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Calendar

We know that
 
Any non-leap year contains 365 days = 52 weeks + 1 day
And leap-year contains 366 days = 52 weeks + 2 days
 
This 1-day and 2 days extra added to any year create changes in the calendar and this is the reason why calendar of Nth year will not be same as N + 1th year.

Before we proceed ahead, we should be very clear about two things:

Which years are leap years?

It takes the earth about 365.2422 days to go around the sun, but a normal calendar year is only 365 days. The extra fraction of a day added up four times makes four years (or, four revolution of earth around sun) 1460.9688 days, but four calendar years would only be 1460 days. That 0.9688 is almost a whole day, so every four years we add an extra day to our calendar, February 29. We call that year leap year. To make things easier, leap years are always divisible by four: 2004 and 2008 will both be leap years.
 
For hundreds of years, people used a calendar called the Julian calendar that followed this rule, adding a leap year every four years. However, because 0.9688 isn’t exactly a whole day, the Julian calendar slowly began to disagree with the real seasons. In 1582, Pope Gregory fixed this problem by ordering everyone to use a new set of rules. These rules are named the Gregorian calendar, after him. They work like this:
 
The Gregorian Calendar
 

Rule

Examples

Every fourth year is a leap year.

2004, 2008, and 2012 are leap years.

However, every centenary year is not a leap year.

1900 and 2100 arenot leap years.

In case of centuries, every four hundred years, there’s a leap year after all.

2000 and 2400 are leap years.


In layman terms, all the century years divisible by 400 will be leap years and all the non-century years divisible by 4 will be leap years. So, leap year next to 2096 AD is 2104 AD and not 2100 AD.
 

Because 2000, 4000, 6000, etc. are
leap years and 1000, 3000, 5000, etc. are not, the number of leap days in each millennium alternates between 242 and 243, with the first, third, etc. millennia (i.e., 1–1000, 2001–3000, etc.) having 242 leap days, and the second, fourth, etc. (i.e., 1001–2000, 3001–4000, etc.) having 243 leap days.

How the days of consecutive years change?

Due to any non-leap year, calendar of next year go ahead by 1 day and due to any leap year, calendar of next year goes ahead by 2 days, but this change in calendar will be there only before 29th February.
 
It can be seen through the example given below:

 

1991

1992

1993

1st January

Sunday

Monday

Wednesday

28th February

Tuesday

Wednesday

Friday

1st March

Wednesday

Friday

Saturday


In the above example, we have assumed that 1st January of 1991 is Sunday. 1991 and 1993 are non-leap years and 1992 is a leap year.


If now we try to find out the symmetricity of calendars, we can see this in the following way:
  1. For any leap-year
Let us see, for example, the case of 1972.

Year

1972

1973

1974

1975

1976

Excess days

2

1

1

1

2


Since no. of excess days are 7, so the days of the year 1972 and year 1977 will be same from 1st of January and 28th of February.
  1. For any (leap-year+1) year

Year

1973

1974

1975

1976

1977

1978

Excess days

1

1

1

2

1

1


Since no. of excess days are 7, so calendar of year 1973 and 1979 will be same for whole year.
  1. For any (leap-year + 2) year

Year

1974

1975

1976

1977

1978

1979

Excess days

1

1

2

1

1

1


Since excess days are 7, so calendar of year 1974 and 1980 will be same till 28th of February.
  1. For any (leap-year + 3) year

Year

1975

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

Excess days

1

2

1

1

1

2

1

1

1

2

1


Since no. of excess days are 14, so calendar of year 1975 and 1986 will be same for whole year.


This whole mechanism can be summed up in following way:
 

Nature of year

No. of years after which 1st January will be same

Leap year

5

Leap year + 1

6

Leap year + 2

6

Leap year + 3

11


So, if 1st January of 1972 and 1st January of 1977 will be on same day.
 
If 1st January of 1973 and 1st January of 1979 will be on same day and so on.

Exception

No century year, which is not a leap year, should be included in this calculation.
 
Example
Sum of dates of last Monday of previous month and 1st Thursday of next month is 38. If both the dates are of the same year, then which month is the current month?
Solution
Sum of dates of last Monday of previous month and 1st Thursday of next month is 38 is possible only if last Monday is 31st and 1st Thursday is 7th. (Since if we take 30 + 8 = 38, then 30 can be last Monday of any month but 8th can not be the 1st Thursday of any month)
 
So, 31st of last month is a Monday. Hence 7th of current month, 14th of current month, 21st of current month and 28th of current month will be a Monday.
 
Now, if current month is a month with 30 days, then 5th of next month will be a Monday, so 7th of next month cannot be a Thursday.
 
If current month is a month with 31 days, then 4th of next month will be a Monday, so 7th of next month will be a Thursday.
 
Finally we can conclude that previous month and current month, both are having 31 days. Since both the dates are of the same year, so current month is August.
 

Finding day of a date by using a reference date

Let us see this with the help of an example: If 9th Dec of 1972 is Sunday, then which day it will be on 14th December 1998?
 
Process: There are several processes to do this calculation: 1. Year method, 2. Days method, 3. Actual calculation method
  1. Year Method – We use the above given table to find out about any of the years.
     
    9.12.1972 – Sunday
     
    1.1.1973 – Tuesday (It is a Leap year + 1 year)
     
    So, 1.1.1979 – Tuesday, (It is a Leap year + 3 year)
     
    So, 1.1.1990 – Tuesday, (It is a Leap year + 2 year)
     
    So, 1.1.1996 – Tuesday
     
    Now, we can find out all the next years one-by-one.
     
    1.1.1997 – Thursday
     
    1.1.1998 – Friday – 31.12.1998 – 24.12.1998 – 17.12.1998
     
    So, 14.12.1998 – Tuesday
  2. Days method – We use the no. of excess days every year to find out the no. of days calendar will move ahead by.
     
    1.1.1973 – Tuesday
     
    Due to 1973, calendar will go ahead by 1 day, similarly due to 1974 – 1 day, due to 1975 – 1 day and due to 1976- 2 days.
     
    So, in four years, calendar will go ahead by 5 days.
     
    Using unitary method, in four years, calendar will move ahead by 5 days.
     
    So, in 24 years calendar will move ahead by 30 days. Hence calendar will move ahead by 2 days.
     
    So, 1.1.1997 will be two days ahead of Tuesday i.e. Thursday.
     
    Now, it is calculation as given in Year Method.
  3. Actual Calculation method – With the help of this method, we can find the actual day of any date of 20th century. To use this method effectively, we need to remember the Month Codes of all the months.
     
    Let us learn this method by finding the date of 15th August 1947 –
     
    At 1st, add the Date + Month code of August + Last two digits of year + Description: 10788.png
     
    (Where [x] represents the greatest integer value of x.)
     
    So, corresponding to 15th August 1947 – 15 + 3 + 47 + 11 = 76
     
    Now, divide this value by 7 to find out the remainder.
     
    If the remainder is 0 then it is a Saturday
     
    If the remainder is 1 then it is a Sunday
     
    If the remainder is 2 then it is a Monday
     
    If the remainder is 3  then it is a Tuesday
     
    If the remainder is 4 then it is a Wednesday
     
    If the remainder is 5 then it is a Thursday
     
    If the remainder is 6 then it is a Friday
     
    Here, remainder is 6, so 15th August 1947 was a Friday.(It should have been ‘Free’day)
List of Month Code:
 

 

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Leap Year

0

3

4

0

2

5

0

3

6

1

4

6

Non-leap year

1

4

4

0

2

5

0

3

6

1

4

6





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