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

The correct option is Only statement 3 is sufficient.

Let us analyze the problem step-by-step to understand why Statement 3 alone is sufficient to ensure that any two consecutive floors have different colours.
The building has five floors, labeled from bottom to top as I, II, III, IV, and V.
We are given that exactly four different colours are available to paint these floors, and each floor is painted with exactly one colour.
Our goal is to ensure the condition: any two consecutive floors must have different colours. This means:
- Floor I and Floor II have different colours.
- Floor II and Floor III have different colours.
- Floor III and Floor IV have different colours.
- Floor IV and Floor V have different colours.

Let us evaluate Statement 3:
Statement 3 states: The first (I) and the fifth (V) floors are painted red.
Let the four available colours be Red (R), and three other colours, say A, B, and C.
According to Statement 3:
- Floor I = Red
- Floor V = Red

Now, we need to paint the remaining three middle floors (II, III, and IV) using the available colours (Red, A, B, C) such that no two adjacent floors share the same colour.
Since Floor I is Red, Floor II cannot be Red. Thus, Floor II must be painted with one of the colours {A, B, C}.
Since Floor V is Red, Floor IV cannot be Red. Thus, Floor IV must be painted with one of the colours {A, B, C}.
Floor III is adjacent to II and IV. To ensure consecutive floors have different colours, Floor III must have a different colour than Floor II and Floor IV.

Let us see if we can always satisfy the condition under Statement 3:
Suppose we choose a colour for Floor II from {A, B, C}, say colour A.
Suppose we choose a colour for Floor IV from {A, B, C}, say colour B (or it could also be A).
- If Floor II is A and Floor IV is B, then Floor III is adjacent to A and B. We can paint Floor III with either Red (since it is not adjacent to I or V) or the remaining colour C. In either case, all adjacent floors will have different colours.
- If Floor II is A and Floor IV is A, then Floor III is adjacent to A on both sides. We can paint Floor III with Red, B, or C. Again, all adjacent floors will have different colours.

Since we have four colours available and only need to satisfy simple inequality constraints between adjacent floors, having the boundary floors (I and V) fixed to the same colour (Red) leaves us with more than enough colours to consistently color the middle floors without any two adjacent floors sharing a colour. Thus, Statement 3 alone is sufficient to guarantee that a valid coloring exists where any two consecutive floors have different colours.