It was Sunday on 1 January 2006. What day of the week was 1 January 2010?

Introduction / Context:This problem tests your ability to track how weekdays shift from year to year, especially across leap years. In calendar aptitude, it is crucial to know how many days a year contributes modulo 7 to the day-of-week change from one year to the next.

Given Data / Assumptions:

• 1 January 2006 is a Sunday.
• We must find the weekday on 1 January 2010.
• We use the Gregorian calendar and its leap year rules.
• Years between 2006 and 2010 are 2006, 2007, 2008 and 2009.

Concept / Approach:Each non-leap year has 365 days, which is 52 weeks + 1 day, so it shifts the weekday by +1. A leap year has 366 days, which is 52 weeks + 2 days, shifting the weekday by +2. We count how many leap and non-leap years occur between the given years and add their contributions modulo 7 to determine the weekday shift from 2006 to 2010.

Step-by-Step Solution:Step 1: List the years from 2006 up to (but not including) 2010: 2006, 2007, 2008, 2009.Step 2: Identify leap years among them. 2008 is divisible by 4 and is not a century year, so it is a leap year. The others (2006, 2007, 2009) are non-leap years.Step 3: Each non-leap year contributes a +1 day shift. Each leap year contributes a +2 day shift.Step 4: Total shift = (3 non-leap years * 1 day) + (1 leap year * 2 days) = 3 + 2 = 5 days.Step 5: Starting day is Sunday on 1 January 2006. Moving forward 5 days: Sunday → Monday (1), Tuesday (2), Wednesday (3), Thursday (4), Friday (5).Step 6: Therefore, 1 January 2010 falls on a Friday.

Verification / Alternative check:A quick cross-check is to note that 4 years is about 1461 days (365 + 365 + 366 + 365). Since 1461 mod 7 = 5, the weekday must shift by 5 days, which is exactly what we computed. Starting from Sunday and moving 5 days forward again gives Friday.

Why Other Options Are Wrong:Sunday would imply no net shift, Monday a shift of 1 day, Tuesday a shift of 2 days, and Saturday a shift of 6 days. None of these match the correct 5-day shift caused by 3 non-leap years and 1 leap year in the interval.

Common Pitfalls:

• Forgetting to treat leap years differently from normal years.
• Accidentally including the year 2010 itself in the shift count.
• Miscounting the number of years or making arithmetic mistakes in adding the shifts.

Final Answer:The day of the week on 1 January 2010 was Friday.