Episode 8 — Aptitude and Reasoning / 8.18 — Calendar
8.18.c Calendar - Solved Examples
Example 1: Leap Year Identification
Problem: Which of the following are leap years: 1600, 1700, 1900, 2000, 2024, 2100?
Solution:
1600: Century year. 1600/400 = 4 (exact) -> LEAP YEAR
1700: Century year. 1700/400 = 4.25 -> NOT a leap year
1900: Century year. 1900/400 = 4.75 -> NOT a leap year
2000: Century year. 2000/400 = 5 (exact) -> LEAP YEAR
2024: Non-century. 2024/4 = 506 (exact) -> LEAP YEAR
2100: Century year. 2100/400 = 5.25 -> NOT a leap year
Leap years: 1600, 2000, 2024
Example 2: Finding the Day (Independence Day)
Problem: What day of the week was 15th August 1947?
Solution:
Step 1: Break down 1947 = 1900 complete years + 47 years
Step 2: Odd days in 1900 years
1900 = 4 * 400 + 300
Odd days = 4 * 0 + 1 = 1
Step 3: Odd days in 47 years
Leap years in 47 years = floor(47/4) = 11
Ordinary years = 47 - 11 = 36
Odd days = 36 * 1 + 11 * 2 = 36 + 22 = 58
58 mod 7 = 2
Step 4: Odd days in Jan-Jul (1947 is not a leap year)
Jan(3) + Feb(0) + Mar(3) + Apr(2) + May(3) + Jun(2) + Jul(3) = 16
16 mod 7 = 2
Step 5: Remaining days = 15
15 mod 7 = 1
Step 6: Total odd days = 1 + 2 + 2 + 1 = 6
6 = FRIDAY
15th August 1947 was a FRIDAY.
Example 3: Finding the Day (Republic Day)
Problem: What day of the week was 26th January 1950?
Solution:
Step 1: 1950 = 1900 + 50
Step 2: Odd days in 1900 years = 1 (as calculated above)
Step 3: Odd days in 50 years
Leap years = floor(50/4) = 12
Ordinary years = 50 - 12 = 38
Odd days = 38 + 24 = 62
62 mod 7 = 6
Step 4: Odd days in completed months before January = 0
Step 5: Days = 26
26 mod 7 = 5
Step 6: Total = 1 + 6 + 0 + 5 = 12
12 mod 7 = 5
5 = THURSDAY
26th January 1950 was a THURSDAY.
Example 4: Finding the Day (Y2K)
Problem: What day was January 1, 2000?
Solution:
Step 1: 2000 = 2000 complete years (but we need up to Dec 31, 1999)
Actually: Consider 1999 complete years + Jan 1
Step 2: Odd days in 1999 years = Odd days in 1600 + 300 + 99
1600 years: 0 odd days
300 years: 1 odd day
99 years: Leap = floor(99/4) = 24, Ordinary = 75
Odd days = 75 + 48 = 123, 123 mod 7 = 4
Step 3: Total for 1999 years = 0 + 1 + 4 = 5
Step 4: No completed months in Jan, date = 1
Odd days = 1
Step 5: Total = 5 + 1 = 6
6 = SATURDAY
January 1, 2000 was a SATURDAY.
Example 5: Finding Day of a Recent Date
Problem: What day was March 14, 2023?
Solution:
Step 1: 2023 = 2000 + 23
Step 2: Odd days in 2000 years
2000 = 5 * 400 -> 5 * 0 = 0
Step 3: Odd days in 23 years
Leap years = floor(23/4) = 5 (years 4, 8, 12, 16, 20)
Ordinary = 18
Odd days = 18 + 10 = 28
28 mod 7 = 0
Step 4: Months - Jan + Feb (2023 is NOT a leap year)
Jan(3) + Feb(0) = 3
Step 5: Date = 14
14 mod 7 = 0
Step 6: Total = 0 + 0 + 3 + 0 = 3
3 = TUESDAY
March 14, 2023 was a TUESDAY.
Example 6: Number of Odd Days
Problem: How many odd days are there in 800 years?
Solution:
800 years = 2 * 400 years
Each 400 years = 0 odd days
Total odd days = 2 * 0 = 0
There are 0 odd days in 800 years.
Example 7: Counting Leap Years
Problem: How many leap years are there between 1901 and 2000?
Solution:
Years divisible by 4 from 1904 to 2000:
1904, 1908, ..., 2000
Count = (2000 - 1904)/4 + 1 = 96/4 + 1 = 24 + 1 = 25
But check century year: 2000 is divisible by 400, so it IS a leap year.
Total leap years = 25
(If the range were 1801-1900: 1804, 1808, ..., 1896
Count = (1896-1804)/4 + 1 = 24. Since 1900/400 != integer, 1900 is NOT a leap year.
Total = 24)
Example 8: Same Day Calculation
Problem: If January 1, 2024 was a Monday, what day is January 1, 2025?
Solution:
2024 is a leap year (2024/4 = 506, not a century year).
Odd days in a leap year = 2
Day of Jan 1, 2025 = Monday + 2 days = WEDNESDAY
Example 9: Day After n Days
Problem: If today is Wednesday, what day will it be after 150 days?
Solution:
150 mod 7 = 21 * 7 + 3 = 3 odd days
Wednesday + 3 = Saturday
After 150 days, it will be SATURDAY.
Example 10: Day Before n Days
Problem: If today is Friday, what day was it 200 days ago?
Solution:
200 mod 7 = 28 * 7 + 4 = 4 odd days
Friday - 4 = Monday
200 days ago, it was MONDAY.
Example 11: Repeating Calendar
Problem: The calendar of 2021 can be used again in which year?
Solution:
2021 is an ordinary year starting on Friday.
We need 7 odd days (net 0) from 2021:
Year | Type | Odd days | Cumulative
2021 | Ordinary | 1 | 1
2022 | Ordinary | 1 | 2
2023 | Ordinary | 1 | 3
2024 | Leap | 2 | 5
2025 | Ordinary | 1 | 6
2026 | Ordinary | 1 | 7 -> 0 mod 7
After 6 years, total odd days = 7 = 0 mod 7.
Check: 2027 should be an ordinary year starting on Friday.
2027 is not divisible by 4, so it is ordinary. Correct!
The 2021 calendar repeats in 2027.
Example 12: Leap Year Calendar Repetition
Problem: In which year will the calendar of 2024 (a leap year) repeat?
Solution:
A leap year calendar repeats after 28 years.
2024 + 28 = 2052
The 2024 calendar will repeat in 2052.
Verification: 2052/4 = 513 (exact), and 2052 is not a century year.
So 2052 is indeed a leap year. Correct!
Example 13: Finding Day of Birth
Problem: Mahatma Gandhi was born on October 2, 1869. What day was it?
Solution:
Step 1: 1869 = 1800 + 69
Step 2: Odd days in 1800 years
1800 = 4 * 400 + 200
= 0 + 3 = 3 odd days
Step 3: Odd days in 69 years
Leap years = floor(69/4) = 17
Ordinary = 52
Odd days = 52 + 34 = 86
86 mod 7 = 2
Step 4: Months (Jan to Sep, 1869 is not a leap year)
Jan(3) + Feb(0) + Mar(3) + Apr(2) + May(3) + Jun(2) + Jul(3) + Aug(3) + Sep(2) = 21
21 mod 7 = 0
Step 5: Date = 2
Step 6: Total = 3 + 2 + 0 + 2 = 7
7 mod 7 = 0
0 = SATURDAY
October 2, 1869 was a SATURDAY.
Example 14: Number of Sundays in a Month
Problem: How many Sundays are in January 2023 if January 1, 2023 is a Sunday?
Solution:
January has 31 days = 4 weeks + 3 days
If Jan 1 = Sunday, then:
Sundays fall on: 1, 8, 15, 22, 29
Total Sundays = 5
The extra 3 days (Jan 29, 30, 31) include one Sunday (Jan 29).
Example 15: Days Between Two Dates
Problem: Find the number of days between March 8, 2023 and November 22, 2023.
Solution:
From March 8 to November 22:
Remaining days in March: 31 - 8 = 23
April: 30
May: 31
June: 30
July: 31
August: 31
September: 30
October: 31
November: 22 (up to Nov 22)
Total = 23 + 30 + 31 + 30 + 31 + 31 + 30 + 31 + 22 = 259 days
Example 16: What Day After Years
Problem: If February 14, 2020 was a Friday, what day is February 14, 2025?
Solution:
Count odd days from 2020 to 2025:
2020 (leap year): 2 odd days
2021 (ordinary): 1 odd day
2022 (ordinary): 1 odd day
2023 (ordinary): 1 odd day
2024 (leap year): 2 odd days
Total odd days = 2 + 1 + 1 + 1 + 2 = 7
7 mod 7 = 0
Day = Friday + 0 = FRIDAY
February 14, 2025 is a FRIDAY.
Example 17: Century Starting Day
Problem: What day was January 1, 1901?
Solution:
Step 1: 1901 = 1900 + 1
Step 2: Odd days in 1900 years = 1 (as before)
Step 3: Odd days in 0 complete years = 0
Step 4: No completed months before January = 0
Step 5: Date = 1
Step 6: Total = 1 + 0 + 0 + 1 = 2
2 = TUESDAY
January 1, 1901 was a TUESDAY.
Example 18: Finding the Year for a Given Day
Problem: In which year (after 2020) will Republic Day (January 26) first fall on a Sunday?
Solution:
January 26, 2020: Let's find this day first.
2020 = 2000 + 20
Odd days in 2000: 0
Odd days in 20 years: Leap = 5, Ordinary = 15
= 15 + 10 = 25, 25 mod 7 = 4
Months: 0 (January)
Date: 26, 26 mod 7 = 5
Total = 0 + 4 + 0 + 5 = 9, 9 mod 7 = 2 = TUESDAY (but actually...)
Wait, let me recalculate. We know Jan 26, 2020 was a Sunday in reality. Let me verify:
Actually Jan 1, 2020 was Wednesday. Jan 26 = Jan 1 + 25 days.
25 mod 7 = 4. Wednesday + 4 = Sunday. Yes, Jan 26, 2020 was SUNDAY.
So we need the next year when Jan 26 is again Sunday.
From 2020: Add odd days per year:
2020 (leap): +2 -> 2021: Sunday + 2 = Tuesday
2021 (ord): +1 -> 2022: Tuesday + 1 = Wednesday
2022 (ord): +1 -> 2023: Wednesday + 1 = Thursday
2023 (ord): +1 -> 2024: Thursday + 1 = Friday
2024 (leap): +2 -> 2025: Friday + 2 = Sunday!
January 26, 2025 is a SUNDAY. (This is correct - Republic Day 2025 was indeed on Sunday.)
Example 19: Months with Same Starting Day
Problem: In the year 2023 (ordinary year), which months start on the same day as January?
Solution:
In an ordinary year, January and October start on the same day.
Verification:
Jan 1 to Oct 1:
Jan(31) + Feb(28) + Mar(31) + Apr(30) + May(31) + Jun(30) + Jul(31) + Aug(31) + Sep(30) = 273
273 mod 7 = 0
Since 273 is exactly divisible by 7, Oct 1 falls on the same day as Jan 1.
In 2023, January and October both start on a Sunday.
Example 20: Maximum Fridays in a Month
Problem: A month has 31 days and starts on Thursday. How many Fridays are in this month?
Solution:
31 days = 4 weeks + 3 days
If the month starts on Thursday:
Thursday: Days 1, 8, 15, 22, 29 (5 Thursdays)
Friday: Days 2, 9, 16, 23, 30 (5 Fridays)
Saturday: Days 3, 10, 17, 24, 31 (5 Saturdays)
The remaining 4 days of the week appear 4 times each.
There are 5 Fridays in this month.
Example 21: Number of Days in a Period
Problem: How many days are there from January 1, 2024 to December 31, 2024 (inclusive)?
Solution:
2024 is a leap year.
Total days = 366
(From Jan 1 to Dec 31 inclusive is the entire year.)
Example 22: Historical Date
Problem: What day was July 4, 1776 (American Independence Day)?
Solution:
Step 1: 1776 = 1600 + 176
Step 2: Odd days in 1600 years = 0
Step 3: Odd days in 176 years
Leap years = floor(176/4) = 44
Ordinary years = 132
Odd days = 132 + 88 = 220
220 mod 7 = 220 - 31*7 = 220 - 217 = 3
Step 4: Months Jan to Jun (1776 IS a leap year, 1776/4 = 444)
Jan(3) + Feb(0+1 for leap) + Mar(3) + Apr(2) + May(3) + Jun(2)
= 3 + 1 + 3 + 2 + 3 + 2 = 14
14 mod 7 = 0
Step 5: Date = 4
Step 6: Total = 0 + 3 + 0 + 4 = 7
7 mod 7 = 0
0 = SUNDAY... but wait.
Let me recheck. Actually, we should not count 1776 as a completed year because the date is IN 1776. The 176 years should be years 1601-1776... Let me reconsider.
Actually, we count years up to 1775 (completed years), then add months and days of 1776.
Step 1: 1775 completed years = 1600 + 175
Step 2: Odd days in 1600 years = 0
Step 3: Odd days in 175 years
Leap years = floor(175/4) = 43
Ordinary = 132
Odd days = 132 + 86 = 218
218 mod 7 = 218 - 31*7 = 218 - 217 = 1
Step 4: Jan to Jun of 1776 (1776 IS a leap year)
Jan(3) + Feb(1, leap) + Mar(3) + Apr(2) + May(3) + Jun(2) = 14
14 mod 7 = 0
Step 5: Date = 4
Step 6: Total = 0 + 1 + 0 + 4 = 5
5 = THURSDAY
July 4, 1776 was a THURSDAY.
Example 23: Odd Days in a Specific Period
Problem: Find the odd days from March 3 to September 12 of the same ordinary year.
Solution:
From March 3 to September 12:
Remaining days in March: 31 - 3 = 28
April: 30
May: 31
June: 30
July: 31
August: 31
September: 12
Total = 28 + 30 + 31 + 30 + 31 + 31 + 12 = 193
193 mod 7 = 27 * 7 + 4 = 4 odd days
Next: 8.18 - Practice MCQs