Weekdays follow a pattern
Ask someone what day of the week 20 July 1969 was and most will shrug. Yet the answer, Sunday, is not a fact you need to memorise, because weekdays follow a strict mathematical pattern. The seven-day week has run without interruption for centuries, and the calendar's rules are regular enough that you can work out any weekday with a little arithmetic. This article explains why and shows a mental method used by mathematicians.
Why dates shift by one weekday each year
A normal year has 365 days, which is 52 weeks plus 1 day. So any date moves forward by one weekday from one year to the next. If 4 April is a Friday this year, it is a Saturday next year. In a leap year, which has 366 days, the shift is two weekdays for dates after 29 February.
The 400-year cycle
The Gregorian calendar repeats exactly every 400 years. A 400-year cycle contains 146,097 days, and that is exactly 20,871 weeks. So after 400 years, every date falls on the same weekday again. This is why a calculator can work out a weekday for any date without a lookup table: the pattern is perfectly periodic.
The Doomsday rule
Mathematician John Conway popularised a mental method called the Doomsday rule. In any given year, a set of easy-to-remember dates all fall on the same weekday, called that year's "doomsday":
- 4/4, 6/6, 8/8, 10/10 and 12/12 (the even months, with the day equal to the month)
- 5/9, 9/5, 7/11 and 11/7
- The last day of February (28 February, or 29 in a leap year)
- 3 January (4 January in a leap year)
Once you know the doomsday of a year, you find any date by counting forward or backward in sevens from the nearest doomsday.