Calendar

The whole topic runs on one idea: an odd day is the remainder left after taking out complete weeks.

• An ordinary year has 365 days = 52 weeks + 1 odd day.
• A leap year has 366 days = 52 weeks + 2 odd days.
• 100 years give 5 odd days, 200 give 3, 300 give 1, and 400 give 0.

Leap year test: divisible by 4 — except a century year, which must be divisible by 400. So 2000 was a leap year, but 1900 and 2100 are not. That exception is the single most common mistake in this topic; candidates count 1900 as a leap year and land one day off.

Three question types:

1. What day was date X? Count the odd days between a date you know and date X, then move that many weekdays forward.
2. Which date is the nth day of a year? Add the month lengths (31, 28 or 29, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31) until you pass n.
3. Which year repeats this calendar? The odd days accumulated in between must total a multiple of 7, and both years must be of the same kind (ordinary or leap).

Counting rule for type 1: when the span from one date to the same date next year includes a 29 February, it is 366 days, so the weekday advances by 2, not 1.

ordinary year 1 odd day · leap year 2 · 100 yr 5, 200 yr 3, 300 yr 1, 400 yr 0 · leap = ÷4, century must be ÷400 · odd days 0 = Sunday … 6 = Saturday

15 August 1947 was a Friday. What day was 15 August 1948? → The span is one year that contains 29 February 1948, so it is 366 days = 52 weeks + 2 odd days. Friday + 2 = Sunday. Sanity check: had 1948 not been a leap year the span would have been 365 days and the answer Saturday — the leap day is worth exactly one extra weekday.