Calendar

1. Core Concepts & Theoretical Blueprint

Calendar problems reduce every date-finding question to counting "odd days" — the remainder of total days divided by 7.

Absolute Core Definition:

Odd Days Reference Table:

Period	Odd Days
1 ordinary year (365 days)	1
1 leap year (366 days)	2
100 years	5
200 years	3
300 years	1
400 years	0

Leap Year Rule:

Day-Code Mapping:

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

Month Odd-Days Table (non-leap year, cumulative from Jan 1):

Month	Days	Odd Days
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

The Universal Trap: 1) Century-year leap-year misapplication (1900 is NOT a leap year). 2) Off-by-one in date-range counting. 3) Forgetting the leap-day adjustment applies only from March onward.

2. Exhaustive Question Typology

                              CALENDAR
                                 |
    ------------------------------------------------------------
    |               |                |               |         |
Type 1:          Type 2:          Type 3:         Type 4:    Type 5:
Find day of      Find day for     Reference-      Leap Year  Same Calendar
week for a       a given date     Century-code    Identification  Repeats
given date       using a known    method                    (Two years with
                 reference date                              identical
                                                               calendar)
    |               |
Type 6:          Type 7:
Number of        Odd days in
Mondays/         a given month
Sundays in a     range/quarter
given month or
year

Type 1 — Find the day of the week for a given date:

• Governing Equation: Total Odd Days = century contribution + year contribution + month contribution + date, mod 7.

Type 2 — Find day given a known reference date:

• Governing Equation: $\text{New Day Code} = (\text{Reference Day Code} + \text{Odd Days in Elapsed Period}) \bmod 7$

Type 3 — Century reference-code method:

• Governing Equation: Century Codes — 1600s/2000s: 0; 1700s: 5; 1800s: 3; 1900s: 1.

Type 4 — Leap year identification:

• Governing Equation: $(Y\bmod 4=0 \text{ and } Y\bmod100\neq0)$ OR $(Y\bmod400=0)$

Type 5 — Same calendar repeats:

• Governing Equation: Sum odd days year by year until cumulative sum is a multiple of 7, matching leap status.

Type 6 — Number of Mondays/Sundays in a given month/year:

• Governing Equation: Extra odd day(s) determine which weekday(s) occur 53 times instead of 52 in a year; a 31-day month has exactly 3 weekdays occurring 5 times.

Type 7 — Odd days across a multi-month span/quarter:

• Governing Equation: Odd Days = (Total Days elapsed) mod 7.

3. Type-wise Practice MCQs with Full Solutions

Type 1 — Find the Day of the Week for a Given Date

MCQ 1. What was the day of the week on 26th January 1950? (A) Wednesday (B) Thursday (C) Friday (D) Saturday

Correct Answer: (A) Solution: 1900s code=1. Years 1901-1949 (48 years, 12 leap years): odd days=48+12=60→60mod7=4. Running=1+4=5. Jan cumulative before Jan=0, add date 26: 26mod7=5. Total=5+5=10→10mod7=3=Wednesday.

MCQ 2. What was the day of the week on 15th August 1947? (A) Friday (B) Thursday (C) Saturday (D) Sunday

Correct Answer: (A) Solution: 1900s code=1. Years 1901-1946 (46 years, 11 leap): 46+11=57→57mod7=1. Running=1+1=2. Jan-Jul cumulative=212 days→212mod7=2. Running=2+2=4. Add date 15: 4+15=19→19mod7=5=Friday.

MCQ 3. What day of the week was 1st January 2000? (A) Saturday (B) Sunday (C) Friday (D) Monday

Correct Answer: (A) Solution: This is a well-established historical/reference fact confirmed by the odd-days method: 1 January 2000 was a Saturday.

Type 2 — Find Day Given a Known Reference Date

MCQ 1. If 1st January 2024 was a Monday, what day was 1st January 2025? (A) Wednesday (B) Tuesday (C) Thursday (D) Monday

Correct Answer: (A) Solution: 2024 is a leap year, contributing 2 odd days. New day code=(1+2)mod7=3=Wednesday.

MCQ 2. If 1st January 2023 was a Sunday, what day was 1st January 2024? (A) Monday (B) Tuesday (C) Sunday (D) Wednesday

Correct Answer: (A) Solution: 2023 is not a leap year (1 odd day). New code=(0+1)mod7=1=Monday.

MCQ 3. If 15th March 2023 was a Wednesday, what day was 15th March 2024 (spanning the Feb 29, 2024 leap day)? (A) Friday (B) Thursday (C) Saturday (D) Sunday

Correct Answer: (A) Solution: The span from 15 Mar 2023 to 15 Mar 2024 covers 366 days (includes Feb 29, 2024) → 366mod7=2. New day=(3+2)mod7=5=Friday.

Type 3 — Century Reference-Code Method

MCQ 1. Using century codes, find the day of the week for 15th August 1947. (A) Friday (B) Thursday (C) Saturday (D) Sunday

Correct Answer: (A) Solution: As computed in Type 1 MCQ 2: Friday.

MCQ 2. Find the day of the week for 4th July 1776 (1700s code=5). (A) Thursday (B) Friday (C) Wednesday (D) Saturday

Correct Answer: (A) Solution: Years 1701-1775 (75 years, 18 leap years): 75+18=93→93mod7=2. Running=5+2=0(mod7). Jan-Jun cumulative for 1776 (a leap year, with Feb 29 occurring before July)=181+1=182→182mod7=0. Running=0+0=0. Add date 4: 0+4=4=Thursday.

MCQ 3. Find the day of the week for 25th December 1600 (1600s code=0). (A) Thursday (B) Saturday (C) Friday (D) Sunday

Correct Answer: (A) Solution: Code=0 for the century, no full elapsed years before 1600 itself. 1600 is a leap year. Jan-Nov cumulative (leap-adjusted)=334+1=335→335mod7=0. Add date 25: 0+25=25→25mod7=4=Thursday.

Type 4 — Leap Year Identification

MCQ 1. Is 2100 a leap year? (A) No (B) Yes (C) Cannot be determined (D) Only for February

Correct Answer: (A) Solution: 2100 is divisible by 100 but NOT by 400, so it is NOT a leap year.

MCQ 2. Is 1996 a leap year? (A) Yes (B) No (C) Cannot be determined (D) Only partially

Correct Answer: (A) Solution: 1996 is divisible by 4, and not a century year, so it IS a leap year.

MCQ 3. Is 2000 a leap year? (A) Yes (B) No (C) Cannot be determined (D) Only for January

Correct Answer: (A) Solution: 2000 is divisible by 400, so it IS a leap year.

Type 5 — Same Calendar Repeats

MCQ 1. The calendar for 2023 (non-leap) will next be identical to which year? (A) 2034 (B) 2029 (C) 2030 (D) 2033

Correct Answer: (A) Solution: Cumulative odd days from 2023 through subsequent years reach a multiple of 7 with matching leap status at 2034.

MCQ 2. The calendar for 2015 will repeat itself in which year (2015 is non-leap)? (A) 2026 (B) 2020 (C) 2021 (D) 2025

Correct Answer: (A) Solution: Cumulative odd days (2015 through 2025) sum to a multiple of 7 with matching leap status, giving 2026.

MCQ 3. For a leap year, what is the standard minimum number of years after which the same calendar (matching both day alignment and leap status) typically repeats, away from century boundaries? (A) 28 years (B) 12 years (C) 11 years (D) 6 years

Correct Answer: (A) Solution: The standard non-century-boundary repeat cycle for a leap year's calendar is 28 years.

Type 6 — Number of Mondays/Sundays

MCQ 1. How many Sundays are there in a non-leap year that begins on a Sunday? (A) 53 (B) 52 (C) 51 (D) 54

Correct Answer: (A) Solution: A non-leap year has 1 odd day. If it begins on Sunday, that odd day is also Sunday, causing 53 Sundays.

MCQ 2. How many Mondays are there in a leap year that begins on a Monday? (A) 53 (B) 52 (C) 51 (D) 54

Correct Answer: (A) Solution: A leap year has 2 odd days. If it begins on Monday, the two odd days are Monday and Tuesday, so Monday occurs 53 times.

MCQ 3. A 31-day month begins on a Wednesday. Which days occur 5 times in this month? (A) Wednesday, Thursday, Friday (B) Monday, Tuesday, Wednesday (C) Sunday, Monday, Tuesday (D) Thursday, Friday, Saturday

Correct Answer: (A) Solution: A 31-day month has 4 full weeks + 3 extra days, matching the first 3 weekdays of the month: Wednesday, Thursday, Friday.

Type 7 — Odd Days Across a Multi-Month Span

MCQ 1. Find the day of the week 90 days after a Wednesday. (A) Tuesday (B) Thursday (C) Wednesday (D) Saturday

Correct Answer: (A) Solution: $90\bmod7=6$. Counting 6 days forward from Wednesday: Thu,Fri,Sat,Sun,Mon,Tue — lands on Tuesday.

MCQ 2. Find the day of the week 200 days after a Monday. (A) Friday (B) Thursday (C) Wednesday (D) Saturday

Correct Answer: (A) Solution: $200\bmod7=4$. Counting 4 days forward from Monday: Tue,Wed,Thu,Fri — lands on Friday.

MCQ 3. Find the day of the week 45 days before a Sunday. (A) Thursday (B) Saturday (C) Friday (D) Monday

Correct Answer: (A) Solution: $45\bmod7=3$. Counting 3 days backward from Sunday: Sat,Fri,Thu — lands on Thursday.

4. High-Yield Speed Tricks & Shortcut Mental Models

Shortcut 1 — The Century Code Table

• Application: Any Type 1/3 problem involving a full date.
• Mental Model: Pre-memorize: 1600s & 2000s → 0, 1700s → 5, 1800s → 3, 1900s → 1.

Shortcut 2 — Month Odd-Days Cumulative Recall

• Application: Any date-to-day-of-week conversion.
• Mental Model: Memorize the cumulative odd-days-before-month sequence for instant lookup.

5. Deep-Dive: Most Frequently Asked Questions

Problem 1 (SSC/RRB Standard): What was the day of the week on 15th August 1947?

Traditional Method (Slow): Full breakdown across 1600s, 300 years, 46 years, months, date — multiple additions (~45-50 seconds).

Exam Shortcut (Fast): Century Code (1900s)=1. Years 1901-1946: 57→mod7=1. Running=2. Month cumulative (before Aug)=2. Running=4. +15=19→mod7=5=Friday. (Under 20 seconds.)

Problem 2 (UPSC/Banking Advanced): The calendar for the year 2023 will be exactly the same as the calendar for which future year, given 2023 is not a leap year?

Step-by-Step Breakdown:

1. Sum odd days year by year starting 2023.
2. Cumulative odd days reach 0 mod 7 at year 2034, with matching non-leap status.
3. Answer: The calendar of 2023 repeats in 2034.

6. Chapter Checklist for Students

• I apply the correct leap year test every time, with no shortcuts on century years.
• I have memorized the century odd-day codes for instant lookup.
• I have memorized the cumulative month odd-days sequence and adjust for leap years post-February.
• I check leap-year status matching when solving "calendar repeats" problems.
• I always confirm whether a date-range question treats both endpoints as inclusive.