BEST CALENDAR TRICKS | ODD DAY CONCEPT.

                                           

Sohel Sahoo:  Hello Everyone;
In this article, we are going to discuss the topic of calendars. To solve questions from this topic, you first need to understand the concept of odd days.The questions from this topic appears off and on in all the major competitive exams.However, this topic, if understood properly, is easier to handle as compared to a few other topics of mathematics.

What is a Calendar?

A Calendar is a chart or series of pages showing the days, weeks, and months of a particular year, or giving particular seasonal information.

In our exams only Gregorian calendars are included which are the ordinary ones.

 Concept of  Odd Days:-

The additional days which couldn’t make a week are called as odd days in calendar problems. 

Basic Structures of a Calendar:

1.   Ordinary year: Any year which 365 days is called an ordinary year.Ex: 1879, 2009, 2019, etc.

The division of the number 365 by 7 gives the quotient 52 and remainder 1 which indicates that an ordinary year has 52 weeks and one extra day. This extra day is referred to as an “odd day” throughout the calendar topics.

2. Leap year: Any year which has 366 days is called a leap year.Ex: 2012, 2016 2020 etc.

A leap year has 366 days, the division of the number 366 by 7 gives the quotient 52 and remainder 2. This indicates that a leap year has 52 weeks and 2 extra days. These two extra days are also referred to as “odd days”.

NOTE:-

  1. An ordinary year has one odd day, whereas a leap year has two odd days.

  2. In an ordinary year 1^st day of the year = Last day of the year

        Such as, 01-01-2017 was Sunday

     31-12-2017 also Sunday.

3.In a leap year the last day of the year will be one more that of 1^st day of the year.

For example, 01-01-2016 was Friday

                       31-12-201 was Saturday.

Evaluation of leap year:-

The leap year occurs every four years, most of the time, but there are cases where the gap between two leap years was 8 years instead of the regular 4 years.

Ex: The year 1896 is a leap year. The next leap year comes in 1904 (1900 is not a leap year).

In order to make this concept easier and faster, any year which is divisible by number 4 completely (remainder becomes zero) is considered as a leap year.

Ex: 1888, 2012, 2016 are leap years as it’s completely divisible by 4. Years like 2009, 2019 etc. are not divisible by 4 completely hence they normal years.

An exception to note:

A year 700 is completely divisible by 4, but it is not considered as a leap year. For a century year, the logic follows that it should always be divisible by 400 not by 4. Even though the year 700 is divisible by 4 but not by 400. Hence, the year 700 cannot be considered as a leap year.

Ex: 400, 800, 1200 etc. are leap years as they are divisible by 400 and years 300, 700, 100 etc. are not leap years as they are not divisible by 400.

Number of odd days in a month:-

January has 31 days irrespective of whether it’s an ordinary year or leap year. The division of the number 31 by 7 provides the remainder 3. Hence the January has 3 odd days. On generalizing, any month which has 31 days has 3 odd days and any month which has 30 days has 2 odd days.

# The only exception happens is in the case of February. The February month of an ordinary year has 28 days, division of 28 by 7 provides zero as remainder. Hence the number of odd days in February of an ordinary year will have 0 odd days and that of leap year will have 1 odd day as February in a leap year has 29 days.

Table: List of number of odd days in every month

The below table depicts the number of odd days in different months of a calendar year:

Month Name	Number of odd days
January	3
February(ordinary/leap)	(0/1)
March	3
April	2
May	3
June	2
July	3
August	3
September	2
October	3
November	2
December	3

Type 1 Problems: Finding the day when another day is given:-

In this section, one has to find out the day of the week of a given date using the day of the week information on the reference date on examination paper.

Steps to follow are given below;

Step-1: First find the difference between the given two years & divide it by 4& take the quotient only.Add the difference and the quotient. This is same as finding ordinary&leap years & taking odd days.

Step-2: Now count the odd days for the months completed & add extra days to complete the date.

Step-3: Finally divide by 7 & find remainder. Now,count the remainder days in the given day.The result will be the required day.

You can understand these steps better with the help of following examples.

Question: If 17^th march 2008 was Monday, what was 1^st April 2012?

Solution: The total number of odd days from 17^th March 2008 to 17^th March 2012.

2008 (leap year)	2 odd days
2009 (ordinary year)	1 odd day
2010 (ordinary year)	1 odd day
2011 (ordinary year)	1 odd day
Total odd days =	5 odd days

Since 17^th march 2008 was Monday and 17^th march 2012 is 5 days more than Monday.

 Then adding 5 odd days to Monday, we get Saturday. Hence 17^th march to April 1^st we have 15 days. Saturday+15=Sunday. Adding 15 days or (15 = 14+1) to Saturday, we get the answer as Sunday.

When the day&month of reference date& asked date are equal:-

Q.1: If  11-04-1717 was Friday  then what would be 11-04-1721?

Sol:-  11-04-1717 = Friday

            11-04-1718 = Saturday

           11-04-1719 = Sunday

         11-04-1720 = Tuesday (*Leap year)

         11-04-1721 = Wednesday

Shortcut: NO. of years = 1721-1717 = 4

           Leap year in these 4 years = Quotient by dividing 4 by 4 = 1

Odd days = 4+1 = 5

[{ No of ordinary years = 3 & leap year =1 in 1717 to 1721. Odd days = 3x1+1x2=5 } Clearly, instead of finding separately we can find the difference & just add 1 odd day for leap year as leap year also in this difference]

Now, count 5 day from Friday & it would be Wednesday.

Q.2: If  23-07-1921 was Wednesday  then what would be 23-07-1941?

Sol:- Year in between 1921 to 1941 is 20.

No. of  Leap years = 5 [ 20/4 = 5 quotient]

Odd days = 20+5 =25

As 25 days can make weeks so divide 25 by 7 & remainder will be 4.

Counting 4 days from wednesday we have sunday.

When the reference date & asked date is completely different:-

We should break the asked date as one date whose day& month are exact matched with reference date & then count the remaining odd days of rest months&days as explained below.

Q. If 15-03-1941 was wednesday then what would be day on 10-09-1947?

Sol:- We can rearrange 10-09-1947 as 15-03-1947 + (Rest days from march 16 to september 10)

           Difference of years= 1947-1941 =6 &leap year =1

          Days in March (31-15) =16

       Odd Days for April to August = 2+3+2+3+3=13

         september 10 = 10

Sum of days = 6+1+16+13+10 = 46

Hence odd days = remainder obtained by dividing 46 by 7 = 4

So the day will be 4 more than wednesday i.e. Monday.

Q. If 27-04-1738 was sunday then what it would be on 16-11-1750?

Sol:- For years,        1750-1738 = 12

                                   Leap years = 3

Odd days in days, April(30-27) = 3

                       and november 16 = 2

Odd days in months, (from may to october) = 3+2+3+3+2+3 = 16

Now, total odd days = 12+3+3+2+16 = 36

So,odd day = remainder by dividing 36 by 7 = 1

Counting 1 day from Sunday we have Monday as our answer.  

Type 2 Problems: Matching the Calendar year with another year:-

Question: Which year in the future will have the same calendar exactly as 2009?

A. 2010

B. 2013

C. 2015

D. 2017

Solution: If the total number of odd days between any years is zero or it’s a multiple of seven. Then, those two years will have the same calendar.

The total number of odd days is listed below:

Year	Odd days
2009	1
2010	1
2011	1
2012	2
2013	1
2014	1
Total number of odd days = 7

Hence, 2015 will have the same calendar as 2009. Option C is the correct answer.

There is also a type in which a date is given & the corresponding day is asked. I have covered this type as a special method . You may check it out at : Finding Day from any given Date in the Calendar .

If you want to learn reason behind this comment down for full explanation & several questions.

Hope this will serve my purpose of writing to reveal the logic &concepts behind each word of mathematics&at the end you shall be affectionated with it.Feel free to contanct us for getting answer of any query.
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