Calculating the Day of the Week for March 31st in a Leap Year

This problem involves finding the day of the week for a specific date in a leap year, given the day of the week for an earlier date in the same year. We need to determine the total number of days between the two dates and use the concept of 'odd days' to find the shift in the day of the week.

Understanding the Problem

We are given:

• The year is a leap year.
• January 1st is a Friday.
• We need to find the day of the week for March 31st.

Days in Each Month for a Leap Year

In a leap year, February has 29 days. The number of days in January and March remains the same.

• January: 31 days
• February (Leap Year): 29 days
• March: 31 days

Calculating the Total Number of Days

We need to find the total number of days from January 1st to March 31st, inclusive. Alternatively, we can count the number of full days after January 1st up to March 31st.

Let's count the number of days from January 1st *to* March 31st. We count the number of days remaining in January, all days in February, and all days in March up to the 31st.

• Days remaining in January (after Jan 1): $31 - 1 = 30$ days
• Days in February: 29 days
• Days in March: 31 days

Total number of days from Jan 2nd to March 31st = $30 + 29 + 31 = 90$ days.

Alternatively, the total number of days from Jan 1st to March 31st, inclusive, is the sum of the days in Jan, Feb, and Mar: $31 + 29 + 31 = 91$ days. If Jan 1st is Day 1, then March 31st is Day 91.

Using the Concept of Odd Days

The day of the week repeats every 7 days. The shift in the day of the week from a starting date to a date $N$ days later is determined by the remainder when $N$ is divided by 7. This remainder is called the number of 'odd days'.

If we consider Jan 1st as our starting point (Day 0, which is Friday), then March 31st is 90 days after Jan 1st. We need to find the number of odd days in 90 days.

Divide the total number of days (90) by 7:

\begin{equation*} 90 \div 7 \end{equation*}

\begin{equation*} 90 = (12 \times 7) + 6 \end{equation*}

The remainder is 6. This means there are 6 odd days.

Determining the Final Day of the Week

Since January 1st is a Friday, we add the number of odd days (6) to Friday.

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

Therefore, the day of the week for March 31st in that leap year is Thursday.

Revision Table: Days in Months & Odd Days

Additional Information: Leap Years and Odd Days

Leap Years: A leap year occurs every 4 years to synchronize the calendar year with the astronomical year. A year is a leap year if it is divisible by 4, except for years divisible by 100 but not by 400. For example, 2000 and 2020 were leap years, but 1900 was not. Leap years have 366 days instead of 365, with the extra day added to February (making it 29 days).

Odd Days: The concept of odd days is crucial for solving calendar-based problems. It represents the number of days that exceed a complete week. For example, 10 days have $10 \div 7 = 1$ week and 3 odd days. These 3 odd days cause a shift of 3 days forward in the day of the week from the starting day.

In this problem, we calculated the total days from Jan 1st to March 31st (91 days inclusive, or 90 days difference). The number of odd days in 90 is 6 ($90 = 12 \times 7 + 6$). Starting from Friday (Jan 1st), we move 6 days forward to reach Thursday (March 31st).