Calculating the Day of the Week for 24 August 1923

Finding the day of the week for any given date, like 24 August 1923, involves using a system based on calculating 'odd days'. Odd days are the number of days remaining after dividing the total number of days by 7 (since there are 7 days in a week). We can calculate the total number of odd days accumulated up to the day before the given date from a fixed reference point.

A common method involves breaking down the date into centuries, years within the century, and days within the year. The number of odd days for centuries and years can be pre-calculated or derived, and the odd days for months are based on the number of days in each month.

Understanding Odd Days

The number of odd days depends on the total number of days. For example:

• 7 days = 1 week + 0 odd days
• 10 days = 1 week + 3 odd days
• Ordinary year (365 days) = 52 weeks + 1 odd day
• Leap year (366 days) = 52 weeks + 2 odd days

Odd Days in Centuries

The number of odd days repeats every 400 years. The odd days for standard centuries are:

So, the odd days pattern for centuries (like 1600, 1700, 1800, 1900, 2000, etc.) is: 1600 (0), 1700 (5), 1800 (3), 1900 (1), 2000 (0), and this pattern 0, 5, 3, 1 repeats for 2400, 2500, 2600, 2700 etc.

Calculating for 24 August 1923

We need to find the total number of odd days up to 23 August 1923.

Step 1: Odd Days from Complete Centuries up to 1900

We consider the period up to 1900 AD (or the period up to the last complete century before the target year, which is 1923). 1900 AD can be written as 1600 years + 300 years.

• Odd days in 1600 years (a multiple of 400) = 0.
• Odd days in 300 years = 1.

Total odd days up to the end of 1900 = $0 + 1 = 1$ odd day.

Step 2: Odd Days from Complete Years from 1901 to 1922

We need to consider the 22 complete years from 1901 to 1922. These 22 years contain ordinary years and leap years.

• To find the number of leap years between 1901 and 1922 (inclusive of 1904, 1908, ... 1920), we divide 22 by 4, which gives 5. The leap years are 1904, 1908, 1912, 1916, 1920. So, there are 5 leap years.
• Number of ordinary years = Total years - Leap years = $22 - 5 = 17$ ordinary years.
• Odd days in 5 leap years = $5 \times 2 = 10$ odd days.
• Odd days in 17 ordinary years = $17 \times 1 = 17$ odd days.

Total odd days in 22 years (1901-1922) = $10 + 17 = 27$ odd days.

Equivalent odd days = $27 \pmod{7} = 6$ odd days.

Step 3: Odd Days from Months in 1923 up to 24 August

1923 is not a leap year (1923 is not divisible by 4). We need to count the days from January 1st to August 24th, 1923.

• Jan: 31 days $\equiv 3$ odd days
• Feb: 28 days $\equiv 0$ odd days (1923 is ordinary year)
• Mar: 31 days $\equiv 3$ odd days
• Apr: 30 days $\equiv 2$ odd days
• May: 31 days $\equiv 3$ odd days
• Jun: 30 days $\equiv 2$ odd days
• Jul: 31 days $\equiv 3$ odd days
• Aug: 24 days $\equiv 3$ odd days ($24 \pmod{7} = 3$)

Total odd days from Jan 1 to Aug 24, 1923 = $3 + 0 + 3 + 2 + 3 + 2 + 3 + 3 = 19$ odd days.

Equivalent odd days = $19 \pmod{7} = 5$ odd days.

Step 4: Total Odd Days

Total odd days up to 24 August 1923 = Odd days up to 1900 + Odd days in 1901-1922 + Odd days in 1923 up to Aug 24.

Total odd days = $1 (\text{from centuries}) + 6 (\text{from years 1901-1922}) + 5 (\text{from months in 1923}) = 12$ odd days.

Equivalent total odd days = $12 \pmod{7} = 5$ odd days.

Step 5: Map Odd Days to Day of the Week

We use the standard mapping:

• 0 odd days = Sunday
• 1 odd day = Monday
• 2 odd days = Tuesday
• 3 odd days = Wednesday
• 4 odd days = Thursday
• 5 odd days = Friday
• 6 odd days = Saturday

Since the total number of odd days is 5, the day of the week on 24 August 1923 is Friday.

Alternative Method using Formula

Another common formula is: $ \text{Day of the week} = (\text{Day} + \text{Month Code} + \text{Year Code} + \text{Century Code}) \pmod{7} $

Where Day is the day of the month (24).

Month Codes (for ordinary year): Jan 0, Feb 3, Mar 3, Apr 6, May 1, Jun 4, Jul 6, Aug 2, Sep 5, Oct 0, Nov 3, Dec 5. For a leap year, Jan code is 6, Feb code is 2, others same.

Year Code: $( \text{Last two digits of year} + \lfloor \text{Last two digits of year} / 4 \rfloor ) \pmod{7}$. For year 1923, last two digits are 23. Leap years in 23 years are $\lfloor 23/4 \rfloor = 5$. Year Code = $(23 + 5) \pmod{7} = 28 \pmod{7} = 0$.

Century Code: For 1900s, the code is 0 (based on the cycle 0, 6, 4, 2 for 1900, 2000, 2100, 2200 etc., corresponding to odd days 1, 0, 5, 3). Let's confirm the century code based on the odd day count: 1900 had 1 odd day, which corresponds to Monday if 0 is Sunday. If we want 0=Sunday, we might need a different set of century codes or adjust the starting point. Let's use the established century codes for the formula method: 1600s=6, 1700s=4, 1800s=2, 1900s=0, 2000s=6.

• Day = 24
• Month Code (Aug, 1923 ordinary year) = 2
• Year Code (Last two digits 23, leap years 5) = $(23 + 5) \pmod{7} = 28 \pmod{7} = 0$
• Century Code (1900s) = 0

Calculation: $(24 + 2 + 0 + 0) \pmod{7} = 26 \pmod{7} = 5$.

Using this formula and these codes, we get a remainder of 5.

Mapping: 0=Sunday, 1=Monday, ..., 6=Saturday.

A remainder of 5 corresponds to Friday.

Both methods consistently indicate that the day of the week on 24 August 1923 was Friday.

Revision Table: Calendar Calculations

Additional Information on Finding Day of the Week

There are several algorithms and methods to determine the day of the week for any given date. The method used above is one of the common approaches based on counting odd days. Other methods include:

• Zeller's Congruence: A well-known algorithm that directly calculates the day of the week using a formula involving the day, month, and year. It's a mathematical formula that yields a number corresponding to the day of the week.
• Doomsday Algorithm: A mental calculation method developed by John Conway. It assigns a specific 'Doomsday' to each year (e.g., April 4th, June 6th, August 8th, October 10th, December 12th are always on the Doomsday). By knowing the Doomsday for a year and fixed Doomsdays for months, one can quickly calculate the day for any date.

These methods rely on the fixed structure of the Gregorian calendar, including the pattern of leap years, which occurs every 4 years, except for years divisible by 100 but not by 400.

The consistency of the calendar allows for these calculations, making it possible to determine historical or future days of the week accurately.