Solving Calendar Problems: Finding the Day of the Week

This problem asks us to find the day of the week for a specific date in the future, given the day of the week for an earlier date. The key to solving such calendar problems is to calculate the total number of "odd days" between the two dates. An odd day is the remainder left after dividing the total number of days by 7 (since there are 7 days in a week).

We are given that 21 May 2009 is a Thursday. We need to find the day on 15 July 2017.

Step 1: Calculate Odd Days from 21 May 2009 to 21 May 2017

First, let's find the number of odd days over the complete years from 21 May 2009 to 21 May 2017. This is a period of exactly 8 years (2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017). We need to identify the leap years within this period.

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

Leap years between 2009 and 2017 (inclusive of the period covered):

• 2012 (Divisible by 4)
• 2016 (Divisible by 4)

There are 2 leap years and 8 - 2 = 6 ordinary years in this period.

Total odd days from 21 May 2009 to 21 May 2017:

• Odd days from ordinary years = $6 \times 1 = 6$
• Odd days from leap years = $2 \times 2 = 4$
• Total odd days = $6 + 4 = 10$

To find the net effect on the day of the week, we find the remainder when 10 is divided by 7:

Odd days = $10 \pmod{7} = 3$

So, 21 May 2017 will be 3 days after Thursday.

Thursday + 3 days = Friday, Saturday, Sunday.

Therefore, 21 May 2017 is a Sunday.

Step 2: Calculate Odd Days from 21 May 2017 to 15 July 2017

Now we need to calculate the number of odd days from 21 May 2017 to 15 July 2017.

• Remaining days in May 2017: May has 31 days. Remaining days = $31 - 21 = 10$ days.
• Days in June 2017: June has 30 days.
• Days in July 2017: We need to count up to 15 July, so 15 days.

Let's find the odd days for each month:

• Odd days in remaining May: $10 \pmod{7} = 3$
• Odd days in June: $30 \pmod{7} = 2$
• Odd days in July (up to 15th): $15 \pmod{7} = 1$

Total odd days from 21 May 2017 to 15 July 2017 = $3 + 2 + 1 = 6$ days.

Step 3: Determine the Final Day of the Week

We found that 21 May 2017 is a Sunday. We need to add the odd days calculated in Step 2 to this day.

Day on 15 July 2017 = Day on 21 May 2017 + Total odd days from May to July

Day on 15 July 2017 = Sunday + 6 days

Counting 6 days after Sunday: Monday, Tuesday, Wednesday, Thursday, Friday, Saturday.

So, 15 July 2017 is a Saturday.

Summary of Odd Days Calculation

Alternative calculation: Add total odd days from both steps:

Total odd days from 21 May 2009 to 15 July 2017 = Odd days (years) + Odd days (months)

Total odd days = $3 + 6 = 9$

Net odd days = $9 \pmod{7} = 2$

Starting day was Thursday. Add 2 days:

Thursday + 2 days = Friday, Saturday.

So, 15 July 2017 is a Saturday.

Revision Table: Key Calendar Concepts

Additional Information on Calendar Calculations

Calendar problems often involve calculating the number of days between two dates and then finding the corresponding day of the week. Here are some helpful points:

• Days in months:
  • 31 days: January, March, May, July, August, October, December
  • 30 days: April, June, September, November
  • 28/29 days: February (29 in a leap year, 28 in an ordinary year)
• Odd days per month:
  • 31 days: $31 \pmod{7} = 3$ odd days
  • 30 days: $30 \pmod{7} = 2$ odd days
  • 28 days: $28 \pmod{7} = 0$ odd days
  • 29 days: $29 \pmod{7} = 1$ odd day
• The cycle of days repeats every 7 days. Adding 7 odd days (or any multiple of 7) brings you back to the same day of the week.
• When calculating odd days across years, be careful to count the leap years correctly. Leap years occur in years divisible by 4, unless it's a century year not divisible by 400 (e.g., 1900 was not a leap year, but 2000 was). In this problem, 2012 and 2016 are leap years.