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 pair of products that could be advertised during the same week?

• A. G and H
• B. H and J
• C. H and O
• D. M and O

Hint:

Since three of the four pairs involve O, and O only ever sits in week 3, check which second product can safely join O there without starving week 4 of its required repeat.

Answer 1

The Correct Option is D

Step 1: Test (AA) G and H together.
If G and H are advertised in the same week, condition 3 still applies to G: that week must also carry J or O. A week only holds two products, and here the two are G and H, neither of which is J or O. So G's own rule is broken the moment G is paired with H, no matter which week it happens in. Option A is impossible.

Step 2: Test (BB) H and J together.
Condition 1 requires H in the week immediately before any week that carries J, not in the same week as J. If H and J share a week, that shared week does not satisfy J's rule by itself; H would still need to appear again the week before, meaning H is advertised twice, once alongside J and once the week before it, two weeks that sit right next to each other. But the repeated product's two appearances are always week 4 plus week 1 or week 2 (condition 2 forces one slot to week 4 and bans week 3), and week 1 or week 2 is never adjacent to week 4. So H can never legally appear in two back-to-back weeks, and pairing H with J in the same week demands exactly that. H and J together is impossible.

Step 3: Test (CC) H and O together.
O only ever sits in week 3, so H and O together means H is advertised in week 3. That forces J into week 4, the only week right after a H-week, which then forces G to join J in week 4 to satisfy condition 3. Week 4 then holds two single-appearance products, G and J, leaving no seat for the product condition 2 requires to repeat there. H and O together is impossible.

Step 4: Test (DD) K and O, for context.
K must be advertised in week 1 or week 2 to satisfy condition 4. Even if K repeats, its second slot is forced to week 4 by condition 2, never week 3. So K can never validly reach week 3, and pairing K with O, who only ever sits in week 3, is never possible either.

Step 5: Test (EE) M and O together.
M and O together means both sit in week 3. Build the rest of the schedule around this: put H and K in week 1, satisfying condition 4 for K; put G and J in week 2, since J needs H the week before (supplied by week 1) and G needs J in the same week; keep O and M in week 3 as assumed; and let week 4 take the repeated product plus L. Since H, K, G, J, O and M are each placed exactly once, the seventh product, L, must be the one that repeats, filling week 4 alongside a second copy of either H or K. This gives a fully valid schedule, for instance week 1: H, K; week 2: G, J; week 3: O, M; week 4: L, H. Every condition holds, so M and O together is genuinely achievable.

Final Answer:
Only option (EE), M and O, can be advertised during the same week without breaking any condition.