Concepts · Calendars

Odd days

Whole weeks change nothing. Only the days left over move the weekday.

Riya has a calendar question in a mock test:

15 August was a Friday. What day of the week was 15 October of the same year?

She counts it out a week at a time: 22 August, 29 August, 5 September, 12 September… all the way to 10 October, a Friday, and then five more days to 15 October: Wednesday. She is right, but it has taken her four minutes, and there are other questions waiting.

A hundred days

That night her little brother, who is eight, has a riddle for her: “Today is Monday. What day will it be in a hundred days?”

Riya starts counting on her fingers again, and stops. Seven days from Monday is Monday. So is fourteen, and seventy, and ninety-eight. Every whole week lands back on Monday; only the days left over move it at all. A hundred days is 14 weeks and 2 days over. Two on from Monday.

“Wednesday,” she says. Her brother is annoyed at how fast that was.

The rule

Divide the number of days by 7. The whole weeks change nothing; the remainder, the odd days, is how far the weekday moves. Count that many days on from the day you know, or back if the date you want is earlier.

For the number of days between two dates, count month by month: what is left of the first month, then the whole months between, then the days into the last month.

Riya’s question, again

1. Days left in August after the 15th: 31 − 15 = 16. All of September: 30. Into October: 15. In all, 16 + 30 + 15 = 61.
2. 61 = 8 weeks + 5. Five odd days.
3. Five on from Friday: Saturday, Sunday, Monday, Tuesday, Wednesday.

Years and centuries

A year of 365 days is 52 weeks and 1 day. So a date falls one weekday later the next year, or two later if a 29 February comes in between, because that year from date to date has 366 days.

For long spans, count whole years from the start of year 1, and four landmarks do the work: the first 100 years give 5 odd days, the first 200 give 3, the first 300 give 1, and the first 400 give 0. (100 years hold 76 ordinary years and 24 leap years: 76 + 2 × 24 = 124 odd days, which is 17 weeks and 5. A century year is a leap year only if it divides by 400, so the fourth hundred has one leap year more and gives 6. Added up, 5, 10, 15 and 21 leave 5, 3, 1 and 0 over whole weeks.) So 2000 whole years give 0 odd days. Another stretch of 100 years can differ: 1901 to 2000 holds 25 leap years.

Why it helps in the paper

Counting weeks one by one, as Riya did, takes a step for every week. One division takes out all the whole weeks at once, and the remainder is all you need.

Try three

A week later

Kabir has a question for her:

2 November 2026 is a Monday. What day of the week is 2 November 2027?

No 29 February falls between them, so the gap is 365 days: 52 weeks and 1 odd day. One on from Monday. Riya says Tuesday before he has finished asking.