Question

Directions for questions 49 to 52: There are exactly ten stores and no other buildings on a straight street in Bistupur Market. On the northern side of the street, from west to east, are stores 1, 3, 5, 7, and 9. On the southern side, also from west to east, are stores 2, 4, 6, 8, and 10. The stores on the northern side sit directly across the street from the stores on the southern side, facing each other in pairs: 1 and 2; 3 and 4; 5 and 6; 7 and 8; 9 and 10.

Each store is decorated with lights in exactly one colour: green, red, or yellow. The lighting follows these rules:
1. No store has the same light colour as a store next to it on the same side of the street.
2. No store has the same light colour as the store directly across the street from it.
3. Yellow lights decorate exactly one store on each side of the street.
4. Store 4 has red lights.
5. Store 5 has yellow lights.

Suppose that yellow lights decorate exactly two stores on the south side of the street and exactly one store on the north side. If all other conditions remain the same, then which one of the following statements MUST be true?

• A. Green lights decorate store 1
• B. Red lights decorate store 7
• C. Red lights decorate store 10
• D. Yellow lights decorate store 2

Hint:

Only three south-side stores are free to take the two yellow slots; check which pairings among them are actually allowed once you rule out two neighbours both being yellow.

Answer 1

The Correct Option is D

Step 1: Note what changes and what stays the same.
This question replaces rule 3 (exactly one yellow store per side) with a new version: the south side now gets exactly two yellow stores, while the north side still gets exactly one. Everything else, including store 4 being red and store 5 being yellow, stays as given.

Step 2: Re-confirm the stores that do not depend on the yellow count.
Store 3 still has to be green: it faces red store 4, so it cannot be red, and it neighbours yellow store 5, so it cannot be yellow. Store 6 still has to be green for the same kind of reason: it neighbours red store 4, so not red, and it faces yellow store 5, so not yellow. These two facts did not use the "how many yellow stores per side" rule at all, so they still hold under the new condition.

Step 3: Work out where the two south-side yellow stores can go.
The south side is stores 2, 4, 6, 8, 10, and we already know store 4 is red and store 6 is green, so neither of those can be one of the two yellow stores. That leaves only three candidates, store 2, store 8, and store 10, and exactly two of these three must be yellow.

Step 4: Rule out store 8 and store 10 being yellow together.
Store 8 and store 10 are neighbours on the south side (store 8 is directly west of store 10), and rule 1 says neighbouring stores cannot share a colour. So store 8 and store 10 cannot both be yellow at the same time. Since exactly two of store 2, store 8, store 10 must be yellow, and store 8 with store 10 together is not allowed, the two yellow stores must include store 2 paired with either store 8 or store 10. Either way, store 2 is yellow in both possibilities.

Step 5: Confirm store 2 is yellow in every valid case, and check the other options.
Since store 2 shows up as yellow whether the second yellow south store is store 8 or store 10, store 2 being yellow is forced. This makes option 4 ("yellow lights decorate store 2") true in every valid arrangement. Option 1 ("green lights decorate store 1") is false outright, since store 1 was already shown to be forced to red by the base rules (it cannot be green because it neighbours green store 3, and it cannot be yellow because the one north-side yellow slot belongs to store 5). Option 2 ("red lights decorate store 7") is not forced, since store 7 could be red or green depending on how the rest of the arrangement works out. Option 3 ("red lights decorate store 10") is not forced either, since store 10 turns out yellow in one of the two valid cases from step 4 (when store 8 is the second yellow store, store 10 does not have to be red).

Final Answer:
Store 2 is the store that must have yellow lights under this new condition. \[ \boxed{\text{Option 4}} \]