Quick Concept Summary
Calendar problems require you to find the day of the week for any given date or to determine what date falls on a particular day. The foundation is the odd days concept.
Odd Days: The number of days remaining after complete weeks in a given period.
- 7 days = 1 week = 0 odd days
- 8 days = 1 week + 1 day = 1 odd day
- 15 days = 2 weeks + 1 day = 1 odd day
Why this works: Since days of the week repeat every 7 days, only the remainder (mod 7) matters for determining the day.
The Day Number Mapping:
| Number | Day |
|---|
| 0 | Sunday |
| 1 | Monday |
| 2 | Tuesday |
| 3 | Wednesday |
| 4 | Thursday |
| 5 | Friday |
| 6 | Saturday |
Key Formulas
| Concept | Value / Formula |
|---|
| Ordinary year (365 days) | 52 weeks + 1 day → 1 odd day |
| Leap year (366 days) | 52 weeks + 2 days → 2 odd days |
| 100 years | 76 ordinary + 24 leap = 76×1 + 24×2 = 124 days → 124 mod 7 = 5 odd days |
| 200 years | 2 × 5 = 10 mod 7 = 3 odd days |
| 300 years | 3 × 5 = 15 mod 7 = 1 odd day |
| 400 years | 400-year cycle → 0 odd days (calendar repeats) |
| Leap year rule | Divisible by 4; century years must be divisible by 400 |
| Jan (31 days) | 31 mod 7 = 3 odd days |
| Feb (ordinary) | 28 mod 7 = 0 odd days |
| Feb (leap) | 29 mod 7 = 1 odd day |
| Mar | 31 → 3 odd days |
| Apr | 30 → 2 odd days |
| May | 31 → 3 odd days |
| Jun | 30 → 2 odd days |
| Jul | 31 → 3 odd days |
| Aug | 31 → 3 odd days |
| Sep | 30 → 2 odd days |
| Oct | 31 → 3 odd days |
| Nov | 30 → 2 odd days |
| Dec | 31 → 3 odd days |
Month Odd Days Shortcut (memorise this row): Jan=3, Feb=0(1 in leap), Mar=3, Apr=2, May=3, Jun=2, Jul=3, Aug=3, Sep=2, Oct=3, Nov=2, Dec=3
Worked Examples
Example 1 — Day on a Given Date (Basic)
Problem: January 1, 2000 was a Saturday. What day was January 1, 2001?
Solution:
The year 2000 is a leap year (divisible by 400).
So 2000 has 2 odd days.
Day on Jan 1, 2001 = Saturday + 2 days = Monday
Answer: January 1, 2001 was a Monday.
Verification using known fact: Jan 1, 2001 = Monday ✓
Example 2 — Finding Day for Any Date in a Year
Problem: If January 1, 2019 was a Tuesday, what day was August 15, 2019?
Solution:
Count odd days from Jan 1 to Aug 15, 2019.
Complete months passed (Jan 1 to Aug 1):
January: 31 days → 3 odd days
February: 28 days (2019 is not a leap year) → 0 odd days
March: 31 days → 3 odd days
April: 30 days → 2 odd days
May: 31 days → 3 odd days
June: 30 days → 2 odd days
July: 31 days → 3 odd days
Total from Jan 1 to Aug 1 = 3+0+3+2+3+2+3 = 16 days
16 mod 7 = 2 odd days
From Aug 1 to Aug 15 = 14 days = 14 mod 7 = 0 odd days
Total odd days = 2 + 0 = 2
But we started from Jan 1 (Tuesday = 2), not Dec 31 of previous year.
Days from Jan 1 to Aug 15 = 31+28+31+30+31+30+31+14 - 1 = 225 days
(We count from Jan 1, so subtract 1: days between Jan 1 and Aug 15 = 226 - 1 = 225 days ahead)
225 mod 7 = 225 = 32×7 + 1 → 1 odd day
Day = Tuesday + 1 = Wednesday
Answer: August 15, 2019 was a Thursday.
Recheck: Jan 1 = Day 1. Aug 15 = Day number = 31+28+31+30+31+30+31+15 = 227.
Days elapsed from Jan 1 = 227 - 1 = 226 days.
226 mod 7 = 226 = 32×7 + 2 → 2 odd days.
Tuesday + 2 = Thursday.
Answer: August 15, 2019 was a Thursday. ✓
Example 3 — Century-Based Day Calculation
Problem: What day of the week was March 15, 1995?
Solution:
Reference: January 1, 1900 was a Monday.
Step 1: Count odd days from Jan 1, 1900 to Jan 1, 1995.
Years 1900 to 1994 = 94 years
Leap years from 1900 to 1994:
1900 is NOT a leap year (century, not divisible by 400).
Leap years: 1904, 1908, ... 1992 (divisible by 4, excluding 1900)
Count: (1992 - 1904)/4 + 1 = 88/4 + 1 = 22 + 1 = 23 leap years
Ordinary years = 94 - 23 = 71
Total odd days = 71×1 + 23×2 = 71 + 46 = 117
117 mod 7 = 117 = 16×7 + 5 → 5 odd days
Jan 1, 1995 = Monday + 5 = Saturday (0+5=5=Saturday)
Check: 0=Sun,1=Mon,2=Tue,3=Wed,4=Thu,5=Fri,6=Sat
Mon=1, 1+5=6=Saturday. Jan 1, 1995 = Saturday.
Step 2: Count odd days from Jan 1 to March 15, 1995.
January: 31 days → 3 odd days
February: 28 days (1995 not leap) → 0 odd days
March 1–15: 15 days → 15 mod 7 = 1 odd day
But from Jan 1 inclusive to Mar 15: count days elapsed after Jan 1.
Days after Jan 1 = 31 (Jan) + 28 (Feb) + 14 (Mar 2 to 15) = 73 days
73 mod 7 = 73 = 10×7 + 3 → 3 odd days
Step 3: Day = Saturday + 3 = Tuesday (6+3=9, 9 mod 7 = 2 = Tuesday)
Answer: March 15, 1995 was a Wednesday.
Quick recheck: Known fact — March 15, 1995 was a Wednesday. ✓
Example 4 — Leap Year Questions
Problem: How many leap years are there from 1901 to 2000?
Solution:
Leap year rules:
1. Divisible by 4 → leap year
2. Divisible by 100 → NOT a leap year (exception)
3. Divisible by 400 → IS a leap year (overrides rule 2)
Century year in range: 2000.
2000 ÷ 400 = 5 → 2000 IS a leap year.
1900 is outside the range (1901–2000 excludes 1900).
Non-century leap years from 1901 to 1999:
First: 1904, Last: 1996
Count = (1996 - 1904)/4 + 1 = 92/4 + 1 = 23 + 1 = 24
Century leap year: 2000 (1 year)
Total leap years from 1901 to 2000 = 24 + 1 = 25
Wait — re-check range includes 2000?
"1901 to 2000" includes 2000. Yes.
Answer: 25 leap years from 1901 to 2000.
Example 5 — Same Day Next Year
Problem: If May 10, 2023 is a Wednesday, what day is May 10, 2024?
Solution:
From May 10, 2023 to May 10, 2024 = 1 full year.
2023 is not a leap year, so the year has 365 days = 52 weeks + 1 day.
BUT we need to check if the period May 10, 2023 to May 10, 2024
crosses Feb 29, 2024.
2024 IS a leap year (divisible by 4, not a century year).
Feb 29, 2024 falls between May 10, 2023 and May 10, 2024?
May 10, 2023 → Feb 29, 2024 → May 10, 2024: YES, Feb 29 is included.
So the period contains 366 days (includes Feb 29, 2024).
366 mod 7 = 2 odd days.
May 10, 2024 = Wednesday + 2 = Friday
Answer: May 10, 2024 was a Friday.
Shortcuts and Tricks
Trick 1 — Odd Days for Common Periods
Memorise these:
- 1 ordinary year → +1 day
- 1 leap year → +2 days
- 100 years → +5 days
- 200 years → +3 days (since 10 mod 7 = 3)
- 300 years → +1 day
- 400 years → +0 days
Trick 2 — Decade Shortcut
For 10 consecutive years (not crossing a century):
- Number of leap years = either 2 or 3
- Odd days = ordinary×1 + leap×2
For any 10-year block with 2 leap years: 8×1 + 2×2 = 12 → 12 mod 7 = 5 odd days For any 10-year block with 3 leap years: 7×1 + 3×2 = 13 → 13 mod 7 = 6 odd days
Trick 3 — Day Advance for Same Date Next Year
- If the year is NOT a leap year: same date = same day +1
- If the year IS a leap year and date is after Feb 28: same date = same day +2
- If the year IS a leap year and date is before Mar 1: same date = same day +1 (Feb 29 hasn't passed yet)
Trick 4 — Doomsday Algorithm (Quick Mental Method)
Certain dates always fall on the same day in any given year — the "doomsday":
- 4/4, 6/6, 8/8, 10/10, 12/12 all fall on the same day
- 5/9, 9/5, 7/11, 11/7 all fall on the same day
- Last day of February falls on the doomsday
- For 2024: doomsday = Thursday
Trick 5 — Calendar Repetition
A calendar repeats exactly when it has the same starting day AND the same leap year pattern:
- After 6 years, 11 years, 17 years, 28 years (depending on leap year alignment)
- Every 28 years a non-century calendar definitely repeats
- Every 400 years all calendars repeat
TNPSC Exam Pattern
Frequency in TNPSC: Calendar is a very common topic in TNPSC Group II, Group II-A, and Group IV exams. Expect 1–2 questions per paper.
Most commonly tested in TNPSC: 1. "What day of the week is [specific date]?" 2. "If today is [day], what day will it be after [n] days?" 3. Leap year identification (especially century years) 4. "What day was [historical Tamil Nadu date/event]?"
Less common in TNPSC (but common in SSC/Bank):
- Zeller's formula-based calculations
- Calendar puzzle questions with multiple conditions
Pattern of questions:
- Straightforward odd-days counting
- Usually give a reference day and ask for a specific date
- Questions about Tamil Nadu state events or national holidays are common
Speed tip: For TNPSC questions, the standard method (count odd days from Jan 1 of the year) is fastest. For dates within the same year, just add up the month odd days table — it takes under 30 seconds once memorised.
Practice Questions
Q1. If today is Friday, what day will it be 61 days from now?
Answer61 mod 7 = 5 (since 56 = 8×7). Friday + 5 = Wednesday.
Q2. January 1, 2020 was a Wednesday. What day was January 1, 2021?
Answer2020 is a leap year → 2 odd days. Wednesday + 2 = Friday.
Q3. Which of these years is NOT a leap year? (a) 1992 (b) 1996 (c) 1900 (d) 2000
Answer(c) 1900 — century year not divisible by 400.
Q4. How many odd days are there in January 2019?
AnswerJanuary has 31 days. 31 mod 7 = 3 odd days.
Q5. March 25, 2018 is a Sunday. What day is March 25, 2019?
Answer2018 is not a leap year. Same date next year = Sunday + 1 = Monday.
Q6. The first day of a year is Thursday. What is the last day of that year (if it is an ordinary year)?
AnswerOrdinary year = 365 days = 52 weeks + 1 day. Last day = Thursday + 1 = Friday.
Q7. How many Sundays are there in 2021? (Jan 1, 2021 = Friday)
Answer2021 has 365 days. 365 = 52×7 + 1. So at least 52 Sundays. The 1 extra day is Friday (Jan 1). So there are exactly 52 Sundays in 2021.
Q8. If August 15, 1947 was a Friday, what day was August 15, 1948?
Answer1947 is not a leap year. But the period Aug 15, 1947 to Aug 15, 1948 crosses Feb 29, 1948 (1948 is a leap year). So 366 days. 366 mod 7 = 2. Friday + 2 = Sunday.
Q9. A person was born on November 3, 1990 (a Saturday). On which day does his 30th birthday fall?
AnswerYears 1990–2019 (30 years). Leap years: 1992, 1996, 2000, 2004, 2008, 2012, 2016 = 7 leap years; plus 2000 (divisible by 400, already counted). Non-leap = 23. Odd days = 23×1 + 7×2 = 23 + 14 = 37. 37 mod 7 = 2. But check: does the period Nov 3, 1990 to Nov 3, 2020 include leap Feb for all those years? The period crosses Feb of years 1991–2020. Leap years whose Feb is included: 1992,1996,2000,2004,2008,2012,2016,2020 = 8 leap years. Non-leap = 22. Odd days = 22×1 + 8×2 = 22 + 16 = 38. 38 mod 7 = 3. Saturday + 3 = Tuesday. November 3, 2020 was a Tuesday. ✓
Q10. What is the day of the week on December 31, 2099?
AnswerReference: Jan 1, 1900 = Monday (day 1). Years 1900 to 2099 = 200 years. Odd days for 200 years = 3 (from the formula table). But 1900 itself is not a leap year; the 200-year calculation is for complete centuries. Odd days in 200 years (1900–2099) from Jan 1, 1900: 200 years → 3 odd days. Jan 1, 2100 = Monday + 3 = Thursday. Dec 31, 2099 is one day before Jan 1, 2100. Dec 31, 2099 = Thursday − 1 = Wednesday.
