Let's find the day of the week on 10 June 2011 by calculating the number of 'odd days'. Odd days are the extra days left after forming complete weeks from a given number of days. We typically use a reference point, and a common method is to calculate the total odd days from a known date (like the beginning of the Gregorian calendar) up to the required date.

Understanding Odd Days for Calendar Calculations

The concept of odd days is crucial for determining the day of the week. Here's how it works:

• A normal year has 365 days. $365 = 52 \times 7 + 1$. So, a normal year has 1 odd day.
• A leap year has 366 days. $366 = 52 \times 7 + 2$. So, a leap year has 2 odd days.
• The number of odd days in a certain number of days is the remainder when the total number of days is divided by 7.

Calculating Odd Days Up to 2010

We need to find the total number of odd days up to the end of the year 2010. We can break this down into centuries and then the remaining years.

• Odd Days in Centuries:
  • 100 years: 5 odd days
  • 200 years: 3 odd days
  • 300 years: 1 odd day
  • 400 years: 0 odd days (since 400 is a leap century)
  This pattern of odd days (5, 3, 1, 0) repeats every 400 years.
• Odd Days up to 2000 years: $2000 = 5 \times 400$. Since 400 years have 0 odd days, 2000 years also have $5 \times 0 = 0$ odd days.
• Odd Days from 2001 to 2010: This period includes 10 full years.
  • We need to count the number of leap years and ordinary years between 2001 and 2010.
  • Leap years in this period are 2004 and 2008 (years divisible by 4). There are 2 leap years.
  • Ordinary years are $10 - 2 = 8$ years.
  • Total odd days in these 10 years = (Number of ordinary years $\times$ 1) + (Number of leap years $\times$ 2)
  • Total odd days = $(8 \times 1) + (2 \times 2) = 8 + 4 = 12$ days.
  • Odd days modulo 7 = $12 \pmod{7} = 5$ odd days.
• Total Odd Days up to 2010: Odd days (up to 2000) + Odd days (2001-2010) = $0 + 5 = 5$ odd days.

Calculating Odd Days in 2011 up to 10 June

Now, let's calculate the odd days from the beginning of 2011 up to 10 June 2011. Note that 2011 is not a leap year.

• January: 31 days $\equiv 31 \pmod{7} = 3$ odd days
• February: 28 days (2011 is not a leap year) $\equiv 28 \pmod{7} = 0$ odd days
• March: 31 days $\equiv 31 \pmod{7} = 3$ odd days
• April: 30 days $\equiv 30 \pmod{7} = 2$ odd days
• May: 31 days $\equiv 31 \pmod{7} = 3$ odd days
• June: 10 days $\equiv 10 \pmod{7} = 3$ odd days

Total odd days in 2011 up to 10 June = $3 + 0 + 3 + 2 + 3 + 3 = 14$ odd days.

Odd days modulo 7 = $14 \pmod{7} = 0$ odd days.

Finding the Day of the Week

We sum the total odd days calculated:

Total odd days = (Odd days up to 2010) + (Odd days in 2011 up to June 10)

Total odd days = $5 + 0 = 5$ odd days.

We map the total number of odd days to the day of the week using the standard convention:

Since the total number of odd days is 5, the day of the week on 10 June 2011 was Friday.

Revision Table: Key Odd Day Values

Additional Information: Leap Year Rules

A year is a leap year if it is divisible by 4, unless it is a century year (divisible by 100). Century years are leap years only if they are also divisible by 400.

• Examples of leap years: 2000 (divisible by 400), 2004, 2008, 2012, 2016, etc.
• Examples of ordinary years: 2001, 2002, 2003, 2010, 2011, 1900 (century year, but not divisible by 400).

Understanding leap years is essential for accurate odd day calculations over longer periods.