Calendar Problems: Find the Day of Any Date with Odd Days

Days of the week repeat every 7 days, so any number of days can be split into whole weeks and a remainder. The remainder is called the number of odd days. Whole weeks never change the weekday, so only the odd days matter.

• An ordinary year has 365 days = 52 weeks + 1 odd day.
• A leap year has 366 days = 52 weeks + 2 odd days.

The method counts the odd days from a fixed starting point up to the date you want, divides by 7, and reads the remainder as a weekday: 0 = Sunday, 1 = Monday, 2 = Tuesday, 3 = Wednesday, 4 = Thursday, 5 = Friday, 6 = Saturday.

A year is a leap year if it is divisible by 4, except century years, which must be divisible by 400. So 2024 and 2000 are leap years, but 1900 is not.

A century has 24 leap years and 76 ordinary years if its last year is not a leap year. That gives 76 + 2 × 24 = 124 days beyond whole weeks, which is 5 odd days.

Because 400 years give 0 odd days, you can drop any multiple of 400 years. For a date in the 1900s, count 1600 years (0 odd days), then 300 years (1 odd day), then the remaining years.

For the part of the year before your date, add the odd days of each completed month. A 31-day month gives 3 odd days, a 30-day month gives 2, and February gives 0 in an ordinary year or 1 in a leap year.

Any calendar confirms that 15 August 1947 was a Friday. Checking a known date like this is a good way to make sure you have not dropped a leap year.

26 January 1950 was indeed a Thursday.

For "days later" questions, skip the calendar and use the remainder directly.

Two years have identical calendars when they begin on the same weekday and are both leap years or both ordinary years. Add up the odd days year by year from the given year until the total is a multiple of 7, and check that the year you land on is of the same type.

You can check: 1 January 2023 and 1 January 2034 both fall on a Sunday.

A leap year's calendar normally repeats after 28 years. 2024 began on a Monday, and so does 2052, the next leap year to do so. The rule can break across a century year that is not a leap year, such as 2100.

For reference, 1 January 2024 fell on a Monday, which is the starting point for that comparison.

• ✘ Treating 1900 or 2100 as a leap year.
• ✘ Counting February with 29 days in an ordinary year.
• ✘ Including the target year itself as a complete year.
• ✘ Matching start days without checking that both years are leap years or both ordinary years.

Other time-based reasoning: angles between clock hands and fast and slow clocks.

Is 1900 a leap year?

No. A century year is a leap year only if it is divisible by 400. So 1600 and 2000 were leap years, but 1700, 1800 and 1900 were not, and 2100 will not be one either.

Which day number stands for Sunday in the odd days method?

With the usual convention, 0 odd days means Sunday, 1 means Monday, and so on up to 6 for Saturday. This works because 1 January of year 1, counted back on the Gregorian calendar, falls on a Monday, one odd day after a Sunday.

All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.