Understanding Calendar Day Calculations

This problem requires us to determine the day of the week for a specific date in the future, given the day of the week for another date in the past. We can solve this using the concept of 'odd days'. Odd days are the number of days remaining after dividing the total number of days by 7.

Key Concepts: Odd Days

• An ordinary year has 365 days = 52 weeks and 1 day. So, an ordinary year has 1 odd day.
• A leap year has 366 days = 52 weeks and 2 days. So, a leap year has 2 odd days.
• To find the odd days for a period, calculate the total number of days in that period and then find the remainder when divided by 7.

Steps to Solve the Calendar Problem

We need to find the number of odd days between 19 July 2000 and 15 June 2012. We can break this period into two parts:

1. From 19 July 2000 to 19 July 2011 (covering full years).
2. From 19 July 2011 to 15 June 2012 (covering months within a year).

Part 1: Odd Days from 19 July 2000 to 19 July 2011

This period covers exactly 11 years. We need to identify the leap years within this period. When counting full year intervals starting from a date (like July 19th), a leap year contributes 2 odd days if February 29th falls within that 1-year interval. The years from 2000 to 2011 are:

• 2000 (starts July 19th) to 2001 (July 19th): This interval contains Feb 2001 (ordinary year). Odd days = 1.
• 2001 (July 19th) to 2002 (July 19th): Ordinary year. Odd days = 1.
• 2002 (July 19th) to 2003 (July 19th): Ordinary year. Odd days = 1.
• 2003 (July 19th) to 2004 (July 19th): This interval contains Feb 2004 (leap year). Odd days = 2.
• 2004 (July 19th) to 2005 (July 19th): Ordinary year. Odd days = 1.
• 2005 (July 19th) to 2006 (July 19th): Ordinary year. Odd days = 1.
• 2006 (July 19th) to 2007 (July 19th): Ordinary year. Odd days = 1.
• 2007 (July 19th) to 2008 (July 19th): This interval contains Feb 2008 (leap year). Odd days = 2.
• 2008 (July 19th) to 2009 (July 19th): Ordinary year. Odd days = 1.
• 2009 (July 19th) to 2010 (July 19th): Ordinary year. Odd days = 1.
• 2010 (July 19th) to 2011 (July 19th): Ordinary year. Odd days = 1.

Total odd days from 19 July 2000 to 19 July 2011 = $1 + 1 + 1 + 2 + 1 + 1 + 1 + 2 + 1 + 1 + 1 = 13$ odd days.

Odd days modulo 7 = $13 \div 7$. Remainder is $6$.

So, there are 6 odd days from 19 July 2000 to 19 July 2011.

Part 2: Odd Days from 19 July 2011 to 15 June 2012

We need to count the number of days from 19 July 2011 up to 15 June 2012. Note that 2012 is a leap year, so February 2012 will have 29 days.

Total number of days from 19 July 2011 to 15 June 2012:

$12 (Jul 2011) + 31 (Aug) + 30 (Sep) + 31 (Oct) + 30 (Nov) + 31 (Dec) + 31 (Jan 2012) + 29 (Feb) + 31 (Mar) + 30 (Apr) + 31 (May) + 15 (Jun) = 332$ days.

Total odd days for this period = $332 \div 7$.

$332 = 47 \times 7 + 3$. The remainder is $3$.

So, there are 3 odd days from 19 July 2011 to 15 June 2012.

Total Odd Days

Total odd days from 19 July 2000 to 15 June 2012 = Odd days (Part 1) + Odd days (Part 2)

Total odd days = $6 + 3 = 9$ odd days.

To find the net change in the day of the week, we take the total odd days modulo 7:

Net odd days = $9 \div 7$. Remainder is $2$.

So, we need to move forward by 2 days from the starting day.

Determining the Final Day

The day on 19 July 2000 was Wednesday.

Moving forward by 2 days from Wednesday:

• Wednesday + 1 day = Thursday
• Thursday + 1 day = Friday

Therefore, the day of the week on 15 June 2012 would be Friday.

Revision Table: Calendar Odd Days

Starting Day: Wednesday
Net change: +2 days
Final Day: Wednesday + 2 = Friday

Additional Information: Leap Year Rules

Understanding leap years is crucial for calendar problems. Here are the rules:

• A year is a leap year if it is divisible by 4, except for centennial years.
• Centennial years (years divisible by 100) are only leap years if they are also divisible by 400.

Examples:

• 1900: Divisible by 100 but not by 400. Not a leap year.
• 2000: Divisible by 100 and by 400. Is a leap year.
• 2012: Divisible by 4 and not a centennial year. Is a leap year.
• 2011: Not divisible by 4. Not a leap year.

In our calculation from 19 July 2000 to 19 July 2011, we counted leap years based on whether the interval crossed February 29th of a leap year. The interval from 19 Jul 2003 to 19 Jul 2004 crosses Feb 2004 (which is a leap year). The interval from 19 Jul 2007 to 19 Jul 2008 crosses Feb 2008 (which is a leap year). The interval from 19 Jul 2000 to 19 Jul 2001 crosses Feb 2001 (not a leap year). The interval from 19 Jul 2011 to 15 June 2012 includes Feb 2012, and 2012 is a leap year, hence Feb has 29 days, contributing to the total day count.