Finding the Day of the Week for 29 June 2010

Let's determine the day of the week for the specific date, 29 June 2010. To do this, we use the concept of 'odd days' in a calendar. Odd days are the number of days remaining after dividing the total number of days by 7 (the number of days in a week).

The calculation involves finding the total number of odd days from a reference point (usually the beginning of the Christian era, Year 0 or Year 1) up to the day before the given date, and then adding the odd days for the days in the current year up to the given date. The sum of odd days is then divided by 7 to find the final odd day count, which corresponds to a specific day of the week.

Calculating Odd Days up to 2009

First, we calculate the number of odd days for the years completed before 29 June 2010, which is up to the end of the year 2009.

• Odd days in 100 years = 5
• Odd days in 200 years = \(2 \times 5 = 10 \equiv 3 \pmod{7}\)
• Odd days in 300 years = \(3 \times 5 = 15 \equiv 1 \pmod{7}\)
• Odd days in 400 years = Odd days in 300 years + Odd days in 100 years + 1 (for the leap year century) = \(1 + 5 + 1 = 7 \equiv 0 \pmod{7}\)

Since the number of odd days in 400 years is 0, the number of odd days in any multiple of 400 years is also 0. This applies to the year 2000, which is a multiple of 400.

• Odd days in 2000 years = 0

Now, let's consider the remaining years from 2001 to 2009. This is a period of 9 years.

We need to identify the number of leap years and ordinary years in this period.

• A year is a leap year if it is divisible by 4, except for century years which must be divisible by 400.
• In the years 2001 to 2009, the leap years are 2004 and 2008. There are 2 leap years.
• The number of ordinary years is \(9 - 2 = 7\) ordinary years.

An ordinary year has 365 days, which is \(52 \times 7 + 1\) day. So, an ordinary year has 1 odd day.

A leap year has 366 days, which is \(52 \times 7 + 2\) days. So, a leap year has 2 odd days.

• Total odd days from 7 ordinary years = \(7 \times 1 = 7\) odd days.
• Total odd days from 2 leap years = \(2 \times 2 = 4\) odd days.
• Total odd days from 2001 to 2009 = \(7 + 4 = 11\) odd days.
• Number of odd days \(11 \pmod{7} = 4\).

Total odd days up to the end of 2009 = Odd days in 2000 years + Odd days in 2001-2009

Total odd days up to 2009 = \(0 + 4 = 4\) odd days.

Calculating Odd Days in 2010 up to 29 June

Now, we calculate the odd days for the months in the year 2010 up to 29 June. The year 2010 is not divisible by 4, so it is an ordinary year.

• January (31 days): \(31 \pmod{7} = 3\) odd days.
• February (28 days in an ordinary year): \(28 \pmod{7} = 0\) odd days.
• March (31 days): \(31 \pmod{7} = 3\) odd days.
• April (30 days): \(30 \pmod{7} = 2\) odd days.
• May (31 days): \(31 \pmod{7} = 3\) odd days.
• June (up to 29 days): \(29 \pmod{7} = 1\) odd day.

Total odd days from January 1, 2010, to June 29, 2010:

\(3 + 0 + 3 + 2 + 3 + 1 = 12\) odd days.

Number of odd days \(12 \pmod{7} = 5\).

Total Odd Days and Determining the Day

Total odd days from the beginning of the calendar up to 29 June 2010 = Odd days up to end of 2009 + Odd days in 2010 up to 29 June.

Total odd days = \(4 + 5 = 9\) odd days.

The final number of odd days is the remainder when the total is divided by 7:

Final odd days = \(9 \pmod{7} = 2\).

We map this final odd day count to the day of the week based on a standard convention (often starting with Sunday as 0 or Monday as 1):

Since the final odd day count is 2, the day of the week for 29 June 2010 is Tuesday.

Conclusion

By calculating the total number of odd days up to the given date, we determined that 29 June 2010 was a Tuesday.

Revision Table - Day of Week Calculation

Alternatively, summing up the intermediate modulo 7 odd days: \(4 + (3+0+3+2+3+1) = 4 + 12 = 16\). \(16 \pmod{7} = 2\).

Additional Information - Calendar Concepts

Understanding how to calculate the day of the week for any given date relies on a few key calendar concepts:

• Ordinary Year: A year with 365 days. It has one odd day (\(365 \div 7 = 52\) weeks and 1 day).
• Leap Year: A year with 366 days. It has two odd days (\(366 \div 7 = 52\) weeks and 2 days). Leap years occur every 4 years, except for years divisible by 100 but not by 400.
• Odd Days: The extra days left after forming complete weeks from a given number of days. The number of odd days is the remainder when the total number of days is divided by 7.
• Reference Point: Calculations often use the beginning of the calendar era (e.g., 00/00/0000 or 01/01/0001) as a starting point with 0 odd days. The odd day count accumulated up to a specific date determines the day of the week. Different calculation methods might use different reference points or base odd day counts for centuries. The method used above establishes the pattern of odd days for centuries (100 yrs = 5, 200 yrs = 3, 300 yrs = 1, 400 yrs = 0).

The final odd day count (0 to 6) is mapped to the days of the week, typically starting with Sunday=0 or Monday=1, depending on the specific method or table used. The method used here corresponds to Sunday=0, Monday=1, Tuesday=2, and so on.