Episode 8 — Aptitude and Reasoning / 8.18 — Calendar
8.18.a Calendar - Concepts and Formulas
1. Basic Calendar Facts
Days in Each Month
| Month | Days | Mnemonic |
|---|---|---|
| January | 31 | Jan - 31 |
| February | 28 (or 29 in leap year) | Feb - 28/29 |
| March | 31 | Mar - 31 |
| April | 30 | Apr - 30 |
| May | 31 | May - 31 |
| June | 30 | Jun - 30 |
| July | 31 | Jul - 31 |
| August | 31 | Aug - 31 |
| September | 30 | Sep - 30 |
| October | 31 | Oct - 31 |
| November | 30 | Nov - 30 |
| December | 31 | Dec - 31 |
Memory aid (Knuckle method):
Make a fist. Starting from the left knuckle:
Knuckle = 31 days Valley = 30 days (except Feb)
Jan Mar May Jul Aug Oct Dec
| | | | | | |
K K K K K K K
\ / \ / \ / \ / \ / \ /
V V V V V V V
Feb Apr Jun Sep Nov
Total days in a year:
Ordinary year: 365 days
Leap year: 366 days
2. Leap Year Rules
A year is a leap year if it satisfies the following conditions:
Rule 1: Non-Century Years
A year that is not a century year is a leap year if:
Year is divisible by 4
Examples:
2024 -> 2024 / 4 = 506 (exact) -> Leap year
2023 -> 2023 / 4 = 505.75 -> Not a leap year
1996 -> 1996 / 4 = 499 (exact) -> Leap year
2018 -> 2018 / 4 = 504.5 -> Not a leap year
Rule 2: Century Years
A century year (ending in 00) is a leap year only if:
Year is divisible by 400
Examples:
2000 -> 2000 / 400 = 5 (exact) -> Leap year
1900 -> 1900 / 400 = 4.75 -> NOT a leap year
1600 -> 1600 / 400 = 4 (exact) -> Leap year
1800 -> 1800 / 400 = 4.5 -> NOT a leap year
2100 -> 2100 / 400 = 5.25 -> NOT a leap year
2400 -> 2400 / 400 = 6 (exact) -> Leap year
Summary Table
| Year Type | Condition | Leap Year? |
|---|---|---|
| Non-century year | Divisible by 4 | Yes |
| Non-century year | Not divisible by 4 | No |
| Century year | Divisible by 400 | Yes |
| Century year | Not divisible by 400 | No |
Count of Leap Years
In a span of 400 years:
Total years = 400
Century years = 4 (of which 1 is a leap year)
Non-century years divisible by 4 = 96
Total leap years in 400 years = 97
Total ordinary years in 400 years = 303
3. The Concept of Odd Days
What are Odd Days?
Odd days are the number of days more than complete weeks in a given period. In other words, it is the remainder when the total number of days is divided by 7.
Odd days = Total days mod 7
Example:
10 days = 1 week + 3 days -> 3 odd days
30 days = 4 weeks + 2 days -> 2 odd days
365 days = 52 weeks + 1 day -> 1 odd day
366 days = 52 weeks + 2 days -> 2 odd days
Day Correspondence Table
| Odd Days | Day of the Week |
|---|---|
| 0 | Sunday |
| 1 | Monday |
| 2 | Tuesday |
| 3 | Wednesday |
| 4 | Thursday |
| 5 | Friday |
| 6 | Saturday |
4. Odd Days in a Year
Ordinary year (365 days):
365 = 52 * 7 + 1
Odd days = 1
Leap year (366 days):
366 = 52 * 7 + 2
Odd days = 2
5. Odd Days in Each Month
| Month | Days (Ordinary) | Odd Days (Ordinary) | Days (Leap) | Odd Days (Leap) |
|---|---|---|---|---|
| January | 31 | 3 | 31 | 3 |
| February | 28 | 0 | 29 | 1 |
| March | 31 | 3 | 31 | 3 |
| April | 30 | 2 | 30 | 2 |
| May | 31 | 3 | 31 | 3 |
| June | 30 | 2 | 30 | 2 |
| July | 31 | 3 | 31 | 3 |
| August | 31 | 3 | 31 | 3 |
| September | 30 | 2 | 30 | 2 |
| October | 31 | 3 | 31 | 3 |
| November | 30 | 2 | 30 | 2 |
| December | 31 | 3 | 31 | 3 |
| Total | 365 | 1 | 366 | 2 |
Cumulative Odd Days (from Jan 1 to end of month, ordinary year):
| End of Month | Cumulative Odd Days (mod 7) |
|---|---|
| January | 3 |
| February | 3 |
| March | 6 |
| April | 1 |
| May | 4 |
| June | 6 |
| July | 2 |
| August | 5 |
| September | 0 |
| October | 3 |
| November | 5 |
| December | 1 |
For a leap year: Add 1 extra odd day from March onwards.
6. Odd Days in Century/Multiple Years
Odd Days in 100 Years
In 100 years:
Leap years = 24 (every 4th year, but 100th year is NOT a leap year)
Ordinary years = 76
Total odd days = 76 * 1 + 24 * 2
= 76 + 48
= 124
= 17 * 7 + 5
= 5 odd days
100 years = 5 odd days
Odd Days in Different Centuries
| Period | Odd Days |
|---|---|
| 100 years | 5 |
| 200 years | 5 + 5 = 10 = 1 week + 3 = 3 odd days |
| 300 years | 5 + 5 + 5 = 15 = 2 weeks + 1 = 1 odd day |
| 400 years | 5+5+5+5+1 = 21 = 3 weeks = 0 odd days |
Note: 400 years has 0 odd days because the 400th year IS a leap year (adding 1 extra day).
This is a critical table to memorize:
| Century | Odd Days |
|---|---|
| 100 years | 5 |
| 200 years | 3 |
| 300 years | 1 |
| 400 years | 0 |
After every 400 years, the cycle repeats.
7. Finding the Day of Any Given Date
Step-by-Step Method
Step 1: Count the number of odd days from a reference point to the given date.
Step 2: Use the odd days table to determine the day.
Reference Point
January 1, 1 AD was a MONDAY (1 odd day from Sunday reference)
Complete Method to Find Day of a Date
To find the day of DD/MM/YYYY:
Step 1: Break the year into completed centuries and remaining years.
For year 2023: 2000 completed years + 23 remaining years
Step 2: Find odd days for completed centuries.
2000 years = 5 * 400 years = 5 * 0 = 0 odd days
Step 3: Find odd days for remaining years.
23 years = 17 ordinary + 6 leap years
(Leap years: 4, 8, 12, 16, 20 = 5 leap years... actually count properly)
Shortcut: In 23 years:
Leap years = floor(23/4) = 5
Ordinary years = 23 - 5 = 18
Odd days = 18 * 1 + 5 * 2 = 18 + 10 = 28 = 4 * 7 + 0 = 0 odd days
Step 4: Find odd days for completed months.
For August 15:
Jan(3) + Feb(0) + Mar(3) + Apr(2) + May(3) + Jun(2) + Jul(3) = 16
16 mod 7 = 2 odd days
(Add 1 more if it is a leap year and month is after February)
Step 5: Add the remaining days.
15 days = 2 weeks + 1 day = 1 odd day
Step 6: Total odd days and find the day.
Total = 0 + 0 + 2 + 1 = 3 odd days
Note: Check if the given year itself is a leap year and adjust for Feb.
Worked Example: Find the day on 15th August 1947
Step 1: 1947 = 1900 years + 47 years
Step 2: Odd days in 1900 years
= Odd days in (400*4 + 300) years
= 4*0 + 1
= 1 odd day
Step 3: Odd days in 47 years
Leap years = floor(47/4) = 11
Ordinary years = 47 - 11 = 36
Odd days = 36*1 + 11*2 = 36 + 22 = 58
58 mod 7 = 2 odd days
Step 4: Odd days from Jan to Jul (1947 is NOT a leap year)
Jan(3) + Feb(0) + Mar(3) + Apr(2) + May(3) + Jun(2) + Jul(3) = 16
16 mod 7 = 2 odd days
Step 5: 15 days = 2 weeks + 1 day = 1 odd day
Step 6: Total odd days = 1 + 2 + 2 + 1 = 6
6 odd days = FRIDAY
15th August 1947 was a FRIDAY.
8. Repeating Calendar Years
When Does a Calendar Repeat?
A calendar repeats when the same day of the week falls on the same date. This requires the total number of odd days between the two years to be a multiple of 7 (i.e., 0 odd days).
Rules for Repeating Calendars
For an ordinary year (starting on a certain day):
The calendar will repeat after adding a total of 7 (or 14, 21...) odd days.
Year type progression: O, O, O, L, O, O, O, L, ...
Odd days per year: 1, 1, 1, 2, 1, 1, 1, 2, ...
Common repetition gaps:
| Year Type | Repeats After |
|---|---|
| Non-leap year (not before a leap year) | 6 or 11 years |
| Non-leap year (just before a leap year) | 11 or 6 years |
| Leap year | 28 years |
Detailed Repetition Pattern
For an ordinary year:
After 1 year: 1 odd day
After 2 years: 2 odd days
After 3 years: 3 odd days
After 4 years: 3+2=5 odd days (4th year is leap)
After 5 years: 5+1=6 odd days
After 6 years: 6+1=7=0 odd days -> REPEATS! (if no leap year interference)
For a leap year:
The exact same calendar (with Feb 29) repeats after 28 years.
Quick Reference: Calendar Repetition
If the year is:
- Non-leap year: Check for repetition after 6, 11, 11, or 6 years
- Leap year: Repeats after exactly 28 years
Example: 2021 calendar repeats in:
2021 is an ordinary year (not a leap year).
2021 -> 2027 (6 years later): Check...
Odd days: 1+1+2+1+1+1 = 7 = 0 -> YES, 2027 has the same calendar as 2021
2021 calendar repeats in 2027.
9. Important Reference Tables
Day Codes (for quick calculation)
| Day | Code |
|---|---|
| Sunday | 0 |
| Monday | 1 |
| Tuesday | 2 |
| Wednesday | 3 |
| Thursday | 4 |
| Friday | 5 |
| Saturday | 6 |
Month Codes (Odd Days from Jan 1 to start of month - ordinary year)
| Month | Code |
|---|---|
| January | 0 |
| February | 3 |
| March | 3 |
| April | 6 |
| May | 1 |
| June | 4 |
| July | 6 |
| August | 2 |
| September | 5 |
| October | 0 |
| November | 3 |
| December | 5 |
For leap year: Add 1 to codes from March onwards.
Century Codes
| Century (first two digits of year) | Odd Days |
|---|---|
| 00 (e.g., 2000s) | 0 |
| 01 (e.g., 0100s) | 5 |
| 02 (e.g., 0200s) | 3 |
| 03 (e.g., 0300s) | 1 |
| Pattern repeats every 400 years | 0, 5, 3, 1 |
10. Special Facts and Observations
Fact 1: First and Last Day of an Ordinary Year
An ordinary year begins and ends on the SAME day of the week.
Example: If Jan 1 is Monday, Dec 31 is also Monday.
Fact 2: First and Last Day of a Leap Year
A leap year begins on one day and ends on the NEXT day.
Example: If Jan 1 is Monday, Dec 31 is Tuesday.
Fact 3: Same Calendar Months Within a Year
In an ordinary year, these months begin on the same day:
- January and October
- February, March, and November
- April and July
- September and December
In a leap year:
- January, April, and July
- February and August
- March and November
- September and December
Fact 4: Maximum/Minimum Occurrences
In any month:
- Any day of the week occurs at least 4 times
- At most 3 days of the week occur 5 times
Fact 5: The 400-Year Cycle
Every 400 years, the calendar repeats exactly.
Total days in 400 years = 146097 = 20871 * 7 (exact multiple of 7)
This means 400 years have exactly 0 odd days.
Fact 6: Doomsday Shortcut
The following dates ALL fall on the same day each year (called the "Doomsday"):
4/4, 6/6, 8/8, 10/10, 12/12
Last day of February (28th or 29th)
7/11, 11/7 (7th of Nov, 11th of Jul)
11. Summary of Key Formulas
| Concept | Formula / Value |
|---|---|
| Odd days in ordinary year | 1 |
| Odd days in leap year | 2 |
| Odd days in 100 years | 5 |
| Odd days in 200 years | 3 |
| Odd days in 300 years | 1 |
| Odd days in 400 years | 0 |
| Leap year condition (regular) | Divisible by 4 |
| Leap year condition (century) | Divisible by 400 |
| Ordinary year calendar repeats | After 6 or 11 years |
| Leap year calendar repeats | After 28 years |
| 400-year cycle | Exact repetition |
| Odd days to day mapping | 0=Sun, 1=Mon, 2=Tue, 3=Wed, 4=Thu, 5=Fri, 6=Sat |