Finding the Day of the Week for 26th November 1994

To determine the day of the week for a specific date, we use the concept of 'odd days'. Odd days are the number of days left after forming complete weeks from a given period.

Here's how we calculate the day for 26th November 1994:

We need to calculate the total number of odd days from the beginning of the calendar epoch (usually considered Year 0 or Year 1, depending on convention; for these types of problems, the logic relies on relative odd days over centuries/years, often starting calculations from a known point like a century divisible by 400) up to the day before the target date (i.e., up to 25th November 1994, or equivalently, the end of 1993 plus the days in 1994 up to 26th November).

Let's break down the period up to 26th November 1994:

1. Years completed before 1994: 1993 years.
2. Days in the year 1994 up to 26th November.

Calculating Odd Days in 1993 Years

We can divide 1993 years into centuries and remaining years:

\(1993 \text{ years} = 1600 \text{ years} + 300 \text{ years} + 93 \text{ years}\)

• Odd days in 1600 years: A century year is a leap year if it is divisible by 400. Odd days in 400 years are 0. Therefore, odd days in $1600 = 4 \times 400$ years is $4 \times 0 = 0$.
• Odd days in 300 years: The number of odd days in 100 years is 5. The number of odd days in 200 years is $2 \times 5 = 10 \equiv 3 \pmod{7}$. The number of odd days in 300 years is $3 \times 5 = 15 \equiv 1 \pmod{7}$.
• Odd days in 93 years (1901 to 1993): We need to find the number of leap years and ordinary years in this period. A year is a leap year if it is divisible by 4, unless it is a century year not divisible by 400. In the years 1901 to 1993, the leap years are 1904, 1908, ..., 1992.
  • Number of leap years = $\lfloor \frac{93}{4} \rfloor = 23$ leap years.
  • Number of ordinary years = Total years - Leap years = $93 - 23 = 70$ ordinary years.
  • Odd days in 93 years = (Number of leap years $\times$ 2) + (Number of ordinary years $\times$ 1)
  • Odd days in 93 years = $(23 \times 2) + (70 \times 1) = 46 + 70 = 116$ odd days.
  • To find the equivalent odd days within a week: $116 \pmod{7}$. $116 = 16 \times 7 + 4$. So, $116 \equiv 4 \pmod{7}$.
  • Odd days in 93 years = 4.

Total odd days up to the end of 1993 = Odd days in 1600 years + Odd days in 300 years + Odd days in 93 years = $0 + 1 + 4 = 5$ odd days.

Calculating Odd Days in 1994 up to 26th November

1994 is not a leap year (1994 is not divisible by 4).

We sum the number of days in each month from January to October and add the days in November up to the 26th. Then we find the total odd days.

• January (31 days): $31 \pmod{7} = 3$ odd days
• February (28 days): $28 \pmod{7} = 0$ odd days (1994 is ordinary)
• March (31 days): $31 \pmod{7} = 3$ odd days
• April (30 days): $30 \pmod{7} = 2$ odd days
• May (31 days): $31 \pmod{7} = 3$ odd days
• June (30 days): $30 \pmod{7} = 2$ odd days
• July (31 days): $31 \pmod{7} = 3$ odd days
• August (31 days): $31 \pmod{7} = 3$ odd days
• September (30 days): $30 \pmod{7} = 2$ odd days
• October (31 days): $31 \pmod{7} = 3$ odd days
• November (26 days): $26 \pmod{7} = 5$ odd days

Total odd days in 1994 up to 26th November = $3 + 0 + 3 + 2 + 3 + 2 + 3 + 3 + 2 + 3 + 5 = 29$ odd days.

To find the equivalent odd days within a week: $29 \pmod{7}$. $29 = 4 \times 7 + 1$. So, $29 \equiv 1 \pmod{7}$.

Odd days in 1994 up to 26th November = 1 odd day.

Total Odd Days and Determining the Day

Total odd days up to 26th November 1994 = Total odd days up to end of 1993 + Odd days in 1994 up to 26th November.

Total odd days = $5 + 1 = 6$ odd days.

We map the total odd days to the day of the week using the following convention:

Since the total number of odd days is 6, the day of the week on 26th November 1994 was Saturday.

Revision Table: Calendar Concepts

Additional Information: Calendar Calculations

Calendar calculations are a common topic in logical reasoning and quantitative aptitude sections of various exams. Understanding the concept of odd days is crucial for solving problems related to finding the day of the week for a given date, determining the number of leap years in a period, or finding the day of the week after a certain number of days.

The cycle of odd days repeats every 400 years. This means the number of odd days accumulated at the end of every 400-year period is 0. This property is very useful when dealing with centuries.

Remembering the number of odd days in standard periods like 100, 200, 300, and 400 years, and the number of odd days in each month, simplifies these calculations significantly.