Calendar Problems: Odd Days Method, Leap Years & Day Finding

Calendar problems are a standard topic in Kerala PSC mental ability sections. Questions ask you to find the day of the week for a given date. The odd days method is the fastest approach. Master the rules and practise the solved problems below.

1. Key Concepts

Term	Definition
Odd days	The number of days more than complete weeks in a given period
1 week	7 days (0 odd days)
Leap year	A year with 366 days (February has 29 days)
Ordinary year	A year with 365 days (February has 28 days)
Odd days in 1 ordinary year	365 / 7 = 52 weeks + 1 day = 1 odd day
Odd days in 1 leap year	366 / 7 = 52 weeks + 2 days = 2 odd days

2. Leap Year Rules

Rule	Detail
Divisible by 4	Generally a leap year
Divisible by 100	NOT a leap year (exception below)
Divisible by 400	IS a leap year
Examples — Leap years	2000, 2004, 2008, 2024, 2400
Examples — NOT leap years	1900, 2100, 1800, 2200

Quick check: 1900 is divisible by 4 AND by 100 but NOT by 400 = not a leap year.

3. Odd Days Coding for Days of the Week

Odd Days	Day
0	Sunday
1	Monday
2	Tuesday
3	Wednesday
4	Thursday
5	Friday
6	Saturday

4. Odd Days in Each Month

Month	Days	Odd Days
January	31	3
February (ordinary)	28	0
February (leap)	29	1
March	31	3
April	30	2
May	31	3
June	30	2
July	31	3
August	31	3
September	30	2
October	31	3
November	30	2
December	31	3

5. Odd Days in Century Blocks

Period	Odd Days
100 years	5 odd days
200 years	3 odd days
300 years	1 odd day
400 years	0 odd days

Why 400 years = 0 odd days: 400 years have 97 leap years and 303 ordinary years. Total odd days = (303 x 1) + (97 x 2) = 303 + 194 = 497. Now 497 / 7 = 71 weeks + 0 days. So 0 odd days.

This means the calendar repeats exactly every 400 years.

6. Step-by-Step Method to Find the Day

Step 1: Split the year into completed centuries + remaining years before the target year.

Step 2: Use the century-block table for completed centuries.

Step 3: Count odd days for the remaining completed years after the last century (each ordinary = 1, each leap = 2).

Step 4: Count odd days for completed months of the target year.

Step 5: Add the date itself.

Step 6: Total all odd days, divide by 7, remainder = day code (use table in Section 3).

Important: “Remaining years” means years AFTER the century up to but NOT including the target year.

7. Solved Problem 1 — What day was 15 August 1947?

Step	Calculation
Completed centuries	1900 years = 1600 + 300 years
1600 years	4 x 400 = 0 odd days
300 years	1 odd day
Remaining 46 years (1901-1946)	Leap: 04,08,…,44 = 11. Ordinary: 35. Odd days = 35 + 22 = 57
57 / 7	= 8 weeks + 1 day = 1 odd day
Jan to Jul 1947	Jan(3) + Feb(0) + Mar(3) + Apr(2) + May(3) + Jun(2) + Jul(3) = 16
16 / 7	= 2 weeks + 2 days = 2 odd days
Date: 15	15 / 7 = 2 weeks + 1 day = 1 odd day
Total	0 + 1 + 1 + 2 + 1 = 5
5 =	Friday

15 August 1947 was a Friday.

8. Solved Problem 2 — What day was 26 January 1950?

Step	Calculation
1600 years	0 odd days
300 years	1 odd day
49 years (1901-1949)	Leap: 04,08,…,48 = 12. Ordinary: 37. Odd days = 37 + 24 = 61
61 / 7	= 5 odd days
Months: none (January not completed)	0
Date: 26	26 / 7 = 5 odd days
Total	0 + 1 + 5 + 0 + 5 = 11
11 / 7	= 4 odd days = Thursday

26 January 1950 was a Thursday.

9. Solved Problem 3 — What day is 1 January 2026?

Step	Calculation
2000 years	5 x 400 = 0 odd days
25 years (2001-2025)	Leap: 04,08,12,16,20,24 = 6. Ordinary: 19. Odd days = 19 + 12 = 31
31 / 7	= 3 odd days
Months: none	0
Date: 1	1 odd day
Total	0 + 3 + 0 + 1 = 4
4 =	Thursday

1 January 2026 is a Thursday.

10. More Solved Problems

Problem 4: What day was 2 October 1869 (Gandhi’s birthday)?

1600 yrs (0) + 200 yrs (3). Remaining 68 yrs (1801-1868): Leap: 04,08,…,68 = 17. Ordinary: 51. Odd = 51+34=85, 85/7=1. Months Jan-Sep: 3+0+3+2+3+2+3+3+2=21, 21/7=0. Date=2. Total=0+3+1+0+2=6=Saturday.

Problem 5: What day was 14 November 1889 (Nehru’s birthday)?

1600 yrs (0) + 200 yrs (3). Remaining 88 yrs (1801-1888): Leap: 04,08,…,88 = 22. Ordinary: 66. Odd = 66+44=110, 110/7=5. Jan-Oct: 3+0+3+2+3+2+3+3+2+3=24, 24/7=3. Date=14, 14/7=0. Total=0+3+5+3+0=11, 11/7=4=Thursday.

Problem 6: If 1 January of a year is Monday, what day is 1 March (non-leap year)?

Jan = 31 days (3 odd) + Feb = 28 days (0 odd) = 3 odd days after Monday = Thursday.

Problem 7: How many odd days in the year 2024?

2024 is a leap year = 2 odd days.

Problem 8: On what dates in April 2026 does Sunday fall?

1 Jan 2026 = Thursday (calculated above). Jan(3) + Feb(0) + Mar(3) = 6 odd days after Thursday. Thursday + 6 = Wednesday. So 1 April 2026 = Wednesday. Sundays: 5, 12, 19, 26 April.

Problem 9: What day was 25 December 2000?

1600 yrs (0) + 300 yrs (1). Remaining 99 yrs (1901-1999): Leap: 24. Ordinary: 75. Odd = 75+48=123, 123/7=4. Months Jan-Nov 2000: 3+1(leap)+3+2+3+2+3+3+2+3+2=27, 27/7=6. Date=25, 25/7=4. Total=0+1+4+6+4=15, 15/7=1=Monday.

Problem 10: What day was 1 January 2000?

1600 yrs (0) + 300 yrs (1). Remaining 99 yrs (1901-1999): Leap: 24. Ordinary: 75. Odd = 75+48=123, 123/7=4. Months: none. Date=1. Total=0+1+4+0+1=6=Saturday.

11. Practice Problems (Try Before Checking)

Problem	Answer
11. Day on 15 August 2026	Saturday
12. Day on 26 January 2000	Wednesday
13. Day on 1 May 1960	Sunday
14. If today is Wednesday, what day is after 100 days?	Friday (100/7 = 2 odd days)
15. In a non-leap year, if 1 Feb is Monday, what day is 1 March?	Monday (28 days = 0 odd days)

12. Quick Reference — Tricks

Trick	Detail
Same calendar repeats	Every 400 years (0 odd days in 400 years)
Non-leap year that starts on same day	After 6 or 11 years (varies)
Feb 28 to Mar 1	Always same day in non-leap year (0 odd days in Feb)
Last day of century	1900: Monday end, 2000: Sunday end, 2100: Saturday end
Every 28 years	Calendar repeats within a century (not across century boundaries)

13. Previous Year Question Patterns

• “If 1 Jan is Tuesday, what day is 31 Dec (non-leap year)?” — Tuesday (365 days = 1 odd day… but 31 Dec = day 365. From Jan 1 to Dec 31 = 364 days = 0 odd days. So same day: Tuesday)
• “How many odd days in 400 years?” — 0
• “1900 is a leap year — True or False?” — False
• “Republic Day 1950 fell on?” — Thursday