Calendars - Important Formulas; Logical Reasoning

Important Formulas - Calendar

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.

100 years ≡ (76 ordinary years + 24 leap years )
≡ (76 x 1 + 24 x 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 x 2) = 10 ≡ 3 odd days

Number of odd days in 300 years = (5 x 3) = 15 ≡ 1 odd days

Number of odd days in 400 years = (5 x 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

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.

(1) Both years must be of the same type. i.e., both years must be ordinary years or both years must be leap years.

(2) 1^st January of both the years must be the same day of the week.