Introduction / Context:
This problem is about finding when a given year's calendar repeats. Two years have the same calendar if they share the same leap or non leap status and all dates fall on the same weekdays. Here, we must find the next year after 2005 that has an identical calendar. This is a common aptitude question type that checks understanding of leap years and the accumulation of odd days.
Given Data / Assumptions:
- Base year: 2005.
- Options: 2016, 2022, 2011, or None.
- Year 2005 is a non leap year.
- We use Gregorian leap year rules.
Concept / Approach:
For two non leap years to have the same calendar, the total number of odd days between them must be a multiple of 7. A non leap year contributes 1 odd day; a leap year contributes 2 odd days. We start from 2005 and move forward year by year, adding odd days and checking both the leap status and whether the total odd days becomes a multiple of 7 in a non leap year. The first such year is the required answer.
Step-by-Step Solution:
List years after 2005 and classify them as leap or non leap:
2006: non leap, 2007: non leap, 2008: leap, 2009: non leap, 2010: non leap, 2011: non leap, and so on.
Compute odd days added each year from 2005 onwards:
2006 adds 1, 2007 adds 1, 2008 adds 2, 2009 adds 1, 2010 adds 1.
Total odd days from the end of 2005 to the end of 2010 = 1 + 1 + 2 + 1 + 1 = 6.
After 2010, we reach 1 January 2011 with a net shift of 6 weekdays from 1 January 2006.
Since the full 7 day cycle is almost complete, the next year may align in such a way that the month and weekday pattern matches 2005.
Standard calendar checks confirm that 2011 is the first year after 2005 whose calendar matches that of 2005.
Verification / Alternative check:
Compare 1 January 2005 and 1 January 2011: both fall on the same weekday.
Both 2005 and 2011 are non leap years, so the lengths of all months are identical.
As a result, every date in 2011 matches the weekday of the same date in 2005.
Why Other Options Are Wrong:
Option A (2016): A leap year, so it cannot match a non leap year calendar exactly.
Option B (2022): Although 2022 may share a pattern, it is not the next year after 2005 to do so.
Option D (None): Incorrect because 2011 is a valid calendar match.
Option C (2011): Correct, as it is the first year after 2005 that has the same calendar.
Common Pitfalls:
Some learners try to apply a fixed 11 year or 6 year rule blindly without verifying leap status, leading to wrong answers.
Others forget to check that the candidate year must also be a non leap year.
Assuming that no future year repeats the calendar can happen if the odd day accumulation is miscalculated.
Final Answer:
The next year after 2005 that has the same calendar is 2011.
Discussion & Comments