Episode 8 — Aptitude and Reasoning / 8.18 — Calendar
8.18.b Calendar - Tips, Tricks and Shortcuts
Trick 1: The Odd Days Master Table (Memorize This!)
Odd days per year: Ordinary = 1, Leap = 2
Odd days per century: 100yr = 5, 200yr = 3, 300yr = 1, 400yr = 0
| Period | Odd Days |
|---|---|
| 1 ordinary year | 1 |
| 1 leap year | 2 |
| 100 years | 5 |
| 200 years | 3 |
| 300 years | 1 |
| 400 years | 0 |
Mnemonic for century odd days: "5-3-1-0" (decreases, then resets)
Trick 2: Month Odd Days Quick Reference
Memorize this sequence for monthly odd days (ordinary year):
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
3 0 3 2 3 2 3 3 2 3 2 3
Memory trick: "303-232-332-323" - group as three-digit numbers.
Or use: "JFMAMJ JASOND" with values "302323 | 332323"... Honestly, the most reliable approach is the knuckle method plus remembering Feb = 0 (or 1 in leap year).
Trick 3: Quick Leap Year Check
Step 1: Is it a century year (ends in 00)?
YES -> Divisible by 400? YES = Leap, NO = Not Leap
NO -> Divisible by 4? YES = Leap, NO = Not Leap
Speed check for divisibility by 4: Check the last two digits only.
2024: 24/4 = 6 -> Leap
1996: 96/4 = 24 -> Leap
2023: 23/4 = 5.75 -> Not Leap
Trick 4: Counting Leap Years in a Range
Between year A and year B (inclusive), the number of leap years:
Step 1: Count years divisible by 4 = floor(B/4) - floor((A-1)/4)
Step 2: Subtract century years not divisible by 400
Step 3: That gives you the leap year count
Simpler approach for ranges not spanning centuries:
Leap years = floor((B - A) / 4) + adjustment
Trick 5: Fast Odd Day Calculation for Remaining Years
For the "remaining years" part (after subtracting completed centuries):
For Y remaining years:
Odd days = Y + floor(Y/4) (then take mod 7)
Y contributes Y odd days (1 per year)
floor(Y/4) adds the extra day for each leap year
Example: 47 remaining years
Odd days = 47 + floor(47/4) = 47 + 11 = 58
58 mod 7 = 2 odd days
Example: 73 remaining years
Odd days = 73 + floor(73/4) = 73 + 18 = 91
91 mod 7 = 0 odd days
Trick 6: The Complete Day-Finding Algorithm
For any date DD/MM/YYYY:
Step 1: Century odd days (for completed centuries)
Step 2: Remaining year odd days = Y + floor(Y/4), mod 7
Step 3: Month odd days (use cumulative table, adjust for leap year)
Step 4: Add the date (DD)
Step 5: Total mod 7 = Day code (0=Sun, 1=Mon, ..., 6=Sat)
Compact example - 26 January 1950:
1900 years: 400*4 + 300 = 0 + 1 = 1 odd day
50 years: 50 + 12 = 62 -> 62 mod 7 = 6 odd days
January: 0 (no completed months before Jan)
Date: 26
Total = 1 + 6 + 0 + 26 = 33
33 mod 7 = 5 = THURSDAY
(January 26, 1950 was indeed a Thursday.)
Trick 7: Same Calendar Year Shortcuts
For non-leap years:
Add the odd days year by year: 1, 1, 1, 2, 1, 1, 1, 2, ...
Calendar repeats when cumulative odd days = 7 (or 14, 21...)
From any non-leap year, next same calendar:
Usually +6 years or +11 years
For leap years:
Leap year calendars repeat exactly every 28 years.
Quick table:
| If the year is... | Calendar repeats after... |
|---|---|
| Ordinary, followed by ordinary | +6 or +11 years |
| Ordinary, year before leap | +6 or +11 years |
| Leap year | +28 years |
Trick 8: Day of Week Progression
Same date, next year (ordinary): Day advances by 1
Jan 1, 2023 = Sunday -> Jan 1, 2024 = Monday
Same date, after leap year: Day advances by 2
Jan 1, 2024 = Monday -> Jan 1, 2025 = Wednesday
Useful for quick mental calculations: "What day will my birthday be next year?"
Trick 9: The Doomsday Shortcut
Certain dates always fall on the same day of the week in any given year. These "Doomsday" dates are:
4/4 (April 4)
6/6 (June 6)
8/8 (August 8)
10/10 (October 10)
12/12 (December 12)
Last day of February (28th or 29th)
7/11 (July 11) and 11/7 (November 7)
Method: Find the Doomsday for the year, then calculate other dates from there.
Trick 10: Months Starting on the Same Day
Ordinary year (same starting day):
Jan = Oct
Feb = Mar = Nov
Apr = Jul
Sep = Dec
Leap year (same starting day):
Jan = Apr = Jul
Feb = Aug
Mar = Nov
Sep = Dec
Use this to quickly find the day for one month if you know another.
Trick 11: Century Year Quick Reference
| Century Year | Leap? | Odd Days to This Point |
|---|---|---|
| 1600 | Yes | 0 (400-year cycle reset) |
| 1700 | No | 5 (100 years) |
| 1800 | No | 3 (200 years) |
| 1900 | No | 1 (300 years) |
| 2000 | Yes | 0 (400-year cycle reset) |
| 2100 | No | 5 |
| 2200 | No | 3 |
| 2300 | No | 1 |
| 2400 | Yes | 0 |
Trick 12: Number of Days Between Two Dates
Method 1: Count remaining days in first month + full months + days in last month
Method 2: Convert both dates to day-of-year number and subtract
Day-of-year (ordinary year):
Jan 1 = Day 1
Feb 1 = Day 32
Mar 1 = Day 60
Apr 1 = Day 91
May 1 = Day 121
Jun 1 = Day 152
Jul 1 = Day 182
Aug 1 = Day 213
Sep 1 = Day 244
Oct 1 = Day 274
Nov 1 = Day 305
Dec 1 = Day 335
Trick 13: Quick Parity Check
Before solving, check if your answer makes sense:
- There are 7 possible days, so approximately 1/7 of all dates fall on any given day
- In any month, each day of the week occurs at least 4 times and at most 5 times
- Feb has exactly 4 of each day in ordinary years
- Feb has 4 of 5 days and 5 of 2 days in leap years
Exam Strategy Tips
Time Management
- Memorize the tables: The odd days table for centuries and months is non-negotiable.
- Practice the algorithm: With practice, you should find the day in under 30 seconds.
- Common dates: Memorize the day for Jan 1 of recent years as reference points.
Common Traps
- Century leap year rule: 1900 is NOT a leap year. 2000 IS a leap year. This is the #1 trap.
- Odd days vs. total days: Don't confuse odd days (remainder) with total days.
- Leap year adjustment: Remember to add 1 extra odd day for months after February in a leap year.
- Inclusive vs. exclusive counting: "From Jan 1 to Mar 1" - is Jan 1 included? Is Mar 1 included? Read carefully.
Verification
- Use known dates to verify your method:
Jan 1, 2000 = Saturday
Jan 26, 1950 = Thursday
Aug 15, 1947 = Friday
Jul 4, 1776 = Thursday
Next: 8.18.c - Solved Examples