The chairman of a large company visits exactly one department on Monday of every week, except that he does not visit any department on the Monday of the third week of every month. When did he visit the Purchase department? Consider the following statements: I. He visited the Accounts department in the second week of September, after having visited the Purchase department on an earlier occasion. II. He visited the Purchase department immediately after visiting the Stores department but before visiting the Accounts department.

Introduction / Context:
This data sufficiency question involves a recurring schedule of visits by a company chairman to various departments. The chairman visits exactly one department each Monday except on the Monday of the third week of every month. The question asks when he visited the Purchase department. Two statements refer to visits to Accounts, Stores, and Purchase, but do not explicitly fix calendar dates beyond a reference to September. We must decide whether these statements uniquely determine the week or date of the Purchase visit.

Given Data / Assumptions:

• The chairman visits one department on each Monday of every week, except that he skips the Monday of the third week of every month.
• Statement I: He visited the Accounts department in the second week of September after having visited the Purchase department on an earlier occasion.
• Statement II: He visited the Purchase department immediately after visiting the Stores department but before visiting the Accounts department.
• Departments can be revisited in different weeks or months; nothing states that each department is visited exactly once.
• Earlier occasion refers to some previous Monday visit, but not necessarily the immediately preceding Monday unless so stated.

Concept / Approach:
We interpret the recurring Monday visits as a sequence of time slots. Each Monday that is not the third Monday of a month has exactly one department visited. The question is whether the references to Accounts in the second week of September and the order involving Stores and Purchase are enough to pin down the exact week number or calendar date for the Purchase visit. If multiple Monday schedules fit the statements, then the data are not sufficient.

Step-by-Step Solution:
Step 1: From statement I, we know that the chairman visited the Accounts department on the Monday of the second week of September. We also know that the Purchase department was visited on an earlier occasion, but this earlier occasion could be the Monday of the first week of September, or any suitable Monday in previous months. Step 2: Statement I alone does not specify whether that earlier occasion was the immediately previous Monday or some Monday many weeks earlier. Therefore, statement I alone cannot fix the date or even the exact week number of the Purchase visit. Step 3: From statement II, we know that in the sequence of visits there is a pattern Stores then Purchase then Accounts, with Purchase immediately after Stores and before Accounts. There is no information about which month or week this sequence occurs, nor whether Accounts in this sequence is the same Accounts visit mentioned in September in statement I. Step 4: Combining statements I and II, we might naturally guess that the Accounts visit in the order Stores, Purchase, Accounts is the same Accounts visit in the second week of September. However, even if we assume this, the data still do not force the exact week of the Purchase visit. Step 5: Suppose the Accounts visit in the second week of September is the one referred to in statement II. Then the Purchase visit occurred at some earlier Monday visit between a Stores visit and that Accounts visit. The Stores and Purchase visits could have occurred in late August or in the first week of September, or even earlier, depending on how the sequence of Monday visits and the third week gaps line up in the calendar. Multiple schedules can satisfy both statements. Step 6: Additionally, nothing prevents the possibility that the chairman visits Accounts more than once. If there are multiple Accounts visits, then the sequence Stores, Purchase, Accounts may correspond to a different run of Mondays than the September Accounts visit, which makes the date of the Purchase visit even more ambiguous. Step 7: Because there are several possible ways to place the Stores, Purchase, and Accounts visits in the calendar while satisfying all the given conditions, the specific time of the Purchase department visit cannot be uniquely determined from these statements.

Verification / Alternative check:
To verify, imagine two different timelines that both satisfy all given information. In the first, the sequence Stores, Purchase, Accounts might be placed so that Purchase falls on the first Monday of September and Accounts on the second Monday. In another, the same three visits could occur entirely in August, with a different Accounts visit taking place in the second week of September. Since both timelines match statements I and II, yet they assign different dates to the Purchase visit, the answer is not uniquely determined. This confirms that the data are insufficient.

Why Other Options Are Wrong:
Option A claims that statement I alone is sufficient, but as seen, it only says that the Purchase visit occurred sometime before the September Accounts visit, which is too vague. Option B claims that statement II alone is sufficient, which is clearly false because II contains no calendar reference. Option C suggests that either statement alone could be sufficient, which is also incorrect. Option D claims that both statements together are necessary and sufficient, which fails because the combined data still allow many possible placements for the Purchase visit. Only option E correctly states that even both statements together are not sufficient.

Common Pitfalls:
Learners often assume that phrases like earlier occasion and immediately after force a very tight schedule without checking whether multiple months and repeated visits are allowed. Another common mistake is to treat the September Accounts visit as the only such visit, although the problem never makes that explicit. In data sufficiency, we must avoid adding extra assumptions beyond what is stated clearly.

Final Answer:
The data even in both statements together are not sufficient to answer the question, so the correct option is E.