Calendar / Date & Day

Rule to Find Leap Year


A year is a leap year if it satisfies either of these conditions: i. Divisible by 4 but not by 100,  ii. Divisible by 400


1. Check if the year is divisible by 4 : i. If no, then not a leap year,  ii. If yes, check divisible by 100 or not


2. Check if the year is divisible by 100 : i. If no, it is a leap year,  If yes, check if divisible by 400


i. If yes, it is a leap year,  ii. If no, not a leap year


Normal Year



  • A normal year has 365 days.

  • 365 ÷ 7 = 52 weeks + 1 day leftover

  • That leftover 1 day shifts the weekday forward by 1 next year.


Ex: If 01.01. 2020 is Wednesday, 1 Jan 2021 → Thursday (+1 day)


Leap Year



  • A leap year has 366 days.

  • 366 ÷ 7 = 52 weeks + 2 days leftover

  • That leftover 2 days shifts the weekday forward by 2 next year.


Ex: If 01.01. 2016 is Friday, 01.01. 2017 → Sunday (+2 days)


Quick Rule



  • Normal year → +1 day

  • Leap year → +2 days


This is why we add 1 or 2 days for each year when calculating the day of the week.


Examples :


Q. If 31 August 2000 was Friday, then which day was 31 August 2009 ?


Options: (a) Monday (b) Tuesday (c) Wednesday (d) Friday


Soln:



  1. Count the leap years between 2000 and 2009 → 2004, 2008 → each gives +2 days

  2. Count the normal years → 2001, 2002, 2003, 2005, 2006, 2007, 2009 → each gives +1 day


    Total days added: 2 + 2 + 7 = 11 → 11 ÷ 7 = 1 remainder 4


     Move 4 days forward from Friday: Friday → Saturday → Sunday → Monday → Tuesday


Ans: Tuesday


Q. If 1 August 2009 was Saturday, then which day was 1 August 2013 ?


Options: (a) Saturday (b) Thursday (c) Friday (d) Sunday


Soln:  



  1. Count years from 2009 → 2013:


    2010, 2011, 2013 → normal years → +1 day each


    2012 → leap year → +2 days


Total shift: 1+1+2+1 = 5 days



  1. Move 5 days forward from Saturday:

  2. Saturday → Sunday → Monday → Tuesday → Wednesday → Thursday


Ans: Thursday


Q. If 28 February 2017 was Tuesday, then which day was 28 February 2019 ?


Options: (a) Thursday (b) Saturday (c) Friday (d) Monday


Soln:  


Count years from 2017 → 2019: 2018 :  normal → +1 day, 2019 : normal → +1 day


Total shift: 1 + 1 = 2 days


Move 2 days forward from Tuesday: Tuesday → Wednesday → Thursday


Ans: Thursday


Q. If 15 March 1996 was Friday, then which day was 15 March 2023 ?


Options: (a) Wednesday (b) Thursday (c) Friday (d) Saturday


Soln:



  1. Count leap years 1996 → 2023 → 1996, 2000, 2004, 2008, 2012, 2016, 2020 → 7 leap years (+2 each)

  2. Normal years → 28 – 7 = 21 (+1 each)


Total shift: 7×2 + 21×1 = 14 + 21 = 35


Divide by 7 → remainder 0 → no change in day


Ans: Friday


2. Odd Days


Odd days = Extra days left after making complete weeks


Odd Days Summary


i. 1 Ordinary Year → 1 odd day


ii. 1 Leap Year → 2 odd days


iii. 100 Years → 5 odd days


iv. 200 Years → 3 odd days


v. 300 Years → 1 odd day


vi. 400 Years → 0 odd days (calendar repeats)


After every 400 years, the calendar repeats exactly.


3. Month-wise Odd Days



  • January → 3 odd days (same in leap year)

  • February → 0 odd days (→ 1 in leap year)

  • March → 3 odd days

  • April → 2 odd days

  • May → 3 odd days

  • June → 2 odd days

  • July → 3 odd days

  • August → 3 odd days

  • September → 2 odd days

  • October → 3 odd days

  • November → 2 odd days

  • December → 3 odd days


Only February changes in a leap year (0 → 1).


4. Day Codes (0–6 Method)



  • Sunday → 0

  • Monday → 1

  • Tuesday → 2

  • Wednesday → 3

  • Thursday → 4

  • Friday → 5

  • Saturday → 6


These codes are used to calculate the exact day of any date.


5. Formula to Find Day of Any Date


Day = (Year Code + Month Code + Date + Century Code) mod 7


Meaning (by point)



  • Year Code → calculated from last two digits of the year

  • Month Code → fixed code for each month

  • Date → given date (1 to 31)

  • Century Code → depends on century (1600, 1700, etc.)

  • mod 7 → divide total by 7 and take remainder


The remainder gives the day code (0–6)


Then match with day codes: 0 = Sunday, 1 = Monday … 6 = Saturday


Century Codes: 1600 → 6, 1700 → 4, 1800 → 2, 1900 → 0, 2000 → 6


These codes help calculate the day for any date in that century.


Year Code (How to Calculate)



  1. Take last two digits of the year = N

  2. Year Code = N + ⌊N/4⌋


Ex: Year = 1987



  • Last two digits = 87

  • Year Code = 87 + ⌊87/4⌋

  • = 87 + 21

  • = 108


6. Key Concepts


A. Same Calendar Condition


Two years will have the same calendar when:



  • Both are leap years or both are non - leap years

  • Odd day difference = 0


Ex: → 2017 and 2023 have the same calendar


B. Day Forward / Backward Rule



  • After N days: (X+N) mod 7  

  • Before N days: (X−N) mod 7


C. Calendar Repeat



  • Normal year → repeats after 6 or 11 years

  • Leap year → repeats after 28 years


Means the exact same dates + same days will match again after this period.


Exam Tricks


1. After 100 days


100 mod 7 = 2 


Day will be 2 days ahead


2. Same calendar as 2025


i. 2025 is normal year


ii. Odd day = 1


iii. Next matching calendar = 2031


3. Find the Day: 15 August 1947


i. Century code → 0


ii. Last two digits → 47


iii. Year code → 47 + 11 = 58


iv. Month code (August) → 3


v. Date → 15


Total Calculation : 0+58+3+15=76


         76 mod 7 = 6


Day Code Result : Day code 6 = Friday