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.

Which one of the following statements MUST be true?

• A. Green lights decorate store 10
• B. Red lights decorate store 1
• C. Red lights decorate store 8
• D. Yellow lights decorate store 8

Hint:

Trace store 1's two constraints separately: what colour blocks it from being green, and what already uses up the one yellow slot on its side of the street.

Answer 1

The Correct Option is B

Step 1: Work out which stores are forced by the rules alone.
No extra condition is added in this question, so we go back to the five original rules. Store 3 faces store 4 (red), so store 3 cannot be red, and store 3 also sits next to store 5 (yellow), so store 3 cannot be yellow. That leaves green as the only choice for store 3. By the same kind of reasoning, store 6 sits next to store 4 (red) and faces store 5 (yellow), so store 6 must also be green.

Step 2: Use the "one yellow per side" rule on the north side.
Store 5 is already yellow, and rule 3 allows only one yellow store on each side of the street. So on the north side, stores 1, 3, 7, and 9 all cannot be yellow (store 3 is already fixed as green anyway).

Step 3: Pin down store 1.
Store 1 sits right next to store 3, which we found is green, so store 1 cannot be green. From step 2, store 1 also cannot be yellow. Green and yellow are both ruled out, so the only colour left for store 1 is red. This holds in every single valid arrangement of the street, since nothing about stores 7, 9, 2, 8, or 10 affects this chain of reasoning.

Step 4: Check why the other statements are not forced.
Store 10 (option 1) only has to differ from store 8 (its neighbour) and store 9 (its facing store); depending on how the rest of the puzzle is filled in, store 10 can end up green, red, or yellow, so it is not always green. Store 8 (options 3 and 4) only has to differ from store 6 (green) and store 7 (whatever colour that turns out to be), so store 8 can be red or yellow depending on the arrangement, meaning neither "always red" nor "always yellow" has to hold.

Final Answer:
Only "red lights decorate store 1" holds in every possible valid arrangement of the street. \[ \boxed{\text{Option 2}} \]