Calculating the Day of the Week: March 24, 2011 to June 24, 2012

This problem requires us to calculate the number of days between two dates and use the concept of odd days to find the day of the week on the target date.

Understanding Odd Days

An odd day is the number of days remaining after dividing the total number of days by 7 (the number of days in a week). Each odd day shifts the day of the week forward by one day.

• A normal year has 365 days = 52 weeks and 1 day. So, a normal year has 1 odd day.
• A leap year has 366 days = 52 weeks and 2 days. So, a leap year has 2 odd days.

Step-by-Step Solution

Step 1: Calculate the day on March 24, 2012

We are given that March 24, 2011 was a Thursday. We need to find the day on March 24, 2012. This is a period of exactly one year.

We need to determine if the year 2012 is a leap year. A year is a leap year if it is divisible by 4, unless it is a century year not divisible by 400.

• 2012 is divisible by 4 ($2012 \div 4 = 503$).
• Therefore, 2012 is a leap year.

The period from March 24, 2011, to March 24, 2012, includes February 29, 2012 (the leap day). Thus, this period is 366 days long.

Number of odd days in this period = Total days $\pmod{7}$

\begin{equation*} \text{Odd days} = 366 \pmod{7} \end{equation*}

\begin{equation*} 366 = 52 \times 7 + 2 \end{equation*}

So, there are 2 odd days.

Since March 24, 2011 was a Thursday, adding 2 odd days gives the day on March 24, 2012:

Thursday + 2 days = Saturday.

So, March 24, 2012 was a Saturday.

Step 2: Calculate the number of days from March 24, 2012 to June 24, 2012

Now we need to find the number of days between March 24, 2012, and June 24, 2012.

Total number of days from March 24, 2012, to June 24, 2012 is the sum of the days remaining in March (from 25th onwards), plus the full days in April and May, plus the days in June up to the 24th.

Total days = Days remaining in March + Days in April + Days in May + Days in June

\begin{equation*} \text{Total days} = 7 + 30 + 31 + 24 \end{equation*}

\begin{equation*} \text{Total days} = 92 \end{equation*}

Now, we calculate the number of odd days in these 92 days:

\begin{equation*} \text{Odd days} = 92 \pmod{7} \end{equation*}

\begin{equation*} 92 = 13 \times 7 + 1 \end{equation*}

So, there is 1 odd day in the period from March 24, 2012, to June 24, 2012.

Step 3: Find the Day on June 24, 2012

We know that March 24, 2012 was a Saturday.

To find the day on June 24, 2012, we add the number of odd days calculated in Step 2 to the day on March 24, 2012.

Day on June 24, 2012 = Day on March 24, 2012 + Odd days

Day on June 24, 2012 = Saturday + 1 day

Day on June 24, 2012 = Sunday.

Therefore, June 24, 2012, was a Sunday.

Revision Table: Key Date Calculation Concepts

Additional Information: Calendar and Odd Days

The concept of odd days is a fundamental tool in calendar-based reasoning problems. It simplifies calculating the day of the week over long periods.

Understanding leap years is crucial because they add an extra day (February 29th), which changes the number of odd days in a year from 1 to 2. This extra day occurs in years divisible by 4, to keep the calendar year synchronized with the astronomical year.

When calculating days between dates spanning across different months, it's important to be careful about the number of days in each month (30, 31, 28/29). Listing the days month by month, as shown in Step 2, helps avoid errors.

Remember that each odd day shifts the day of the week forward. For example, if you have 3 odd days, the day shifts forward by 3 days (e.g., Monday becomes Thursday).