Calendars

At a glance 10 rules

1. 1Odd days$\text{odd days} = \text{days} \bmod 7$$\text{ordinary year: } 1,\ \text{leap year: } 2$
2. 2Leap yearsA year is a leap year if it divides by 4, but a year ending in 00 only if it divides by 400: 2000 was a leap year, 1900 and 2100 are not.
3. 3Odd days, centuries from year 1

   100

   5

   200

   3

   300

   1

   400

   0

4. 4Odd days in monthsA 31-day month gives 3, a 30-day month 2, February 0 (1 in a leap year). January to December: 3, 0, 3, 2, 3, 2, 3, 3, 2, 3, 2, 3.
5. 5Day from scratchAdd the odd days of the whole centuries, the whole years, the whole months and the date itself. The total mod 7 is the day: 0 Sunday, 1 Monday, up to 6 Saturday.
6. 6Same date, another yearA date moves on 1 weekday a year, or 2 when a 29 February falls between the two dates.
7. 7Doomsday dates4/4, 6/6, 8/8, 10/10, 12/12, 9/5, 5/9, 7/11, 11/7 and the last day of February all fall on the same weekday in any year.
8. 8Months that start alikeOrdinary year: Jan-Oct, Feb-Mar-Nov, Apr-Jul and Sep-Dec begin on the same weekday. Leap year: Jan-Apr-Jul, Feb-Aug, Mar-Nov and Sep-Dec.
9. 9Repeat calendarsSame calendar needs the same leap status and a multiple of 7 odd days in between: after 28 years for a leap year, else 6 or 11. A 00 year that is not leap breaks this.
10. 1053 of a weekdayA year is 52 weeks and 1 day (2 in a leap year), so the weekday of 1 January, and in a leap year also of 2 January, occurs 53 times.