Solving Calendar Problems for Competitive Exams | Easy Guide
Solving Calendar Problems for Competitive Exams | Easy Guide
Introduction
Calendar-based questions are a common part of quantitative aptitude and reasoning sections in competitive exams. These questions require knowledge of days, dates, leap years, and cyclic patterns of the calendar. By understanding a few key concepts and formulas, you can solve these problems quickly and accurately.
Basic Concepts
1. Odd Days
- The concept of odd days is the key to solving calendar problems.
- Odd days refer to the extra days that exceed complete weeks when calculating a given number of days.
- 1 week = 7 days, so the remainder when dividing by 7 gives the odd days.
2. Leap Year vs. Ordinary Year
- Ordinary Year = 365 days = (52 full weeks + 1 odd day)
- Leap Year = 366 days = (52 full weeks + 2 odd days)
- A Leap Year is divisible by 4 (except for century years, which must be divisible by 400).
3. Century Odd Days
- 100 years = 5 odd days
- 200 years = 3 odd days
- 300 years = 1 odd day
- 400 years = 0 odd days (since 400 is divisible by 400, it’s a leap year cycle completion)
Key Formulas & Tricks
1. Formula for Odd Days in a Given Period
To find the number of odd days from a given date:
- Odd Days = (Total days in period) ÷ 7 (Remainder is odd days)
2. Day Code for Each Month
Each month has a fixed day code, which helps determine the day of the week:
| Month | Code |
|---|---|
| January | 0 (1 in leap year) |
| February | 3 (4 in leap year) |
| March | 3 |
| April | 6 |
| May | 1 |
| June | 4 |
| July | 6 |
| August | 2 |
| September | 5 |
| October | 0 |
| November | 3 |
| December | 5 |
3. Day Code for a Given Year
(Last two digits of the year + 4Last two digits + Month Code + Century Code + Date) mod7
The remainder gives the weekday:
| Remainder | Day |
|---|---|
| 0 | Sunday |
| 1 | Monday |
| 2 | Tuesday |
| 3 | Wednesday |
| 4 | Thursday |
| 5 | Friday |
| 6 | Saturday |
4. Century Codes
- 1600, 2000 → 6 (Saturday)
- 1700, 2100 → 4 (Thursday)
- 1800, 2200 → 2 (Tuesday)
- 1900, 2300 → 0 (Sunday)
Types of Calendar Questions & Solutions
1. Finding the Day of the Week for a Given Date
Example: What day of the week was 15th August 1947?
Solution:
- Last two digits of the year = 47
- Divide by 4: ⌊47/4⌋ = 11
- Month code for August = 2
- Century code for 1900s = 0
- Date given = 15
(47+11+2+0+15)mod7=75mod7=5
Answer: Friday
2. Finding the Same Calendar Year
Example: In which year after 2010 will the calendar be the same?
Solution:
A year will have the same calendar if:
- It has the same odd days as 2010.
- 2010 is an ordinary year (1 odd day).
We check subsequent years:
- 2011 → 1 odd day
- 2012 → 2 odd days (Leap Year)
- 2013 → 1 odd day
- 2014 → 1 odd day
- 2015 → 1 odd day
- 2016 → 2 odd days (Leap Year)
- 2017 → 1 odd day
- 2018 → 1 odd day
- 2019 → 1 odd day
- 2020 → 2 odd days (Leap Year)
- 2021 → 1 odd day
Answer: 2021 (Same calendar as 2010)
3. Finding Leap Years in a Given Period
Example: How many leap years are there from 2000 to 2100?
Solution:
Leap years are divisible by 4, so:
- Leap years = 2004, 2008, 2012, …, 2096
- Total = 25 leap years
Answer: 25
Shortcut Tricks
✅ Use 7-day cycles: Instead of calculating full years, focus on odd days.
✅ Memorize century codes: Quick recall helps in solving year-based problems faster.
✅ Use leap year rules: Century years need divisibility by 400.
✅ Use the cyclic property of days: A calendar repeats every 400 years exactly.
Conclusion
Calendar problems can be solved quickly with a systematic approach using odd days, century codes, and cyclic patterns. With practice, you can master this topic and improve your speed in competitive exams.
