A five-storeyed building with floors from I to V is painted using four different colours and only one colour is used to paint a floor. Consider the following statements : 1. The middle three floors are painted in different colours. 2. The second (II) and the fourth (IV) floors are painted in different colours. 3. The first (I) and the fifth (V) floors are painted red. To ensure that any two consecutive floors have different colours

(a) Only statement 2 is sufficient
(b) Only statement 3 is sufficient ✓ UPSC's answer
(c) Statement 1 is not sufficient, but statement 1 along with statement 2 is sufficient
(d) Statement 3 is not sufficient, but statement 3 along with statement 2 is sufficient

Why the answer is (b)

• Five floors are painted with four different colours, one colour per floor, so all four colours are used and exactly one colour appears on two floors.

• Statement 3 says floors I and V are both red, so red is the repeated colour. Floors II, III and IV must then carry the other three colours, all different from each other and from red, so no two consecutive floors match. Statement 3 alone is sufficient.

• Statement 2 alone (II and IV differ) still allows the repeated colour on two adjacent floors such as I and II; statement 1 alone (II, III and IV all different) still allows I = II or IV = V.

• Hence option (b).

Why the other options are wrong

(a) Only statement 2 is sufficient: Statement 2 alone still allows the repeated colour on two adjacent floors such as I and II.
(c) Statement 1 is not sufficient, but statement 1 along with statement 2 is sufficient: Statements 1 and 2 together still allow floors I and II to share a colour.
(d) Statement 3 is not sufficient, but statement 3 along with statement 2 is sufficient: Statement 3 is sufficient on its own; statement 2 is not needed.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2019, held on 2 June 2019. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.