Calendar

A Solar year consists of 365 days 5 hours 48 minutes and 48 seconds. It was due to this reason that the calendar system prevailing now-a-days, an year is taken for 365 days. In order to get rid of this day an extra day is added in every 4th year which is called a 'leap year'. This calendar is known as Julian Calendar, arranged 47 BC by Julius Caesar.

In India, Vikrami and a number of other calendars were in use till 1952. On the basis of report of a committee formed, Govt of India adopted the National Calendar based on Saka with Chaitra as the first month. The days of this Vikrami calendar have permanent correspondence with those of Gregorian calendar - Chaitra beginning on March 22 and March 21 in ordinary and leap years respectively.

Remember:
(a) an ordinary year contains 365 days i.e., 52 weeks and 1 odd day
(b) a leap year contains 366 days i.e., 52 weeks and 2 odd days
(c) a leap year must be divisible by 4 but if complete century then by 400
(d) 100 years contain 5 odd days
(e) 01.01.0001 was Monday - hence count Sunday = 0, Monday = 1…odd days
(f) 400 years contain NIL odd days.
(g) An ordinary year leaves one odd day while a leap year leaves 2 odd days

(1) Odd Days:
Number of days more than the complete weeks are called odd days in a given period.
(2) Leap Year:
A leap year has 366 days.
In a leap year, the month of February has 29 days.

Every year divisible by 4 is a leap year, if it is not a century.

Examples: 1952, 2008, 1680 etc., are leap years.
1991, 2003 etc., are not leap years
Every 4th century is a leap year and no other century is a leap year.

Examples: 400, 800, 1200 etc., are leap years.
100, 200, 1900 etc., are not leap years (3) Ordinary Year:
The year which is not a leap year is an ordinary year.

An ordinary year has 365 days (4) Counting odd days and calculating day of any particular date
1 ordinary year ? 365 days ? (52 weeks + 1 day)
Hence number of odd days in 1 ordinary year = 1.
1 leap year ? 366 days ? (52 weeks + 2 days)
Hence number of odd days in 1 leap year = 2.
years ? (76 ordinary years + 24 leap years)
? (76 × 1 + 24 × 2) odd days ? 124 odd days.
? (17 weeks + 5 days) ? 5 odd days.
Hence number of odd days in 100 years = 5.
Number of odd days in 200 years = (5 × 2) = 10 ? 3 odd days
Number of odd days in 300 years = (5 × 3) = 15 ? 1 odd days
Number of odd days in 400 years = (5 × 4 + 1) = 21 ? 0 odd days
Similarly, the number of odd days in all 4th centuries (400, 800, 1200 etc.,) = 0

Mapping of the number of odd day to the day of the week

No. of odd days
Week day 0
Sunday 1
Monday 2
Tuesday 3
Wednesday 4
Thursday 5
Friday 6
Saturday (5) Additional Notes

Last day of a century cannot be Tuesday or Thursday or Saturday.
For the calendars of two different years to be the same, the following conditions must be satisfied:
(a) Both years must be of the same type. i.e., both years must be ordinary years or both years must be leap years;
(b) 1st Day of the year i.e., 1st January of both the years must be the same day of the week.

Example: What day was 15th August 1947
Solution
1947 means 1946 complete number of years
1946 = 1600 + 300 + 46 + leap years
therefore # of odd days = 0 + 15 + 46 + 11 = 72 days
# of odd days in January = 31

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