Question

During a four-week period, each one of seven previously unadvertised products, G, H, J, K, L, M and O, will be advertised. A different pair of these products will be advertised during each of the four weeks. Exactly one of these seven products will be advertised during two of the four weeks; none of the other six products is repeated in any pair. The following conditions must hold.

1. J is not advertised during a given week unless H is advertised during the week immediately before it.
2. The product that is advertised twice is advertised during week 4 but is not advertised during week 3.
3. G is not advertised during a given week unless either J or O is also advertised during that same week.
4. K is advertised during one of the first two weeks.
5. O is one of the products advertised during week 3.

Which one of the following is a product that could be advertised in any of the four weeks?

• A. H
• B. J
• C. K
• D. L

Hint:

Look for the one product that is never named in any of the five given conditions, since that is the one free to move to any week.

Answer 1

The Correct Option is D

Step 1: Scan the five conditions for which products are actually tied to specific weeks.
Condition 5 nails O to week 3 alone. Condition 4 nails K to week 1 or week 2. Condition 1 ties J to a week right after one that has H. Condition 3 ties G to a week that also carries J or O. Only L is never mentioned in any of the five conditions, which hints that L is the free agent, but each option needs to be checked properly rather than assumed.

Step 2: Test H in week 3.
Suppose H is advertised in week 3, alongside O. Condition 1 then only lets J be advertised in the week right after a H-week, and the only such week is week 4. So J would be forced into week 4. Condition 3 then needs G to share a week with J or O; since O's week is already full with H and O, G would have to join J in week 4. That fills week 4 with two single-appearance products, J and G, leaving no seat for the product that condition 2 says must repeat there. Neither G nor J can be that repeated product, since both are only ever allowed into a week that satisfies their own linking rule, and neither can validly repeat in weeks 1, 2 or 3 without breaking that rule again. So H in week 3 leaves week 4 with no valid repeat, which is not allowed. H cannot be advertised in all four weeks.

Step 3: Test J in week 1.
Condition 1 says J needs H in the week immediately before it. Week 1 has no week before it, so J can never satisfy this rule if placed in week 1. J is permanently barred from week 1, so J cannot be advertised in all four weeks either.

Step 4: Test K in week 3.
Condition 4 requires K to be advertised in week 1 or week 2 at least once. If K's only appearance were week 3, condition 4 would already be broken. If K is instead the repeated product, condition 2 forces its second appearance into week 4, not week 3, so a repeated K still cannot touch week 3. Either way, week 3 is off limits to K, so K also cannot be advertised in all four weeks.

Step 5: Test L.
None of the five conditions restrict L to any particular week, and concrete valid schedules confirm this in practice. One valid arrangement, week 1: H, L; week 2: J, K; week 3: G, O; week 4: L, M, places L in week 1 and week 4. Another, week 1: H, K; week 2: L, J; week 3: G, O; week 4: L, M, places L in week 2 and week 4. A third, week 1: H, K; week 2: J, G; week 3: O, L; week 4: M, K, places L in week 3. Since valid schedules exist with L occupying week 1, week 2, week 3 and week 4 in turn, L is confirmed free to be advertised in any of the four weeks.

Final Answer:
L is the product that could be advertised in any of the four weeks, so option (DD) is correct.