In a certain year, the first day of a month is a Tuesday and the last day of that same month is a Monday. Which month can it be?

Introduction / Context:
This is a calendar reasoning question. You are told the day of the week for the first and last days of a month, and must identify which month has that pattern. Such questions test your understanding of how days of the week cycle through a month depending on how many days the month contains.

Given Data / Assumptions:
- The first day of an unnamed month is a Tuesday.
- The last day of that same month is a Monday.
- We are to choose among the months January, February, March and August.
- Unless otherwise stated, we usually assume a non leap year when reasoning about February.

Concept / Approach:
Days of the week repeat every 7 days. If a month has N days, and the first day is a given weekday, then the last day is the weekday that is N − 1 days after the first day (because the count starts at day 1). In modulo 7 arithmetic, we can express the day difference between the first and last days as (N − 1) mod 7. The question tells us that the first day is Tuesday and the last day is Monday, so the net shift in weekdays is from Tuesday to Monday, which is effectively a shift of minus one day, or equivalently plus six days modulo 7.

Step-by-Step Solution:
Step 1: Let N be the number of days in the month. Step 2: The first day of the month is Tuesday. After N − 1 days, we reach the last day. The weekday shift is (N − 1) mod 7. Step 3: The last day is Monday, which is one day before Tuesday. That means the overall shift from first to last day is −1 day, which is the same as +6 days modulo 7. Step 4: Thus we require (N − 1) ≡ 6 (mod 7), or equivalently N ≡ 0 (mod 7). This means that the month must have a number of days that is a multiple of 7. Step 5: Among the months listed, January has 31 days, March has 31 days, and August has 31 days. None of these is a multiple of 7. February in a non leap year has 28 days, and 28 is a multiple of 7 (28 = 4 × 7). Step 6: Therefore, only February with 28 days can satisfy the condition that N is a multiple of 7 and hence produce a shift from Tuesday to Monday between the first and last days.

Verification / Alternative check:
We can verify by direct counting using February as an example. If February 1 is a Tuesday and February has 28 days, then February 8 is also a Tuesday, February 15 is a Tuesday, and February 22 is a Tuesday. Then February 28 falls on a Monday. This matches the pattern described in the question. For a 31 day month, starting on Tuesday, the last day would be four days later (because 30 mod 7 is 2, so the net shift is +2), leading to a Thursday rather than a Monday, which contradicts the problem statement.

Why Other Options Are Wrong:
- January (31 days) cannot produce a last day that is Monday when the first day is Tuesday, because the shift from day 1 to day 31 is 30 days, which is equivalent to +2 weekdays, not −1.
- March, also with 31 days, fails for the same reason as January.
- August, again with 31 days, similarly does not result in a Monday on the last day if the first day is Tuesday.

Common Pitfalls:
Some students attempt to test each option by intuitively cycling through weekdays without using modular arithmetic, which can be error prone. Others forget that the shift is determined by N − 1 days rather than N days. Applying the simple condition N ≡ 0 (mod 7) keeps the reasoning clean and avoids off by one mistakes.

Final Answer:
Only February, with 28 days in a non leap year, can have its first day on a Tuesday and its last day on a Monday. Therefore the correct answer is February.