It was Sunday on Jan. 1, 2006. What was the day of the week on Jan.1, 2010?
Friday
Understanding the Calendar Problem
This problem requires calculating the day of the week for a future date (January 1, 2010) given a starting date (January 1, 2006) and its corresponding day (Sunday). The calculation involves counting the total number of days between these dates, paying special attention to leap years, and determining the shift in the day of the week.
Step-by-Step Date Calculation
We are given that January 1, 2006, was a Sunday. We need to determine the day of the week for January 1, 2010.
1. Counting the Years and Identifying Leap Years
The time period spans from January 1, 2006, to January 1, 2010. The full years that fall within this period are 2006, 2007, 2008, and 2009. To calculate the day shift, we need to know how many days are in each of these years, specifically identifying any leap years.
A leap year occurs every 4 years (e.g., 2004, 2008, 2012). A year is a leap year if it is divisible by 4, unless it is divisible by 100 but not by 400.
- 2006: Not divisible by 4. It is a common year with 365 days.
- 2007: Not divisible by 4. It is a common year with 365 days.
- 2008: Divisible by 4 (2008 / 4 = 502). It is a leap year with 366 days.
- 2009: Not divisible by 4. It is a common year with 365 days.
2. Calculating Odd Days for Each Year
The number of 'odd days' helps us find the change in the day of the week. An odd day is the remainder when the total number of days in a period is divided by 7.
- For a common year (365 days): The number of odd days is $365 \pmod{7} = 1$.
- For a leap year (366 days): The number of odd days is $366 \pmod{7} = 2$.
Let's calculate the odd days for each year from Jan 1, 2006, to Jan 1, 2010:
| Year | Type of Year | Number of Days | Odd Days ($ \text{Days} \pmod{7} $) |
|---|---|---|---|
| 2006 | Common | 365 | $365 \pmod{7} = 1$ |
| 2007 | Common | 365 | $365 \pmod{7} = 1$ |
| 2008 | Leap | 366 | $366 \pmod{7} = 2$ |
| 2009 | Common | 365 | $365 \pmod{7} = 1$ |
3. Summing the Total Odd Days
To find the total shift in the day of the week from January 1, 2006, to January 1, 2010, we add up the odd days from each year in the period.
Total Odd Days = (Odd days in 2006) + (Odd days in 2007) + (Odd days in 2008) + (Odd days in 2009)
Total Odd Days = $1 + 1 + 2 + 1 = 5$
4. Determining the Day of the Week on Jan 1, 2010
The starting date, January 1, 2006, was a Sunday. We need to advance the day of the week by the total number of odd days calculated (which is 5).
Starting Day: Sunday
Advance by: 5 days
Counting forward 5 days from Sunday:
- Sunday + 1 day = Monday
- Sunday + 2 days = Tuesday
- Sunday + 3 days = Wednesday
- Sunday + 4 days = Thursday
- Sunday + 5 days = Friday
Therefore, January 1, 2010, was a Friday.
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