Identify the image given in the options which, when put in place of the…

Concept: In a positional-cycling series the set of symbols never changes from one step to the next; only the seat each symbol occupies changes. Every symbol advances by exactly one place along a single fixed closed path that runs through all the available seats. Once that path has been read off from consecutive steps, every later step is completely determined by it.

Application: Each box in this series carries the same five letters — D, O, P, S and U — in five seats: the top-left corner, the top-right corner, the centre, the bottom-left corner and the bottom-right corner. Recording the four given steps seat by seat:

Comparing each step with the one after it shows where the letter in every seat travels:

1. The letter in the centre moves to the top-left corner: U sits at the centre in step 1 and at the top-left in step 2; P does the same from step 2 to step 3; O does the same from step 3 to step 4.

2. The letter at the top-left corner moves to the top-right corner: D from step 1 to step 2, U from step 2 to step 3, P from step 3 to step 4.

3. The letter at the top-right corner moves to the bottom-right corner: S from step 1 to step 2, D from step 2 to step 3, U from step 3 to step 4.

4. The letter at the bottom-right corner moves to the bottom-left corner: O from step 1 to step 2, S from step 2 to step 3, D from step 3 to step 4.

5. The letter at the bottom-left corner moves to the centre: P from step 1 to step 2, O from step 2 to step 3, S from step 3 to step 4.

So the closed path is centre → top-left → top-right → bottom-right → bottom-left → centre, and every letter advances exactly one seat along it at each step.

Applying that single advance to step 4, which holds O at the top-left, P at the top-right, S at the centre, D at the bottom-left and U at the bottom-right:

• S leaves the centre and takes the top-left corner.

• O leaves the top-left corner and takes the top-right corner.

• P leaves the top-right corner and takes the bottom-right corner.

• U leaves the bottom-right corner and takes the bottom-left corner.

• D leaves the bottom-left corner and takes the centre.

Hence the box that replaces the question mark carries S at the top-left corner, O at the top-right corner, D at the centre, U at the bottom-left corner and P at the bottom-right corner.

Cross-check: the same path also settles the arrangements that do not fit.

• Running the path backwards from the result returns step 4 exactly: S goes back from the top-left to the centre, O from the top-right to the top-left, P from the bottom-right to the top-right, U from the bottom-left to the bottom-right and D from the centre to the bottom-left, giving O, P, S, D, U once more.

• All five letters D, O, P, S and U appear exactly once, as they do in every given step; no letter is dropped, repeated or replaced.

• An arrangement holding S at the top-left, D at the top-right, P at the centre, U at the bottom-left and O at the bottom-right breaks the path, because the letter that reaches the centre must be the one standing at the bottom-left of step 4, which is D and not P.

• An arrangement holding U at the top-left, O at the top-right, S at the centre, P at the bottom-left and D at the bottom-right keeps S in the centre for a second step running, whereas the path moves every letter one seat at each step.

• An arrangement holding S at the top-left, O at the top-right, U at the centre, P at the bottom-left and D at the bottom-right puts U in the centre, although U stands at the bottom-right of step 4 and the path sends that letter to the bottom-left corner.