CALENDARS AND TECHNIQUES
Key Concepts and Formulas
1. Days of the Week
- Reference Day: Remembering a reference day for a known date helps. For instance, January 1, 1900, was a Monday.
- Day Codes: Each day of the week can be assigned a code:
- Sunday = 0
- Monday = 1
- Tuesday = 2
- Wednesday = 3
- Thursday = 4
- Friday = 5
- Saturday = 6
2. Leap Year Rules
- A year is a leap year if:
- It is divisible by 4.
- However, if it is a century year (ending in 00), it must be divisible by 400.
- Example: 2000 is a leap year, 1900 is not.
3. Odd Days Concept
- Odd days are the number of days more than the complete weeks in a given period.
- Calculating Odd Days:
- 1 ordinary year = 365 days = 52 weeks + 1 day = 1 odd day
- 1 leap year = 366 days = 52 weeks + 2 days = 2 odd days
- 100 years = 76 ordinary years + 24 leap years = (761 + 242) odd days = 124 odd days = 17 weeks + 5 days = 5 odd days
- Thus, 100 years have 5 odd days.
4. Finding the Day of the Week for Any Date
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Steps:
- Break the date into components: Date, Month, Year.
- Calculate the total number of odd days:
- Odd days in the year part.
- Odd days in the month part (use known month codes).
- Odd days in the days part.
- Sum the odd days and take modulo 7 to find the day of the week.
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Example Calculation:
- To find the day of the week on 15th August 1947:
- Years: 1947 years.
- From 1600 to 1900 = 300 years = (3 centuries * 5 odd days) + (1 century * 5 odd days) = 20 odd days.
- 1947-1900 = 47 years. Using leap year calculation, there are 11 leap years and 36 ordinary years, giving (112 + 361) = 58 odd days = 8 weeks + 2 odd days = 2 odd days.
- Month part for January to July and August 15 (Month codes for standard use: Jan = 3, Feb = 0, Mar = 3, Apr = 2, May = 3, Jun = 2, Jul = 3, Aug = 3) = 3 + 0 + 3 + 2 + 3 + 2 + 3 + 15 = 31 days = 4 weeks + 3 odd days = 3 odd days.
- Total odd days = 20 (centuries) + 2 (47 years) + 3 (month part) = 25 odd days = 3 weeks + 4 odd days = 4 odd days.
- Thus, 15th August 1947 was a Friday.
- To find the day of the week on 15th August 1947:
Let's start with an example. Most people look forward to celebrating the country's Independence Day on the 15th of August each year. This year, in 2024, Independence Day will fall on a Thursday. Suppose you want to plan your celebrations for Independence Day in 2025 ahead of time. Without checking the calendar, can you determine which day of the week August 15, 2025, will be?
Here's a quick method to solve such problems. Let's break down the problem clearly.
Question: If August 15, 2024, is a Thursday, what day will it be on August 15, 2025?
To solve this, we first need to determine if 2025 is a leap year. Checking for a leap year is your initial step. A year is a leap year if it is divisible by 4. If there is no remainder, the year is a leap year. However, if there is a remainder, the year is not a leap year. Since 2025 is not divisible by 4, it is not a leap year. This also means February 2025 will have only 28 days.
When we move from August 15, 2024, to August 15, 2025, month by month, we will not encounter February 29 because 2025 is not a leap year. In such cases, where there is no February 29 in the transition from one year to the next, we simply add 1 to the day of the week to find the answer
So, if its Thursday on 15th August 2024, it will be (Thursday + 1 = Friday) on 15th August 2025
Let's look at another example:
Question: If August 15, 2027, is a Sunday, what day of the week will it be on August 15, 2028?
To solve this, we need to move from August 15, 2027, to August 15, 2028, month by month to see if we encounter February 29, 2028. First, note that 2028 is divisible by 4, making it a leap year with 366 days and an extra day in February, specifically February 29, 2028. As we move from August 15, 2027, to August 15, 2028, we do encounter February 29, so we add 2 days to Sunday to find our answer. Therefore, if August 15, 2027, is a Sunday, August 15, 2028, will be a Tuesday
Practice Problems
Problem 1: Determine the day of the week on 26th January 1950.
Problem 2: If today is Monday, what day will it be 100 days from today?
Problem 3: Find the number of leap years between 1800 and 2000.
