Solving Calendar Day Problems

This problem asks us to find the day of the week for a specific date in 2023, given the day of the week for a date in 2015. To solve this, we use the concept of "odd days".

An odd day is the number of days remaining after dividing the total number of days in a period by 7. Since there are 7 days in a week, the day of the week repeats every 7 days. Therefore, adding or subtracting a number of odd days tells us how many days forward or backward in the week we need to count.

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

Step 1: Calculate Odd Days from 28 May 2015 to 28 May 2023

We need to find the total number of odd days over the years from 28 May 2015 to 28 May 2023. This period spans exactly 8 years (2023 - 2015 = 8).

Let's identify the leap years within this period. A year is a leap year if it is divisible by 4, except for century years not divisible by 400. For the period between two specific dates, a leap year contributes an extra day if February 29th falls within that period.

The years are 2016, 2017, 2018, 2019, 2020, 2021, 2022, 2023.

• 2016 is divisible by 4, and Feb 29, 2016 is between May 28, 2015 and May 28, 2023. So, 2016 is a leap year in this context.
• 2020 is divisible by 4, and Feb 29, 2020 is between May 28, 2015 and May 28, 2023. So, 2020 is a leap year in this context.

The other years (2017, 2018, 2019, 2021, 2022, 2023) are normal years in this context.

Number of normal years = 8 total years - 2 leap years = 6 normal years.

Total odd days from 28 May 2015 to 28 May 2023:

\(\text{Odd days} = (\text{Number of normal years} \times 1) + (\text{Number of leap years} \times 2)\)

\(\text{Odd days} = (6 \times 1) + (2 \times 2) = 6 + 4 = 10\)

To find the net odd days, we find the remainder when 10 is divided by 7:

\(\text{Net odd days} = 10 \pmod{7} = 3\)

So, 28 May 2023 will be 3 days after 28 May 2015. Given 28 May 2015 was Thursday:

Thursday + 1 day = Friday

Thursday + 2 days = Saturday

Thursday + 3 days = Sunday

So, 28 May 2023 is a Sunday.

Step 2: Calculate Odd Days from 28 May 2023 to 24 April 2023

Now we need to find the day on 24 April 2023, given the day on 28 May 2023. We are moving backward in the calendar within the same year.

Let's count the number of days from 24 April 2023 up to 28 May 2023 (going forward first, then adjust):

• Number of days in April from 24th to 30th: \(30 - 24 = 6\) days
• Number of days in May from 1st to 28th: \(28\) days

Total number of days from 24 April to 28 May 2023 = \(6 + 28 = 34\) days.

Calculate the odd days for these 34 days:

\(\text{Odd days} = 34 \pmod{7}\)

\(34 = 4 \times 7 + 6\)

So, there are 6 odd days between 24 April and 28 May 2023.

Since we are going backward from 28 May 2023 (Sunday) to 24 April 2023, we need to subtract these 6 odd days from Sunday. Subtracting 6 days is the same as adding \(7 - 6 = 1\) day.

Starting from Sunday on 28 May 2023:

Sunday - 1 day = Saturday

Sunday - 2 days = Friday

Sunday - 3 days = Thursday

Sunday - 4 days = Wednesday

Sunday - 5 days = Tuesday

Sunday - 6 days = Monday

Alternatively, adding 1 day to Sunday: Sunday + 1 day = Monday.

So, 24 April 2023 is a Monday.

Summary of Calculations

Starting day: 28 May 2015 is Thursday.

Day on 28 May 2023 = Thursday + 3 days = Sunday.

Day on 24 April 2023 = Sunday - 6 days = Monday.

Therefore, the day of the week on 24 April 2023 will be Monday.

Revision Table: Key Concepts for Calendar Problems

Additional Information on Calendar Day Calculation

Understanding how to calculate odd days is fundamental to solving calendar-based reasoning problems. This method allows you to determine the day of the week for any given date, provided you have a reference date.

• Counting Odd Days Month-wise: The number of odd days in a month depends on the number of days in that month:
  • 31-day months (Jan, Mar, May, Jul, Aug, Oct, Dec): \(31 \pmod{7} = 3\) odd days.
  • 30-day months (Apr, Jun, Sep, Nov): \(30 \pmod{7} = 2\) odd days.
  • February: 28 days (\(28 \pmod{7} = 0\) odd days) in a normal year; 29 days (\(29 \pmod{7} = 1\) odd day) in a leap year.
• Moving Backwards: When calculating days backward in time, subtract the net odd days or add the equivalent positive odd days (\(7 - \text{net odd days}\)).
• Leap Year Check: A year is a leap year if it is divisible by 4, unless it is a century year (like 1900, 2000) in which case it must be divisible by 400 to be a leap year. The year 2000 was a leap year (divisible by 400), but 1900 was not (divisible by 4 but not 400).

Practice with different date ranges and starting days will help you become proficient in these calculations.