The second day of a certain month falls on a Sunday.\nThe very next month has 31 days.\nWhat will be the day of the week on the last day of that next month?

Introduction / Context:
Calendar questions often require reasoning about how weekdays shift between months. In this problem, you are told that the 2nd day of one month is a Sunday and that the following month has 31 days. You are asked to find the weekday on the last day of that next month. The twist is that the length of the first month is not given explicitly.

Given Data / Assumptions:

Day 2 of Month 1 is Sunday, so Day 1 of Month 1 is Saturday.
Month 2 is the next month after Month 1 and has 31 days.
The exact length of Month 1 is not specified; it could be 28, 29, 30, or 31 days depending on which month it is.
We assume the standard Gregorian calendar month lengths.

Concept / Approach:
The weekday of the first day of Month 2 depends on how many days Month 1 has, because the weekday advances by (length of Month 1) mod 7. Once we know the weekday of Day 1 of Month 2, the weekday of its last day (Day 31) is 30 days later, so it shifts by 30 mod 7 = 2 days. If different valid lengths for Month 1 give different starting weekdays for Month 2, then the last day of Month 2 cannot be uniquely determined.

Step-by-Step Solution:
Step 1: Determine Day 1 of Month 1. If Day 2 is Sunday, then Day 1 is Saturday. Step 2: Consider possible lengths of Month 1. Month 1 could have 28, 29, 30, or 31 days (examples: February, April, June, August, etc.). Step 3: Compute weekday of Day 1 of Month 2 for each case. If Month 1 has 28 days: shift = 28 mod 7 = 0, so Month 2 starts on Saturday. If Month 1 has 29 days: shift = 1, so Month 2 starts on Sunday. If Month 1 has 30 days: shift = 2, so Month 2 starts on Monday. If Month 1 has 31 days: shift = 3, so Month 2 starts on Tuesday. Step 4: Find the last day of Month 2 in each case. Month 2 has 31 days, so Day 31 is 30 days after Day 1, i.e., weekday shift = 30 mod 7 = 2 days. Starting Saturday → last day Monday; starting Sunday → last day Tuesday; starting Monday → last day Wednesday; starting Tuesday → last day Thursday.

Verification / Alternative check:
You can match these possibilities to actual month pairs (e.g., February–March, April–May) and see that, depending on which month pair you choose, the last day of the 31-day month can be different weekdays. Therefore, more than one answer is consistent with the given information, confirming that the weekday is not uniquely determined.

Why Other Options Are Wrong:
Choosing Friday, Saturday, or Sunday assumes a unique specific weekday, but we have shown that the last day could be Monday, Tuesday, Wednesday, or Thursday depending on the length of Month 1. None of the fixed weekday options can be guaranteed.

Common Pitfalls:
A common mistake is to assume a particular month (for example, March or January) without being told, and then solve using that single assumption. Another error is to forget that 29-day February can occur in leap years, adding another case. Always check whether the data uniquely determine the answer; if different plausible scenarios give different results, the correct conclusion is that it cannot be determined uniquely.

Final Answer:
The weekday of the last day of the next month cannot be fixed; it is not uniquely determined from the given information.