# Calendars

You have to find the day of the week on a particular given date. The process of finding it depends upon the **number of odd days**, which are quite different from the odd numbers.

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

**Ordinary Year:** An ordinary year has 365 days.

**Leap Year: **Every year (except century) which is divisible by 4 is called a leap year whereas century is a leap year by itself when it is divisible by 400. For example, 1964, 1968, 1972, 1984, and so on, are all leap years whereas 1986, 1990, 1994, 1998, and so on, are not leap years.

Further, the centuries 1200, 1600, 2000 and so on, are all leap years as they are divisible by 400 whereas 900, 1300, 1500 and so on, are not leap years.

### Counting of Odd Days

**1.** One ordinary year = 365 days = 52 weeks + 1 day. Therefore, an ordinary year has 1 odd day.

**2.** One leap year = 366 days = 52 weeks + 2 days. Therefore, a leap year has 2 Odd days.

**3.** 100 years = 76 ordinary years + 24 leap years

= 76 odd days + 24 × 2 odd days = 124 odd days = 17 weeks + 5 days

Therefore, a century (100 years) has 5 odd days.

**4.** 200 years contain 10 and therefore 3 odd days.

**5.** 300 years contain 15 and therefore 1 odd day.

**6.** 400 years contain (20 + 1) and therefore 0 odd days. Similarly, each one of 800, 1200, 1600, and so on contain 0 odd days.

**7.** (7n + m) odd days, where m less than or equal to 7 is equivalent to m odd days.

**Working Rule #1**

To find the day of the week on a particular date when reference day is given:

Step I: Find the net number of odd days for the period between the reference date and the given date (Exclude the reference day but count the given date for counting the number of net odd days).

Step II: The day of the week on the particular date is equal to the number of net odd days ahead of the reference day (if the reference day was before this date) but behind the reference day (if this date was behind the reference day).

**Working Rule #2**

To find the day of the week on a particular date when no reference day is given:

Step I: Count the net number of odd days on the given date.

Step II: Write Sunday for 0 odd day, Monday for 1 odd day, Tuesday for 2 odd days, and so on.

### Examples

**Example 1:** January 11, 1997 was a Sunday. What day of the week was on January 7, 2000?

Total number of days between January 11, 1997 and January 7, 2000

= (365 - 11) in 1997 + (365 days in 1998) + (365 days in 1999) + (7 days in 2000)

= (50 weeks + 4 odd days) + (52 weeks + 1 odd day) + (52 weeks + 1 odd day) + (7 odd days)

= 13 days = 1 week + 6 odd days

Hence, January 7, 2000 would be 6 days ahead of Sunday, i.e., it was on Saturday.

**Example 2:** What day of the week was on June 5, 1999?

June 5, 1999 means 1998 years + first five months up to May of 1999 + 5 days of June

1600 years have 0 odd day. 300 years have 1 odd day.

98 years have 24 leap years + 74 ordinary years = (24 × 2) + (74 × 1) days

= 122 days = 17 weeks + 3 odd days

Thus, 1998 years have 4 odd days.

January 1, 1999 to May 31, 1999 has 3 + 0 + 3 + 2 + 3 = 11 odd days or 4 odd days.

Total number of odd days on June 5, 1999 = 4 + 4 + 5 = 13 odd days or 6 odd days.

Hence, June, 5 1999 was Saturday.