## Calendar

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## Important formula for Calendar

Odd Days: Odd days are the number of days, more than the complete number of weeks in a given period.

Normal Year: Year having 365 days (365 days = 52 weeks + 1 day). A normal year has 1 odd day.

Leap Year: Year having 366 days (365 days = 52 weeks + 2 day). A leap year has 2 odd days. A year that is divisible by 4 is a leap year.
Example: 1988, 1992, 1996, 2004, 2008, 2012 etc. are leap years.

Century year is considered to be a leap year if it is divisible by 400.
Example: 1200, 1600, 2000 etc. are leap years. Century years divisible by 4 but not by 400 are not considered to be leap.

### Counting of Odd days:

1) Every 100 years have 5 odd days.
100 years = 76 normal years + 24 leap years
Total odd days in 100 years = 76 + 48 = 124 odd days = Remainder obtained when 124 is divided by 7 = 5 days

2) Every 200 years have 3 odd days.

3) Every 300 years have 1 odd days.

4) Every 400 years have 0 odd days.

5) A month having 31 days has 3 odd days.

6) A month having 30 days has 2 odd days.

7) For a normal year month of February has 0 odd day and for a leap year month of February has in 1 odd day.

8) For first 6 months i.e from January to June we have 6 odd days in a normal year and 0 odd days in a leap year. This is helpful when you want to find a day after 30th June.

9) For first 9 months i.e from January to September we have 0 odd days in a normal year and 1 odd days in a leap year. This can be helpful when you want to find a day after 30th September.

NOTE: First day of the century must Tuesday, Thursday or Saturday.
Last day of the century cannot be Tuesday, Thursday or Saturday.

### Day Gain/Loss:

Ordinary Year (+1 day or -1 day)
When we proceed forward by 1 year, then 1 day is gained.
Example 1: August 9, 2013 is Friday, then August 9, 2014 has to be Friday + 1 day = Saturday.

When we move backward by 1 year, then 1 day is lost.
Example 2: December 25, 2011 is Wednesday, then December 25, 2012 has to be Wednesday + 2 days = Friday because 2012 is a leap year.

Leap Year (+2 day or -2 day)
When we proceed forward by 1 leap year, then 2 days are gained.
Example 3: August 9, 2013 is Friday, then August 9, 2014 has to be Friday + 1 day = Saturday.
When we move backward by 1 leap year, then 2 days are lost.

Example 4: December 24, 2013 is Tuesday, then December 24, 2012 has to be Tuesday - 1 day = Monday ( The day must have crosses February 29 for adding 2 days otherwise add 1 day ).

Example 5: January 26, 2011 is Wednesday, then January 26, 2012 has to be Wednesday + 1 day = Thursday.
Example 6: March 23, 2011 is Wednesday, then March 23, 2012 has to be Wednesday + 2 days = Friday ( As February 29 has been crossed ).

### Find a Particular Day on the Basis of a Given Day and Date

Step 1: Find the total number of odd days between the given date and the date for which the day is to be determined.
Step 2: The day to be determined, will be that day of the week which is equal to the total number of odd days and this number is counted forward from the given day, in case the given day comes before the day to be determined. But, if the given day comes after the day to be determined, then the same counting is done backward from the given day.

Example 1: If January 5, 1991 was Saturday, then what day of the week was it on March 4, 1992.

Solution: Odd days from January 5, 1991 to January 5, 1992 = 1
Odd days from January 6, 1992 to January 31, 1992 = 5
Odd days in February 1992 = 1
Odd days from March 1, 1992 to March 4, 1992 = 4
Total Odd days = 1 + 5 + 1 + 4 = 11. When 11 is divided by 7 remainder is 4 therefore, March 4, 1992 will be 4 days after Saturday, which is Wednesday.

Example 2: What day of the week was it on November 5, 1987, if it was Monday on April 4, 1988.

Solution: Odd days from November 5, 1987 to April 4, 1988 = (30 - 5) days of November 1987 + 31 days of December 1987 + 31 days of January 1988 + 29 days of February 1988 + 31 days of March 1988 + 4 days of April 1988.
= 25 + 31 + 31 + 29 + 31 + 4 = 151 days = 21 weeks and 4 days
Total Odd days = 4 days.
November 5, 1987 will be 4 days before Monday i.e. Thursday.

### To Find a Particular Day without Given Day and Date

Step 1: Find the total number of odd days upto the date for which the day is to be determined.
Step 2: Required day will be according to the following table.

## Basic formula for Calendar

Number of Odd Days Day
1 Monday
2 Tuesday
3 Wednesday
4 Thursday
5 Friday
6 Saturday
0 Sunday

Example 3: Find the day of the week on January 26, 1950.

Solution: Odd days till January 26, 1950 = Odd days for 1600 yr + Odd days for 300 yr + Odd days for 49 yr + Odd days of 26 days of January 1950.
= 0 + 1 + (12 × 2 + 37) + 5 = 67 days = 9 weeks and 4 days = 4 odd days.
Therefore, January 26, 1950 is Thursday.
NOTE: 49 yr have 12 leap year and 37 ordinary years.

Example 4: Mahatama Gandhi was born on October 2, 1869. What was the day of the week.

Solution: Odd days till the year 1868 = Odd days for 1600 yr + Odd days for 200 yr + Odd days for 68 yr
= 0 + 3 + (17 × 2 + 51) = 3 + 85 = 88 days = 12 weeks and 4 days = 4 odd days.
Now, total number of Odd days in year 1869 till October 2, 1869
January = 3; February = 0; March = 3; April = 2; May = 3; June = 2; July = 3; August = 3; September = 2; October = 2
23 days = 3 weeks + 2 days.
Total number of odd days = 4 + 2 = 6 days.
Therefore, October 2, 1869 is Saturday.