400-year Cycle of Doomsdays
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 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 highlighted.
| -200J 500J 1200J 1900J -400 00 400 800 1200 1600 2000 |
-00J 700J 1400J 2100J -300 100 500 900 1300 1700 2100 |
200J 900J 1600J 2300J -200 200 600 1000 1400 1800 2200 |
400J 1100J 1800J 2500J -100 300 700 1100 1500 1900 2300 |
|
|---|---|---|---|---|
| 00 | Tu | Su | Fr | We |
| 01 29 57 85 | We | Mo | Sa | Th |
| 02 30 58 86 | Th | Tu | Su | Fr |
| 03 31 59 87 | Fr | We | Mo | Sa |
| 04 32 60 88 | Su | Fr | We | Mo |
| 05 33 61 89 | Mo | Sa | Th | Tu |
| 06 34 62 90 | Tu | Su | Fr | We |
| 07 35 63 91 | We | Mo | Sa | Th |
| 08 36 64 92 | Fr | We | Mo | Sa |
| 09 37 65 93 | Sa | Th | Tu | Su |
| 10 38 66 94 | Su | Fr | We | Mo |
| 11 39 67 95 | Mo | Sa | Th | Tu |
| 12 40 68 96 | We | Mo | Sa | Th |
| 13 41 69 97 | Th | Tu | Su | Fr |
| 14 42 70 98 | Fr | We | Mo | Sa |
| 15 43 71 99 | Sa | Th | Tu | Su |
| 16 44 72 | Mo | Sa | Th | Tu |
| 17 45 73 | Tu | Su | Fr | We |
| 18 46 74 | We | Mo | Sa | Th |
| 19 47 75 | Th | Tu | Su | Fr |
| 20 48 76 | Sa | Th | Tu | Su |
| 21 49 77 | Su | Fr | We | Mo |
| 22 50 78 | Mo | Sa | Th | Tu |
| 23 51 79 | Tu | Su | Fr | We |
| 24 52 80 | Th | Tu | Su | Fr |
| 25 53 81 | Fr | We | Mo | Sa |
| 26 54 82 | Sa | Th | Tu | Su |
| 27 55 83 | Su | Fr | We | Mo |
| 28 56 84 | Tu | Su | Fr | We |
| 2000 2400 |
2100 2500 |
2200 2600 |
2300 2700 |
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.
Frequency in the 400-year cycle (leap years are widened again):
- 44 × Thursday, Saturday (non-leap years)
- 43 × Monday, Tuesday, Wednesday, Friday, Sunday (non-leap years)
- 15 × Monday, Wednesday (leap years)
- 14 × Friday, Saturday (leap years)
- 13 × Tuesday, Thursday, Sunday (leap years)
Adding common and leap years:
- 58 × Monday, Wednesday, Saturday
- 57 × Thursday, Friday
- 56 × Tuesday, Sunday
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.
Read more about this topic: Doomsday Rule
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