If 1st January 2013 was a Tuesday, then on which day of the week did 31st December 2013 fall in the same year?

Introduction / Context:
This problem asks you to relate the first day of a year to the last day of the same year. It is a classic calendar question used in exams to test understanding of how many days there are in an ordinary year and how that affects the weekday pattern between 1st January and 31st December.

Given Data / Assumptions:

• 1st January 2013 was a Tuesday.
• We are asked to find the day of the week on 31st December 2013.
• 2013 is a non-leap (ordinary) year, with 365 days.
• We use the standard 7-day week structure.

Concept / Approach:
The key idea is to understand how the number of days in a year shifts the weekday. A non-leap year has 365 days, which equals 52 weeks plus 1 extra day (because 365 = 52 * 7 + 1). This means that from 1st January of an ordinary year to 1st January of the next year, the day of the week moves one step forward. Between 1st January and 31st December of the same year, we span 364 days, which is exactly 52 weeks, so the weekdays match.

Step-by-Step Solution:
Step 1: Note that 2013 is not divisible by 4, so it is not a leap year and has 365 days. Step 2: From 1st January 2013 to 31st December 2013, the number of days elapsed is 364 (since 31st December is the 365th day, there are 364 days between these two dates). Step 3: Express 364 in terms of weeks: 364 = 52 * 7, which is exactly 52 complete weeks. Step 4: Because the number of days between the two dates is a multiple of 7, there is no net shift in the day of the week. Step 5: Therefore, 31st December 2013 falls on the same day of the week as 1st January 2013. Step 6: Since 1st January 2013 was a Tuesday, 31st December 2013 is also a Tuesday.

Verification / Alternative check:
You can verify this logic by thinking about the structure of a non-leap year: 52 weeks account for 364 days, which bring you to the same weekday, and the remaining one day pushes 1st January of the next year forward by one weekday. As a result, 1st January 2014 would be Wednesday, but within 2013 the 1st and 31st of the year align on the same weekday. A calendar or any date-checking tool confirms that 31st December 2013 is indeed a Tuesday.

Why Other Options Are Wrong:
Sunday: This would require a shift of five days backward from the correct weekday, not supported by the total-day calculation.
Monday: This implies a shift of one day backward, which contradicts the exact multiple-of-7 structure between 1st January and 31st December in an ordinary year.
Wednesday: This would be one day ahead of Tuesday and would occur if the difference in days was 365 instead of 364, which is not the case here.
Thursday: This would be two days ahead of Tuesday, which again does not fit the actual number of days between the two dates.

Common Pitfalls:
A common mistake is to assume that because the year has 365 days, the last day must be the next weekday after the first day; however, that reasoning applies to the next year's 1st January, not to 31st December of the same year. Another error is miscounting the days between the start and end dates, or forgetting that the day count between them is 364, an exact multiple of 7. Always distinguish clearly between the interval within the year and the jump to the next year.

Final Answer:
Thus, 31st December 2013 fell on a Tuesday.