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Tropical Year and Civil Year

We all know that earth moves round the sun. Earth takes 365.2422 days (corrected up to 4 decimal places) to complete one revolution. This period is called a tropical year. Since the tropical year does not contain exact number of days, it is not suitable for the purpose of civil life. Hence, another kind of year called civil year was introduced. This civil year originally consisted of 365 days.
 
It is evident that one civil year was shorter than one tropical year by 0.2422 days. So, 4 such civil years were shorter than 4 tropical years by 0.9688 days. This error was partly set right by the Roman Emperor Julius Caesar, with the help of an astronomer Sosigenes by introducing the concept of leap year. He suggested that one civil year in every 4 consecutive years should consist of 366 days.
 
On effecting the above correction, 4 civil years were in excess of 4 tropical years by 0.0312 of a day (because 1 day is added instead of 0.9688 of a day). Thus, 400 civil years exceeded 400 tropical years by 3.12 days. This error was again corrected in the year 1582 A.D., during the period of Pope Gregory XIII. He suggested that 3 leap years out of 100 leap years in a period of 400 years must be converted into ordinary years. On account of this correction, a century year is not a leap year unless it is divisible by 400. For instance, the year 1900, 2100, 2200, 2300 are not leap years. But the year 2000, 2400 and 2800 are leap years as per Gregorian correction. Consequent to this correction effected by Pope Gregory, the calendar is named as Gregorian calendar.
 
In spite of all these ingenious corrections, there is still a small error in Gregorian calendar. Four thousand civil years are in excess of 4000 tropical years by 1.2 days.

How Does the Number of Odd Days Help us in Finding the Day on a Given Date?

After effecting the corrections to the calendar as narrated above, the first day of Gregorian calendar (1 January 0001) was shifted to Monday. So, while finding out the day on a particular date, the number of odd days should be counted from Monday. If the number of odd days is 1, the particular date was on Monday. If the number of odd days is 2, the particular date was on Tuesday and so on. If the number of odd days is 6, the particular date was on Saturday and if the number of odd days is 0 (zero), the particular date was on Sunday.
 
The above concept can be better understood by considering an example.
 
We shall find out the day on which 12 January 1979 fell.
 
We know that the number of odd days in a normal year is 1 and the number of odd days in a leap year is 2. In the first century, there were 24 leap years and 76 normal years (because the year 100 was a normal year as per Gregorian correction). So, the total number of odd days in the first century works out to 5 as detailed below.

Number of odd day in 76 normal years 76 × 1 = 76.
 
Number of odd day in 24 leap years 24 × 2 = 48.
 
Total number of odd days in first 100 years = 124.

On converting the total number of odd days to weeks, we get 17 weeks and 5 odd days.
Total number of odd days in the first two centuries = 5 × 2 = 10, which is equal to 1 week and 3 days. Total number of odd days in the first three centuries = 5 × 3 = 15, which is equal to 2 weeks and 1 day.
 
In the fourth century, there are 25 leap years and 75 normal years. So, the number of odd days in fourth century is 6. Hence, the total number of odd days in the first 4 centuries (400 years) is 1 + 6 = 7, i.e., 0 (zero).
 
So, the number of odd days in every block of 400 years, commencing from the first day of Gregorian calendar, is zero.
 
Now, we shall calculate the total number of odd days from the first day of the Gregorian calendar (i.e., from 1 January 0001) till 12th January 1979.
 
Number of odd days in the first 1600 years Nill
Number of odd days in the next 300 years 1
Number of odd days in the next 78 years (19 leap years and 59 normal years) 6
Number of odd days in 12 days of January 1979 5

The total number of odd days till 12 January 1979 = 12, which is equal to 1 week and 5 odd days.
 
If we count 5 days from Monday, we get Friday. So, 12 January 1979 was Friday.

Following Points Are to Be Remembered While Solving Calendar Problems

  1. An ordinary year (normal year) has 365 days. Number of odd days in a normal year is 1.
  2. A leap year has 366 days. Number of odd days in a leap year is 2.
  3. In one normal century, there are 24 leap years and 76 normal years. So, the total number of odd days in a century is 5 days.
  4. In four centuries, the total number of leap years is 97 because the fourth century year is a leap year. So, the total number of odd days in four centuries is zero (97 × 2 + 303 × 1 = 497), which is equal to exactly 71 weeks).
  5. Commencing from the first day of Gregorian calendar (i.e., from 1 January 0001), the total number of odd days in one century is 5, in two centuries it is 3, in three centuries it is 1 and in four centuries it is 0. This cycle repeats for every four centuries.

Some Basic Information to Be Remembered

If 1 March 1983 was a Tuesday, then

If 1 February 1983 was a Tuesday, then

1 March 1982 was a Monday

1 February 1982 was a Monday

1 March 1981 was a Sunday

1 February 1981 was a Sunday

1 March 1980 was a Saturday

1 February 1980 was a Friday

1 March 1979 was a Thursday

1 February 1979 was a Thursday

1 March 1978 was a Wednesday

1 February 1978 was a Wednesday

1 March 1984 was a Thursday

1 February 1984 was a Wednesday

1 March 1985 was a Friday

1 February 1985 was a Friday

 

 

Month

No. of Days

Odd Days

January

31

3

February

28/29

0/1 (ordinary/leap year)

March

31

3

April

30

2

May

31

3

June

30

2

July

31

3

August

31

3

September

30

2

October

31

3

November

30

2

December

31

3





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