1. Introduction to Calendar Concepts

The calendar system is based on a cycle of days, weeks, months, and years. Aptitude problems involving calendars focus on finding specific days, leap years, odd days, and relationships between dates.

2. Key Concepts

• Odd Days: The number of days left after complete weeks in a given period.
• Leap Year: A year divisible by 4 is a leap year. Exceptions: Years divisible by 100 but not divisible by 400.
• Normal Year: A year with 365 days (52 weeks and 1 odd day).

Odd Days:

• 1 normal year → 1 odd day
• 1 leap year → 2 odd days
• 100 years → 5 odd days
• 200 years → 3 odd days
• 300 years → 1 odd day
• 400 years → 0 odd days

3. Days of the Week

The days of the week cycle as follows:

• Sunday (0 odd days)
• Monday (1 odd day)
• Tuesday (2 odd days)
• Wednesday (3 odd days)
• Thursday (4 odd days)
• Friday (5 odd days)
• Saturday (6 odd days)

4. Finding Day on a Given Date

Steps:

1. Break the problem into parts: centuries, years, months, and days.
2. Calculate the total odd days from each part.
3. Find the remainder when total odd days are divided by 7.
4. Map the remainder to a weekday.

4.1 Example

Problem: Find the day on 15th August 1947.

Solution:

1. Start from the nearest known base year (e.g., 1900).
2. Odd Days:
• From 1900 to 1947 = 47 years (11 leap years, 36 normal years):

Odd days = \( (36 × 1) + (11 × 2) = 36 + 22 = 58 \).

Reduce modulo 7: \( 58 \div 7 = 2 \) odd days.

• From Jan to August in 1947:
• Jan (3), Feb (0 - leap year), Mar (3), Apr (2), May (3), Jun (2), Jul (3), Aug 15 → 15 days.
• Total = \( 3+0+3+2+3+2+3+15 = 31 \).

Odd days = \( 31 \div 7 = 3 \) odd days.

3. Add odd days: \( 2 (years) + 3 (months) = 5 \).
4. Day = Friday.

5. Calendar Repetition

Calendars repeat after a specific number of years if the number of odd days is the same.

• For normal years: Repeats after 6 years.
• For leap years: Repeats after 28 years.

Example: The calendar of 2023 will repeat in 2029 (6 years).

6. Finding Days for Centuries

Use the odd day concept for centuries:

• 100 years → 5 odd days → Friday
• 200 years → 3 odd days → Wednesday
• 300 years → 1 odd day → Monday
• 400 years → 0 odd days → Sunday

Example: If 1st Jan 2000 is Saturday, 1st Jan 2400 will be Sunday (0 odd days in 400 years).

7. Date Problems: Weeks and Decades

• Add Weeks: Add multiples of 7 days → the day remains the same.
• Adding Years: 1 year = 1 odd day, leap year = 2 odd days.
• Adding Decades: Each decade = 2 leap years and 8 normal years → \( (8 × 1) + (2 × 2) = 12 \) odd days → 12 mod 7 = 5 odd days.

Example: If 1st Jan 2023 is Sunday, find the day on 1st Jan 2033:

• 10 years → 5 odd days.
• Sunday + 5 days = Friday.

8. Problems on Leap Years

Key Rules:

• Check divisibility by 4 → Leap Year.
• Years divisible by 100 are not leap years unless divisible by 400.

Example: 1900 is not a leap year, but 2000 is.

9. Miscellaneous Problems

9.1 Day Difference Between Two Dates

Steps:

1. Find total days between dates (years + months + days).
2. Divide by 7 and find the remainder.
3. Map to the corresponding day.

9.2 Example

Problem: Find the day on 15th August 2047 if 15th August 2023 is Tuesday.

Solution:

• Total years = 2047 - 2023 = 24 years.
• Count leap years: \( 24 \div 4 = 6 \).
• Odd days = \( 18 (normal years) × 1 + 6 (leap years) × 2 = 18 + 12 = 30 \).
• Odd days = \( 30 \div 7 = 2 \).
• Tuesday + 2 days = Thursday.

10. Summary of Key Formulas

• Odd Days: The remainder when days are divided by 7.
• Angle Between Days: Leap year = 2 odd days; normal year = 1 odd day.
• Calendar Repeats: Normal year → 6 years; Leap year → 28 years.
• Centuries Odd Days: 100 → 5, 200 → 3, 300 → 1, 400 → 0.