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
Also we
should understand that odd days can range from 06 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 –
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

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
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 –
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).
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

thk u sir ...very useful
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