Calendar / Date & Day

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Rule to Find Leap Year

A year is a leap year if it satisfies either of these conditions:

1. Divisible by 4 but not by 100

2. Divisible by 400

i.       Check if the year is divisible by 4

o   If no, then not a leap year

o   If yes, go to step 2

ii. Check if the year is divisible by 100

o   If no, it is a leap year

o   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