authors can be seen in the page history. As with the Calendar Wikia, the text of Wikipedia is available under Creative Commons License. See Wikia:Licensing.


The ISO week date system is a leap week calendar system that is part of the ISO 8601 date and time standard. The system is used (mainly) in government and business for fiscal years, as well as in timekeeping.

The system uses the same cycle of 7 weekdays as the Gregorian calendar. Weeks start with Monday. ISO years have a year numbering which is approximately the same as the Gregorian years, but not exactly (see below). An ISO year has 52 or 53 full weeks (364 or 371 days). The extra week is called a leap week, a year with such a week a leap year.

A date is specified by the ISO year in the format YYYY, a week number in the format ww prefixed by the letter W, and the weekday number, a digit d from 1 through 7, beginning with Monday and ending with Sunday. For example, 2006-W52-7 (or in its most compact form 06W527) is the Sunday of the 52nd week of 2006. In the Gregorian system this day is called December 31, 2006.

The system has a 400-year cycle of 146,097 days (20,871 weeks), with an average year length of exactly 365.2425 days, just like the Gregorian calendar. Since non-leap years have 52 weeks, in every 400 years there are 71 leap years.

Relation with the Gregorian calendar

The ISO year number deviates from the number of the Gregorian year on, if applicable, a Friday, Saturday, and Sunday, or a Saturday and Sunday, or just a Sunday, at the start of the Gregorian year (which are at the end of the previous ISO year) and a Monday, Tuesday and Wednesday, or a Monday and Tuesday, or just a Monday, at the end of the Gregorian year (which are in week 01 of the next ISO year). In the period 4 January–28 December and on all Thursdays the ISO year number is always equal to the Gregorian year number.

Mutually equivalent definitions for week 01 are:

  • the week with the year's first Thursday in it
  • the week with the year's first working day in it (if Saturdays, Sundays, and 1 January are no working days)
  • the week with January 4 in it
  • the first week with the majority (four or more) of its days in the starting year
  • the week starting with the Monday in the period 29 December - 4 January
  • the week with the Thursday in the period 1 - 7 January
  • If 1 January is on a Monday, Tuesday, Wednesday or Thursday, it is in week 01. If 1 January is on a Friday, Saturday or Sunday, it is in week 52 or 53 of the previous year.

Note that while most definitions are symmetric with respect to time reversal, one definition in terms of working days happens to be equivalent.

The last week of the ISO year is the week before week 01; in accordance with the symmetry of the definition, equivalent definitions are:

  • the week with the year's last Thursday in it
  • the week with December 28 in it
  • the last week with the majority (four or more) of its days in the ending year
  • the week starting with the Monday in the period 22 - 28 December
  • the week with the Thursday in the period 25 - 31 December
  • the week ending with the Sunday in the period 28 December - 3 January
  • If 31 December is on a Monday, Tuesday, or Wednesday, it is in week 01, otherwise in week 52 or 53.

The following years have 53 weeks:

  • years starting with Thursday
  • leap years starting with Wednesday

Examples

  • 2005-01-01 is 2004-W53-6
  • 2005-01-02 is 2004-W53-7
  • 2005-12-31 is 2005-W52-6
  • 2007-01-01 is 2007-W01-1 (both years 2007 start with the same day)
  • 2007-12-30 is 2007-W52-7
  • 2007-12-31 is 2008-W01-1
  • 2008-01-01 is 2008-W01-2 (Gregorian year 2008 is a leap year, ISO year 2008 is 2 days shorter: 1 day longer at the start, 3 days shorter at the end)
  • 2008-12-29 is 2009-W01-1
  • 2008-12-31 is 2009-W01-3
  • 2009-01-01 is 2009-W01-4
  • 2009-12-31 is 2009-W53-4 (ISO year 2009 is a leap year, extending the Gregorian year 2009, which starts and ends with Thursday, at both ends with three days)
  • 2010-01-03 is 2009-W53-7

Examples where the ISO year is three days into the next gregorian year

  • "{{ISOWEEKDATE|2009|12|31}}" gives "2009-W53-4" [1]
  • "{{ISOWEEKDATE|2010|1|1}}" gives "2009-W53-5" [2]
  • "{{ISOWEEKDATE|2010|1|2}}" gives "2009-W53-6" [3]
  • "{{ISOWEEKDATE|2010|1|3}}" gives "2009-W53-7" [4]
  • "{{ISOWEEKDATE|2010|1|4}}" gives "2010-W01-1" [5]

Examples where the ISO year is three days into the previous gregorian year

  • "{{ISOWEEKDATE|2008|12|28}}" gives "2008-W52-7" [6]
  • "{{ISOWEEKDATE|2008|12|29}}" gives "2009-W01-1" [7]
  • "{{ISOWEEKDATE|2008|12|30}}" gives "2009-W01-2" [8]
  • "{{ISOWEEKDATE|2008|12|31}}" gives "2009-W01-3" [9]
  • "{{ISOWEEKDATE|2009|1|1}}" gives "2009-W01-4" [10]

The system does not need the concept of month and is not well connected with the Gregorian system of months: some months January and December are divided over two ISO years.

Week number

Anchor dates with a fixed week number in any year other than a leap year starting on Thursday (DC)
W01 W02 W03 W04 W05 W06 W07 W08 W09 W10 W11 W12 W13 W14 W15 W16 W17 W18 W19 W20 W21 W22 W23 W24 W25 W26 W27 W28 W29 W30 W31 W32 W33 W34 W35 W36 W37 W38 W39 W40 W41 W42 W43 W44 W45 W46 W47 W48 W49 W50 W51 W52 W53
Jan04 Jan11 Jan18 Jan25 Feb01 Feb08 Feb15 Feb22 Mar01 Mar08 Mar15 Mar22 Mar29 Apr05 Apr12 Apr19 Apr26 May03 May10 May17 May24 May31 Jun07 Jun14 Jun21 Jun28 Jul05 Jul12 Jul19 Jul26 Aug02 Aug09 Aug16 Aug23 Aug30 Sep06 Sep13 Sep20 Sep27 Oct04 Oct11 Oct18 Oct25 Nov01 Nov08 Nov15 Nov22 Nov29 Dec06 Dec13 Dec20 Dec27 —

The day of the week for these days are related to Doomsday because for any year, the Doomsday is the day of the week that the last day of February falls on. These dates are one day after the Doomsdays, except that in January and February of leap years the dates themselves are Doomsdays. In leap years the week number is the rank number of its Doomsday.

All other month dates can fall in one of two weeks, except for 29 December through 2 January which can be in W52, W53 or W01.

Mapping of week dates to month dates in common years
Type Jan01 Jan02 Jan03 Jan04 Jan05 Jan06 Jan07 Jan08 Jan09 Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16 Jan17 Jan18 Jan19 Jan20 Jan21 Jan22 Jan23 Jan24 Jan25 Jan26 Jan27 Jan28 Jan29 Jan30 Jan31 Feb01 Feb02 Feb03 Feb04 Feb05 Feb06 Feb07 Feb08 Feb09 Feb10 Feb11 Feb12 Feb13 Feb14 Feb15 Feb16 Feb17 Feb18 Feb19 Feb20 Feb21 Feb22 Feb23 Feb24 Feb25 Feb26 Feb27 Feb28 Mar01 Mar02 Mar03 Mar04 Mar05 Mar06 Mar07 Mar08 Mar09 Mar10 Mar11 Mar12 Mar13 Mar14 Mar15 Mar16 Mar17 Mar18 Mar19 Mar20 Mar21 Mar22 Mar23 Mar24 Mar25 Mar26 Mar27 Mar28 Mar29 Mar30 Mar31 Apr01 Apr02 Apr03 Apr04 Apr05 Apr06 Apr07 Apr08 Apr09 Apr10 Apr11 Apr12 Apr13 Apr14 Apr15 Apr16 Apr17 Apr18 Apr19 Apr20 Apr21 Apr22 Apr23 Apr24 Apr25 Apr26 Apr27 Apr28 Apr29 Apr30 May01 May02 May03 May04 May05 May06 May07 May08 May09 May10 May11 May12 May13 May14 May15 May16 May17 May18 May19 May20 May21 May22 May23 May24 May25 May26 May27 May28 May29 May30 May31 Jun01 Jun02 Jun03 Jun04 Jun05 Jun06 Jun07 Jun08 Jun09 Jun10 Jun11 Jun12 Jun13 Jun14 Jun15 Jun16 Jun17 Jun18 Jun19 Jun20 Jun21 Jun22 Jun23 Jun24 Jun25 Jun26 Jun27 Jun28 Jun29 Jun30 Jul01 Jul02 Jul03 Jul04 Jul05 Jul06 Jul07 Jul08 Jul09 Jul10 Jul11 Jul12 Jul13 Jul14 Jul15 Jul16 Jul17 Jul18 Jul19 Jul20 Jul21 Jul22 Jul23 Jul24 Jul25 Jul26 Jul27 Jul28 Jul29 Jul30 Jul31 Aug01 Aug02 Aug03 Aug04 Aug05 Aug06 Aug07 Aug08 Aug09 Aug10 Aug11 Aug12 Aug13 Aug14 Aug15 Aug16 Aug17 Aug18 Aug19 Aug20 Aug21 Aug22 Aug23 Aug24 Aug25 Aug26 Aug27 Aug28 Aug29 Aug30 Aug31 Sep01 Sep02 Sep03 Sep04 Sep05 Sep06 Sep07 Sep08 Sep09 Sep10 Sep11 Sep12 Sep13 Sep14 Sep15 Sep16 Sep17 Sep18 Sep19 Sep20 Sep21 Sep22 Sep23 Sep24 Sep25 Sep26 Sep27 Sep28 Sep29 Sep30 Oct01 Oct02 Oct03 Oct04 Oct05 Oct06 Oct07 Oct08 Oct09 Oct10 Oct11 Oct12 Oct13 Oct14 Oct15 Oct16 Oct17 Oct18 Oct19 Oct20 Oct21 Oct22 Oct23 Oct24 Oct25 Oct26 Oct27 Oct28 Oct29 Oct30 Oct31 Nov01 Nov02 Nov03 Nov04 Nov05 Nov06 Nov07 Nov08 Nov09 Nov10 Nov11 Nov12 Nov13 Nov14 Nov15 Nov16 Nov17 Nov18 Nov19 Nov20 Nov21 Nov22 Nov23 Nov24 Nov25 Nov26 Nov27 Nov28 Nov29 Nov30 Dec01 Dec02 Dec03 Dec04 Dec05 Dec06 Dec07 Dec08 Dec09 Dec10 Dec11 Dec12 Dec13 Dec14 Dec15 Dec16 Dec17 Dec18 Dec19 Dec20 Dec21 Dec22 Dec23 Dec24 Dec25 Dec26 Dec27 Dec28 Dec29 Dec30 Dec31
A W52 W01 W02 W03 W04 W05 W06 W07 W08 W09 W10 W11 W12 W13 W14 W15 W16 W17 W18 W19 W20 W21 W22 W23 W24 W25 W26 W27 W28 W29 W30 W31 W32 W33 W34 W35 W36 W37 W38 W39 W40 W41 W42 W43 W44 W45 W46 W47 W48 W49 W50 W51 W52
B W52 W01 W02 W03 W04 W05 W06 W07 W08 W09 W10 W11 W12 W13 W14 W15 W16 W17 W18 W19 W20 W21 W22 W23 W24 W25 W26 W27 W28 W29 W30 W31 W32 W33 W34 W35 W36 W37 W38 W39 W40 W41 W42 W43 W44 W45 W46 W47 W48 W49 W50 W51 W52
B* W53
C W53 W01 W02 W03 W04 W05 W06 W07 W08 W09 W10 W11 W12 W13 W14 W15 W16 W17 W18 W19 W20 W21 W22 W23 W24 W25 W26 W27 W28 W29 W30 W31 W32 W33 W34 W35 W36 W37 W38 W39 W40 W41 W42 W43 W44 W45 W46 W47 W48 W49 W50 W51 W52
D W01 W02 W03 W04 W05 W06 W07 W08 W09 W10 W11 W12 W13 W14 W15 W16 W17 W18 W19 W20 W21 W22 W23 W24 W25 W26 W27 W28 W29 W30 W31 W32 W33 W34 W35 W36 W37 W38 W39 W40 W41 W42 W43 W44 W45 W46 W47 W48 W49 W50 W51 W52 W53
E W01 W02 W03 W04 W05 W06 W07 W08 W09 W10 W11 W12 W13 W14 W15 W16 W17 W18 W19 W20 W21 W22 W23 W24 W25 W26 W27 W28 W29 W30 W31 W32 W33 W34 W35 W36 W37 W38 W39 W40 W41 W42 W43 W44 W45 W46 W47 W48 W49 W50 W51 W52 W01
F W01 W02 W03 W04 W05 W06 W07 W08 W09 W10 W11 W12 W13 W14 W15 W16 W17 W18 W19 W20 W21 W22 W23 W24 W25 W26 W27 W28 W29 W30 W31 W32 W33 W34 W35 W36 W37 W38 W39 W40 W41 W42 W43 W44 W45 W46 W47 W48 W49 W50 W51 W52 W01
G W01 W02 W03 W04 W05 W06 W07 W08 W09 W10 W11 W12 W13 W14 W15 W16 W17 W18 W19 W20 W21 W22 W23 W24 W25 W26 W27 W28 W29 W30 W31 W32 W33 W34 W35 W36 W37 W38 W39 W40 W41 W42 W43 W44 W45 W46 W47 W48 W49 W50 W51 W52 W01
Mapping of week dates to month dates in leap years
Type Jan01 Jan02 Jan03 Jan04 Jan05 Jan06 Jan07 Jan08 Jan09 Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16 Jan17 Jan18 Jan19 Jan20 Jan21 Jan22 Jan23 Jan24 Jan25 Jan26 Jan27 Jan28 Jan29 Jan30 Jan31 Feb01 Feb02 Feb03 Feb04 Feb05 Feb06 Feb07 Feb08 Feb09 Feb10 Feb11 Feb12 Feb13 Feb14 Feb15 Feb16 Feb17 Feb18 Feb19 Feb20 Feb21 Feb22 Feb23 Feb24 Feb25 Feb26 Feb27 Feb28 Feb29 Mar01 Mar02 Mar03 Mar04 Mar05 Mar06 Mar07 Mar08 Mar09 Mar10 Mar11 Mar12 Mar13 Mar14 Mar15 Mar16 Mar17 Mar18 Mar19 Mar20 Mar21 Mar22 Mar23 Mar24 Mar25 Mar26 Mar27 Mar28 Mar29 Mar30 Mar31 Apr01 Apr02 Apr03 Apr04 Apr05 Apr06 Apr07 Apr08 Apr09 Apr10 Apr11 Apr12 Apr13 Apr14 Apr15 Apr16 Apr17 Apr18 Apr19 Apr20 Apr21 Apr22 Apr23 Apr24 Apr25 Apr26 Apr27 Apr28 Apr29 Apr30 May01 May02 May03 May04 May05 May06 May07 May08 May09 May10 May11 May12 May13 May14 May15 May16 May17 May18 May19 May20 May21 May22 May23 May24 May25 May26 May27 May28 May29 May30 May31 Jun01 Jun02 Jun03 Jun04 Jun05 Jun06 Jun07 Jun08 Jun09 Jun10 Jun11 Jun12 Jun13 Jun14 Jun15 Jun16 Jun17 Jun18 Jun19 Jun20 Jun21 Jun22 Jun23 Jun24 Jun25 Jun26 Jun27 Jun28 Jun29 Jun30 Jul01 Jul02 Jul03 Jul04 Jul05 Jul06 Jul07 Jul08 Jul09 Jul10 Jul11 Jul12 Jul13 Jul14 Jul15 Jul16 Jul17 Jul18 Jul19 Jul20 Jul21 Jul22 Jul23 Jul24 Jul25 Jul26 Jul27 Jul28 Jul29 Jul30 Jul31 Aug01 Aug02 Aug03 Aug04 Aug05 Aug06 Aug07 Aug08 Aug09 Aug10 Aug11 Aug12 Aug13 Aug14 Aug15 Aug16 Aug17 Aug18 Aug19 Aug20 Aug21 Aug22 Aug23 Aug24 Aug25 Aug26 Aug27 Aug28 Aug29 Aug30 Aug31 Sep01 Sep02 Sep03 Sep04 Sep05 Sep06 Sep07 Sep08 Sep09 Sep10 Sep11 Sep12 Sep13 Sep14 Sep15 Sep16 Sep17 Sep18 Sep19 Sep20 Sep21 Sep22 Sep23 Sep24 Sep25 Sep26 Sep27 Sep28 Sep29 Sep30 Oct01 Oct02 Oct03 Oct04 Oct05 Oct06 Oct07 Oct08 Oct09 Oct10 Oct11 Oct12 Oct13 Oct14 Oct15 Oct16 Oct17 Oct18 Oct19 Oct20 Oct21 Oct22 Oct23 Oct24 Oct25 Oct26 Oct27 Oct28 Oct29 Oct30 Oct31 Nov01 Nov02 Nov03 Nov04 Nov05 Nov06 Nov07 Nov08 Nov09 Nov10 Nov11 Nov12 Nov13 Nov14 Nov15 Nov16 Nov17 Nov18 Nov19 Nov20 Nov21 Nov22 Nov23 Nov24 Nov25 Nov26 Nov27 Nov28 Nov29 Nov30 Dec01 Dec02 Dec03 Dec04 Dec05 Dec06 Dec07 Dec08 Dec09 Dec10 Dec11 Dec12 Dec13 Dec14 Dec15 Dec16 Dec17 Dec18 Dec19 Dec20 Dec21 Dec22 Dec23 Dec24 Dec25 Dec26 Dec27 Dec28 Dec29 Dec30 Dec31
AG W52 W01 W02 W03 W04 W05 W06 W07 W08 W09 W10 W11 W12 W13 W14 W15 W16 W17 W18 W19 W20 W21 W22 W23 W24 W25 W26 W27 W28 W29 W30 W31 W32 W33 W34 W35 W36 W37 W38 W39 W40 W41 W42 W43 W44 W45 W46 W47 W48 W49 W50 W51 W52 W01
BA W52 W01 W02 W03 W04 W05 W06 W07 W08 W09 W10 W11 W12 W13 W14 W15 W16 W17 W18 W19 W20 W21 W22 W23 W24 W25 W26 W27 W28 W29 W30 W31 W32 W33 W34 W35 W36 W37 W38 W39 W40 W41 W42 W43 W44 W45 W46 W47 W48 W49 W50 W51 W52
CB W53 W01 W02 W03 W04 W05 W06 W07 W08 W09 W10 W11 W12 W13 W14 W15 W16 W17 W18 W19 W20 W21 W22 W23 W24 W25 W26 W27 W28 W29 W30 W31 W32 W33 W34 W35 W36 W37 W38 W39 W40 W41 W42 W43 W44 W45 W46 W47 W48 W49 W50 W51 W52
DC W01 W02 W03 W04 W05 W06 W07 W08 W09 W10 W11 W12 W13 W14 W15 W16 W17 W18 W19 W20 W21 W22 W23 W24 W25 W26 W27 W28 W29 W30 W31 W32 W33 W34 W35 W36 W37 W38 W39 W40 W41 W42 W43 W44 W45 W46 W47 W48 W49 W50 W51 W52 W53
ED W01 W02 W03 W04 W05 W06 W07 W08 W09 W10 W11 W12 W13 W14 W15 W16 W17 W18 W19 W20 W21 W22 W23 W24 W25 W26 W27 W28 W29 W30 W31 W32 W33 W34 W35 W36 W37 W38 W39 W40 W41 W42 W43 W44 W45 W46 W47 W48 W49 W50 W51 W52 W53
FE W01 W02 W03 W04 W05 W06 W07 W08 W09 W10 W11 W12 W13 W14 W15 W16 W17 W18 W19 W20 W21 W22 W23 W24 W25 W26 W27 W28 W29 W30 W31 W32 W33 W34 W35 W36 W37 W38 W39 W40 W41 W42 W43 W44 W45 W46 W47 W48 W49 W50 W51 W52 W01
GF W01 W02 W03 W04 W05 W06 W07 W08 W09 W10 W11 W12 W13 W14 W15 W16 W17 W18 W19 W20 W21 W22 W23 W24 W25 W26 W27 W28 W29 W30 W31 W32 W33 W34 W35 W36 W37 W38 W39 W40 W41 W42 W43 W44 W45 W46 W47 W48 W49 W50 W51 W52 W01

Advantages

  • The date directly tells the weekday.
  • All years start with a Monday and end with a Sunday.
  • When used by itself without using the concept of month, all years are the same except that leap years have a leap week at the end.
  • The weeks are the same as in the Gregorian calendar.

Disadvantages

Each equinox and solstice varies over a range of at least seven days. This is because each equinox and solstice may occur any day of the week and hence on at least seven different ISO week dates. For example, there are summer solstices on 2004-W12-7 and 2010-W11-7.

It cannot replace the Gregorian calendar, because it relies on it to define the new year day (Week 1 Day 1).

Leap year cycle

The three types of week leap year are D, DC, and ED.

Dominical letters and Doomsdays
DL Doomsday 1 January
A or BA Tuesday Sunday
B or CB Monday Saturday
C or DC Sunday Friday
D or ED Saturday Thursday
E or FE Friday Wednesday
F or GF Thursday Tuesday
G or AG Wednesday Monday
400-year cycle of years by dominical letter
Centuries 1600 1700 1800 1900
2000 2100 2200 2300
Years
00 BA C E G
85 57 29 01 G B D F
86 58 30 02 F A C E
87 59 31 03 E G B D
88 60 32 04 DC FE AG CB
89 61 33 05 B D F A
90 62 34 06 A C E G
91 63 35 07 G B D F
92 64 36 08 FE AG CB ED
93 65 37 09 D F A C
94 66 38 10 C E G B
95 67 39 11 B D F A
96 68 40 12 AG CB ED GF
97 69 41 13 F A C E
98 70 42 14 E G B D
99 71 43 15 D F A C
  72 44 16 CB ED GF BA
  73 45 17 A C E G
  74 46 18 G B D F
  75 47 19 F A C E
  76 48 20 ED GF BA DC
  77 49 21 C E G B
  78 50 22 B D F A
  79 51 23 A C E G
  80 52 24 GF BA DC FE
  81 53 25 E G B D
  82 54 26 D F A C
  83 55 27 C E G B
  84 56 28 BA DC FE AG
Years
Centuries 1600 1700 1800 1900
2000 2100 2200 2300
ISO leap years in one 400-year cycle (28-year subcycles arranged horizontally)
DC D D ED D
+004 +009 +015 +020 +026
+032 +037 +043 +048 +054
+060 +065 +071 +076 +082
+088 +093 +099
+105 +111 +116 +122
+128 +133 +139 +144 +150
+156 +161 +167 +172 +178
+184 +189 +195
+201 +207 +212 +218
+224 +229 +235 +240 +246
+252 +257 +263 +268 +274
+280 +285 +291 +296
+303 +308 +314
+320 +325 +331 +336 +342
+348 +353 +359 +364 +370
+376 +381 +387 +392 +398

There are 13 28-year subcycles with 5 leap years each, and 6 remaining leap years in the remaining 36 years (the absence of leap days in the Gregorian calendar in 2100, 2200, and 2300 interrupts the subcycles). The leap years are 27 times 5 years apart, 43 times 6 years, and once 7 years. (A slightly more even distribution would be possible: 26 times 5 years apart, and 45 times 6 years.)

The Gregorian years corresponding to the 71 ISO leap years can be subdivided as follows:

Thus 27 ISO years are 5 days longer than the corresponding Gregorian year, and 44 are 6 days longer. Of the other 329 Gregorian years (neither starting nor ending with Thursday), 70 are Gregorian leap years, and 259 are non-leap years, so 70 ISO years are 2 days shorter, and 259 are 1 day shorter.

Alternative leap year rules

KARL PALMEN

The best leap week calendars to convert to Gregorian would be those based on the ISO week or a similar week numbering scheme. I thought of another that interlocks with 28-year cycle that such calendars have, which is interrupted by a dropped leap day. 

002 008 014 020 026
030 036 042 048 054
058 064 070 076 082
086 092 098 104 110 116 122
126 132 138 144 150
154 160 166 172 178
182 188 194 200 206 212 218
222 228 234 240 246
250 256 262 268 274
278 284 290 296 302 308 314
318 324 330 336 342
346 352 358 364 370
374 380 386 392 398

In each row, the leap years are six years apart and first of each row is four years after the last of the previous row. Each row covers 28 years unless it contains a dropped Gregorian leap day, in which case it covers 40 years. The rows are synchronised to the 28-year cycles that occur in any week-number calendar like ISO week date. In particular, the 2nd year of each row is in the Gregorian calendar a leap year starting on Tuesday. A simpler leap week rule would have rows alternating between 28 and 34, but this would be harder to convert to/from the Gregorian calendar.

The following variation matches all even-numbered years with 53 ISO weeks. Red years are one year later than ISO and the blue years are a year early. Gregorian leap years are shown in bold. 

014 020 026 032 038
042 048 054 060 066
070 076 082 088 094 100 106
110 116 122 128 134
138 144 150 156 162
166 172 178 184 190
194 200 206 212 218 224 230
234 240 246 252 258
262 268 274 280 286
290 296 302 308 314 320 326
330 336 342 348 354
358 364 370 376 382
386 392 398 004 010

list of ISO 53-week years

As with my original proposal every row has years six years apart. The first year of each row is four years after the previous. Each row has five, unless it spans a dropped leap year, then it has seven. Unlike my original proposal, it is not symmetrical.

To get a leap week rule where every leap week occurs in a Gregorian leap year and furthermore every Gregorian leap year that also has 53 ISO weeks is included amongst them. Take the previously mentioned proposal make the leap weeks in the first and third columns occur two years later, the leap week in the 5th and 7th column occur two years earlier. Then move 100 to 96 and 200 to 204. 

016 020 --- 028 032 036 ---
044 048 --- 056 060 064 ---
072 076 --- 084 088 092 096 <<< 104 ---
112 116 --- 124 128 132 ---
138 144 --- 152 156 160 ---
166 172 --- 180 184 188 ---
196 >>> 204 208 212 216 --- 224 228 ---
236 240 --- 248 252 256 ---
264 268 --- 276 280 284 ---
292 296 --- 304 308 312 --- 320 324 ---
332 336 --- 344 348 352 ---
360 364 --- 372 376 380 ---
388 392 --- 000 004 008 ---

The Gregorian leap years with 53 ISO weeks remain in bold. Other changes can be made to the years not shown in bold. For example, the 4th row could be changed to 

108 --- 116 120 --- 128 132 ---

CHRISTOPH PÄPER

  • Every year that has a leap week, also has a leap day. 
  • Leap years are divisible by four.

There are 26 leap days per cycle in years without a 53rd week.

​distribution as evenly as possible under these constraints

round down: 

000  004  008   –   016  020   –  
028  032  036   –   044  048   –  
056  060  064   +   072  076   –  
084  088  092   –   100  104   –  
112  116  120   –   128  132   –  
140  144   –   152  156  160   –  
168  172   –   180  184  188   –  
196  200   +   208  212  216   –  
224  228   –   236  240  244   –  
252  256   –   264  268   –   276
280  284   –   292  296   –   304 
308  312   –   320  324   +   332 
336  340   –   348  352   –   360 
364  368   –   376  380   –   388 
392   –

round up: 

000   –   008  012   –   020  024  
 –   032  036  040   –   048  052  
 –   060  064  068   +   076  080 
 –   088  092  096   –   104  108 
 –   116  120  124   –   132  136 
 –   144  148   –   156  160  164 
 –   172  176   –   184  188  192 
 +   200  204   –   212  216  220 
 –   228  232   –   240  244  248 
 –   256  260   –   268  272   – 
280  284  288   –   296  300   – 
308  312  316   –   324  328   +  
336  340  344   –   352  356   –  
364  368  372   –   380  384   –  
392  396

The former matches 27, the latter just 20 of the 71 long years according to ISO 8601.

One would now need to select three of the empty places, which do not contain a “leap+leap year”, to remain empty for leap-day years, e.g. – as marked above with plus signs – 

068, 204, 328

072, 196, 332

Other week numbering systems

For an overview of week numbering systems see week number. The US system has weeks from Sunday through Saturday, and partial weeks at the beginning and the end of the year. An advantage is that no separate year numbering like the ISO year is needed, while correspondence of lexicographical order and chronological order is preserved.

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