Calendar questions are among the most scoring and logic-driven topics in Quantitative Aptitude. But many learners find them confusing because, unlike straightforward arithmetic, calendar questions involve rules about years, centuries, leap years, and the concept of “odd days.”
If you understand these three ideas clearly:
✔ Leap year rules
✔ Odd days calculation
✔ Mapping odd days to actual weekdays
Then you can solve ANY day-finding question effortlessly.
This Calendar – Odd Days & Day Calculation Guide covers all concepts, formulas, shortcuts, tables, examples, and FAQs, making it a perfect one-stop resource for competitive exams.
Quick Overview: Calendar Formulas
| Concept / Situation | Considered | Used | Formula (With Meaning of Symbols Inside Row) |
|---|---|---|---|
| Odd Days Definition | Total days in a period | Remove complete weeks | Odd Days = Total Days – (Number of weeks × 7) (Odd days = leftover days determining weekday) |
| Ordinary Year | 365 days | 1 odd day | 1 Ordinary Year = 365 = 52 weeks + 1 → 1 Odd Day |
| Leap Year | 366 days | 2 odd days | 1 Leap Year = 366 = 52 weeks + 2 → 2 Odd Days |
| Leap Year Rule (Normal Year) | Year divisible by 4 | Check leap year | If (Year % 4 = 0) & NOT century → Leap Year |
| Leap Year Rule (Century Year) | Century numbers | Check leap year | Century is leap year only if divisible by 400 (e.g., 1600, 2000) |
| Odd Days in 100 Years | 76 ordinary + 24 leap | 5 odd days | (76×1 + 24×2) = 124 days = 17 weeks + 5 odd days |
| Odd Days in 200 Years | Twice of 100 years | 3 odd days | 2 × 5 = 10, 10 = 1 week + 3 odd days |
| Odd Days in 300 Years | Thrice of 100 years | 1 odd day | 3 × 5 = 15, 15 = 2 weeks + 1 odd day |
| Odd Days in 400 Years | 400-year cycle | 0 odd days | 400-year block always gives 0 odd days |
| Day Mapping Table | Odd day number | Convert to weekday | 0–6 → Sun–Sat (0 = Sunday, 1 = Monday…6 = Saturday) |
| Finding Weekday of a Date | Date, Month, Year | Total odd days | Weekday = Odd Day Value → Map using Day Table |
Odd-Day Calculation Formulas
Odd days indicate how many days exceed complete weeks in a given time period. Since weekdays work on a 7-day cycle, odd days help identify the correct weekday.
1. Formula to Calculate Odd Days From Total Days
Whenever you know total days, find odd days using:
Odd Days = Total Days mod 7
Why this works:
Because every complete set of 7 days forms a full week and does not affect the weekday.
Example:
900 days → 900 mod 7 = 4 odd days
→ Day shifts by 4 days.
2. Odd Days in an Ordinary Year (365 Days)
An ordinary year has:
365 = 52 weeks × 7 + 1 day
Therefore:
✔ Ordinary Year → 1 Odd Day
Used When:
Finding the day in years like 2001, 2002, 1997, 2023, etc.
3. Odd Days in a Leap Year (366 Days)
A leap year has:
366 = 52 weeks × 7 + 2 days
Therefore:
✔ Leap Year → 2 Odd Days
Used When:
Year has 29 days in February.
Leap Year Identification Formula
Before calculating odd days, you must identify whether a year is leap or not.
1. Leap Year (Non-Century Year)
A year is a leap year if:
Year % 4 = 0
AND
It is NOT a century year.
Examples:
1948, 2004, 2020 → Leap Years
2. Leap Year (Century Year)
A century year is leap only if:
Year % 400 = 0
Examples:
1600, 2000 → Leap Years
1800, 1900 → Not Leap Years
Why this matters:
Incorrect leap-year assumption → wrong odd-day total → wrong weekday.
Century Odd-Day Formula
These formulas allow quick conversion of long spans of years into odd days.
1. Odd Days in 100 Years
100 years = 76 ordinary + 24 leap
Odd days = 76×1 + 24×2 = 124
124 mod 7 = 5 odd days
Therefore:
✔ 100 Years = 5 Odd Days
2. Odd Days in 200 Years
2 × 100-year odd days = 2 × 5 = 10
10 mod 7 = 3 odd days
Therefore:
✔ 200 Years = 3 Odd Days
3. Odd Days in 300 Years
3 × 100-year odd days = 3 × 5 = 15
15 mod 7 = 1 odd day
Therefore:
✔ 300 Years = 1 Odd Day
4. Odd Days in 400 Years
400 years contain exactly 0 odd days.
Therefore:
✔ 400 Years = 0 Odd Days
Why:
Total 400-year days = multiple of 7
Hence no leftover odd day.
Month-wise Odd Day Contribution Formula
To find a day for a given date, irregular month lengths are important.
1. Odd Days in Months (Ordinary Year)
| Month | Days | Odd Days (mod 7) |
|---|---|---|
| Jan | 31 | 3 |
| Feb | 28 | 0 |
| Mar | 31 | 3 |
| Apr | 30 | 2 |
| May | 31 | 3 |
| Jun | 30 | 2 |
| Jul | 31 | 3 |
| Aug | 31 | 3 |
| Sep | 30 | 2 |
| Oct | 31 | 3 |
| Nov | 30 | 2 |
| Dec | 31 | 3 |
2. Odd Days in February (Leap Year)
Leap Year February = 29 days → 1 odd day
Date Odd-Day Formula
For the date given:
Odd Days = Date mod 7
Example:
17th → 17 mod 7 = 3 odd days
Day Mapping Formula (Odd Day → Weekday)
| Odd Days | Weekday |
|---|---|
| 0 | Sunday |
| 1 | Monday |
| 2 | Tuesday |
| 3 | Wednesday |
| 4 | Thursday |
| 5 | Friday |
| 6 | Saturday |
This maps your final odd days to actual weekdays.
Combined Day-Finding Formula
To find day on DD/MM/YYYY, compute total odd days from:
- Complete years passed
- Complete months passed
- Given date
- Add leap year corrections
Formula:
**Total Odd Days = (Odd days from years)
- (Odd days from months)
- (Odd days from dates)**
Then:
Weekday = Total Odd Days mod 7
This is the universal formula used in RRB, SSC, IBPS, UPSC exams.
Smart Tips and Practical Tricks for Solving Calendar – Odd Days Problems
Mastering Calendar questions becomes simple once you understand how years, months, and odd days interact. Most students make mistakes not because the formulas are difficult, but because they apply them without understanding how days accumulate across years and centuries. This section breaks down the most important concepts into clear, actionable tips so you can calculate weekdays faster and more accurately in exams.
1. Always Identify Whether the Given Year Is Leap or Ordinary
Many students directly start calculating days without checking leap-year rules.
This leads to incorrect odd-day counts.
Leap Year Rules:
- Year divisible by 4 → Leap Year (non-century)
- Century year divisible by 400 → Leap
- All other century years → Not Leap
Example:
2004 → Leap
1900 → Not Leap
2000 → Leap
This one step ensures correct day calculations for any date.
2. Break Long Years Into Blocks for Faster Odd-Day Calculation
Handling years like 1876 or 2025 is easier when the year is broken into blocks.
Use the Standard Odd-Day Blocks:
- 100 years → 5 odd days
- 200 years → 3 odd days
- 300 years → 1 odd day
- 400 years → 0 odd days
This instantly reduces large calculations into manageable parts.
Example:
To calculate odd days till year 300:
→ 300 years = 1 odd day
3. Count Total Odd Days, Not Total Days
Students often calculate total days, then divide by 7.
That wastes time.
Focus only on the leftover days:
Odd Days = Total Days mod 7
This gives you the weekday shift directly.
Example:
365 days → 365 mod 7 = 1 odd day
4. Memorize Month-Wise Day Distribution for Quick Calculations
Most day-finding questions require month-based day counting.
Quick Month Pattern:
- 31-day months → odd days = 3
- 30-day months → odd days = 2
- February → 0 (ordinary) or 1 (leap)
Memorizing this pattern saves 5–10 seconds per question.
5. Use the Day Mapping Table Correctly
Odd days convert directly to weekdays:
0 → Sun
1 → Mon
2 → Tue
3 → Wed
4 → Thu
5 → Fri
6 → Sat
It prevents wrong answers during final step conversion.
6. Always Check Whether the Question Involves a Century Year
Century years are NOT leap years unless divisible by 400.
Example:
1800, 1900, 2100 → Not Leap
1600, 2000, 2400 → Leap
Skipping this rule leads to incorrect odd day totals.
7. Practice Common Exam Patterns
Calendar questions frequently follow these patterns:
- Day of specific date (e.g., 15 Aug 1947)
- Day after “n” days
- Leap year identification
- Counting odd days up to a given year
- Repetition of calendar year
- Matching birthdays and weekday shifts
- Day on 1st January of a future year
Recognizing these patterns helps you answer almost mechanically.
FAQs About Calendar – Odd Days & Day Finding
Q1. Why do we use odd days in calendar problems?
Odd days represent the number of leftover days after removing complete weeks. Since the weekly cycle repeats every 7 days, odd days help identify the exact weekday.
Example:
10 days = 1 week + 3 odd days
If Monday is the starting day, the answer is Thursday.
Q2. Why is a leap year counted as 2 odd days?
Because a leap year has 366 days, which gives:
366 = 52 weeks + 2 extra days
These 2 extra days are the odd days.
Example:
If Jan 1 is Monday in a leap year, Dec 31 will be Wednesday.
Q3. Why are century years like 1800 or 1900 not leap years?
Century years follow a special rule:
Only centuries divisible by 400 are leap years.
1800 % 400 ≠ 0 → Not leap
1900 % 400 ≠ 0 → Not leap
2000 % 400 = 0 → Leap
Q4. How do we find the day of the week using odd days?
- Calculate total odd days
- Convert odd days using day mapping table
- Match odd day number with weekday (0–6 = Sun–Sat)
Example:
Odd days = 4 → Thursday
Q5. Why does 400 years have 0 odd days?
400 years contain exactly 97 leap years and 303 ordinary years.
The total number of days becomes divisible by 7, resulting in 0 odd days.
Example:
1600, 2000, 2400 all begin on the same weekday.
Q6. Why do calendars repeat every 400 years?
Because 400 years produce 0 odd days.
When odd days reset, the full calendar repeats.
Q7. Why is February special in calendar calculations?
February is the only month with changing days:
- 28 days (ordinary year)
- 29 days (leap year)
This change affects the total odd days of the year.
Example:
In leap year, February contributes 1 odd day.
Q8. Why do students make mistakes in day-finding questions?
Because they forget:
✔ Leap-year rule
✔ Century rule
✔ Month odd-day distribution
✔ Day mapping (0–6)
Skipping even one rule results in incorrect weekday answers.
Q9. How does the odd-day method make calendar questions easier?
Odd days convert complex year counts into simple remainders (0–6).
This avoids long calculations and gives the weekday directly.
Q10. Why is regular practice important in calendar topics?
Practice helps you mentally:
✔ Identify leap/ordinary years quickly
✔ Break years into 100-year blocks
✔ Recall month-wise odd days
✔ Convert odd days into weekdays instantly
This reduces the entire problem to a few seconds.