A, B, C, D, E and F, not necessarily in that order are sitting on six chairs regularly placed around a round table. It is observed that A is between D and F, C is opposite D, and D and E are not on neighbouring chairs. Which one of the following pairs must be sitting on neighbouring chairs?

To solve this, we place the six people around the circular table based on the given conditions.

First, place D and F with A sitting between them. This establishes a sequence of D, A, F.

Second, place C opposite D. In a six-chair arrangement, being opposite means there are two chairs between the individuals. Counting two chairs away from D in either direction puts C opposite D. This places C next to F.

Third, we are told D and E are not on neighboring chairs. D is already next to A. The only other chair next to D is currently empty. Since E cannot sit there, E must sit in the only remaining chair that is not next to D. This position is between C and F.

Finally, the only person left is B. The only available chair for B is the one next to D.

The final clockwise or counter-clockwise arrangement is D, A, F, E, C, B.

Looking at the options, C and E are sitting on neighboring chairs in this arrangement. Therefore, option B is correct.