Basic Concepts: Calendars | CSAT Preparation - UPSC PDF Download

The year consists of 365 days, 5 hours, 48 minutes (52 weeks and 1 odd day). An extra day is added once in every fourth year which was called the leap year, which has 366 days (52 weeks and 2 odd days).To find the day of any given date of the year, you need to understand the calendar calculations:
1) 1st January 1 AD was Monday therefore; we must count days from Sunday. This means the 0th day was Sunday, so the 7th day was Sunday again and so on and so forth.
2) The day gets repeated after every seventh day (concept of a week), if today is Monday, then 28th day from now will also be Monday as it a multiple of 7 (28/7 = 4, so four weeks). Here the 30 day will be calculated by 30/7, which is 4 weeks and 2 days, these two days are called odd days. With starting day as Monday and two odd days, the day will be Wednesday; this point is the most critical in calendars. The other of looking at it is since the 28th day is Monday, so the 30th day will be Wednesday. But you have to understand and use the concept of odd days as the question may be about thousands of years.
3) In a normal year there are 365 days so 52 weeks and 1 odd day, in a leap year there are 366 days so 52 weeks and 2 odd days.
4) In 100 years there are 24 leap years and 76 normal years, so the number of odd days are 24(2) + 76 = 124, which is 17 weeks + 5 odd days, so 100 years have 5 odd days.
5) In 200 years the number of odd days is twice the number in 100 years which is 10, which is one week and 3 odd days, so 200 years have 3 odd days. In 300 years, the number of odd days is 15, which is two weeks and 1 odd day, so 300 years have one odd day.
6) 400 year is a leap year; similarly, the multiples of 400 are also leap years.
7) In 400 years, the number of odd days become 20 + 1(from the leap year), so total days are 21, which is three weeks and 0 odd days. In 400 years there are 0 odd days.

Important Formulas:
Odd Days:
We are supposed to find the day of the week on a given date. For this, we use the concept of 'odd days'. In a given period, the number of days more than the complete weeks are called odd days.

Leap Year:
i. Every year divisible by 4 is a leap year if it is not a century.
ii. Every 4th century is a leap year and no other century is a leap year.
Note: A leap year has 366 days.
Examples:
1) Each of the years 1948, 2004, 1676 etc. is a leap year.
2) Each of the years 400, 800, 1200, 1600, 2000 etc. is a leap year.
3) None of the years 2001, 2002, 2003, 2005, 1800, 2100 is a leap year.

Ordinary Year:
The year which is not a leap year is called an ordinary year. An ordinary year has 365 days.

Counting of Odd Days:
1 ordinary year = 365 days = (52 weeks + 1 day.)
1 ordinary year has 1 odd day

1 leap year = 366 days = (52 weeks + 2 days)
1 leap year has 2 odd days
100 years = 76 ordinary years + 24 leap year
= (76 x 1 + 24 x 2) odd days = 124 odd days
= (17 weeks + days) 5 odd days.

A number of odd days in 100 years = 5
A number of odd days in 200 years = (5 x 2) 3 odd days.
A number of odd days in 300 years = (5 x 3) 1 odd day.
A number of odd days in 400 years = (5 x 4 + 1) 0 odd day.
Similarly, each one of 800 years, 1200 years, 1600 years, 2000 years etc. has 0 odd days.