Calendar

It is pertinent to define the term “odd days” before commencing the discussion on this chapter. “Odd days” as a concept is understood by all of us and we use it very frequently without trying to find a name for it.

The number of days in excess of sets of weeks in any number of days is called odd days.

Alternately, the remainder on dividing any number of days with 7 is said to be the number of odd days in that many days.

Eg: 10 days ➙ 1 week, 3 days or 10/7 gives a remainder of 3 ➙ no of odd days = 3

4 days ➙ 0 week, 4 days or 4/7 gives a remainder of 4 ➙ no of odd days = 4

30 days ➙ 4 weeks, 2 days or 30/7 gives a remainder of 2 ➙ no of odd days = 2

Basically odd days allows us to calculate the day on a particular date with reference to another date. If for instance, the present day is Tuesday, 100 days later will be Tuesday + 2 = Thursday (100 days is equivalent to 14 weeks and 2 days. Any number of exact weeks will give us the same day; hence we add the remainder after dividing with 7).

Also we should understand that odd days can range from 0-6 only (as our divisor is 7). Hence if it is said that an event will take place after 5 weeks and 42 days from the present day, which is a Monday; it implies that the event will occur on a Monday + remainder of 42/7 (i.e. 0) = Monday. (If number of odd days is in excess of 6, we divide it further by 7 to find the final remainder)

Leap Year

A year consisting of 366 days with 29 of them featuring in the month of February is known as a leap year. A normal year on the other hand has 365 days with 28 days in the month of February.

A gap of one year need not necessarily be 1^st Jan – 31^st Dec and can begin from any date of the year. If we compare two one year durations – 

i) 23 Feb’2015 – 22 Feb’2016

ii) 4 Mar’2015 – 3 Mar’2016

We begin by comparing the part of the year which defines a year to be normal/leap year which is Feb end –

In the first case we pass through Feb end of 2015 – as 2015 is not a leap year, this gap is of 365 days and equivalent to a normal year gap. In the second case we pass through Feb end of 2016 – as 2016 is a leap year, this one year gap is of 366 days and is equivalent to a leap year gap.

The distribution of odd days month wise and annually is as follows:

Month	Normal Year		Leap Year
	no of days	equivalent odd days	no of days	equivalent odd days
January	31	3	31	3
February	28	0	29	1
March	31	3	31	3
April	30	2	30	2
May	31	3	31	3
June	30	2	30	2
July	31	3	31	3
August	31	3	31	3
September	30	2	30	2
October	31	3	31	3
November	30	2	30	2
December	31	3	31	3
TOTAL	365	29 or 1	366	30 or 2

From the above we can conclude that any one year gap is a gap of 1 odd day or 2 odd days which can be verified by the number of days in the month of February of the year in which we are passing through February-end during that 1 year duration.

Pattern of Years

As we know that a day is one complete rotation of Earth while revolving around the Sun and as the number of rotations during one full revolution around the Sun is not a whole number, the need arises to accommodate for the approximations by having leap years.

The general perception of every 4^th year being a leap year is not entirely correct. While every 4^th year (i.e. every year which is a multiple of 4) is a leap year; the rule changes when referred to century years like 1800/1900/2000. Although multiples of 100 are also multiples of 4, in case of hundredth years, only multiples of 400 are leap years.

So while 1600 and 2000 are leap years, 1700, 1800 and 1900 are not leap years.

The leap year pattern is as follows –

Century Years (100,200,300,400….)	every multiple of 400 years is a leap year	eg. 400, 800,1200, 1600, 2000
Other Years (1,2,3,4,….)	every multiple of 4 years (except century years) is a leap year	eg. 4, 8, 12, 16….96, 104,108…..

Calculation of Day for any given date

While calculating the day of the week for any given date we find the number of odd days between the first day of 1AD till the given date. For this we need to know that the first day of the week is Monday and hence 1^st January 1AD was a Monday.

Putting together what has been mentioned above, we can find the day of the week for any given date. Illustrating the same by finding the day of the week for 15^th August 1947 –

Description	Period	Odd Days
Till 1947, 1600 years have gone by with 0 odd days. (As there are no odd days in 400 years, there will be no odd days in multiples of 400)	1600 years	0
After 1600, another 300 years have gone by. Every 100 years has 24 leap years and 76 normal years (as 100th year is not a leap year). Number of odd days in next 100 years: [(24 x 2) + (76 x 1)] = 124 which when further divided by 7 leaves a remainder of 5.	1601 - 1700 years	5
Same as above	1701 - 1800	5
Same as above	1801 - 1900	5
After 1900, 46 complete years have gone by with 11 leap years (every 4th year) and 35 (46 - 11) normal years with following number of odd days: [(11 x 2) + (35 x 1)] = 57 which when further divided by 7 leaves a remainder of 1.	1901 - 1946	1
Jan to Jul, no off odd days will be as discussed above	Jan 1947	3
	Feb 1947	0
	Mar 1947	3
	Apr 1947	2
	May 1947	3
	Jun 1947	2
	Jul 1947	3
Till 15th there is 1 odd day in August (15/7 leaves a remainder of 1)	Aug 1947	1
Total odd days	till 15 Aug 1947	33

 

33 days when further divided by 7 leaves a remainder of 5. Hence 15^th Aug 1947 was 5^th day of the week i.e. Friday (counting from 1 as Monday).