Step 1: Reframing the Problem: Instead of visually matching notches directly, we can think of the square PQRS as being made up of a grid of small equal unit cells, the way a chessboard is made of small squares. Panel I occupies a certain set of these unit cells, and the true second piece from Panel II must occupy exactly the remaining unit cells so that together the two pieces cover every cell of the grid exactly once.
Step 2: A Cell Counting and Position Check: Using this grid idea, we can check each candidate in Panel II in two ways, first by seeing whether its overall size or cell count is consistent with being the leftover portion of the square once Panel I's cells are removed, and second by checking that its cell positions along the shared cut boundary exactly mirror the boundary cells of Panel I so nothing is left uncovered or doubled up.
Step 3: Detailed Explanation: For shape (i), several of its filled cells duplicate cells that Panel I already occupies near the cut line, so placing it next to Panel I would double count cells instead of filling the empty ones, which rules it out.
For shape (ii), the overall cell count looks plausible, but tracing the boundary cell by cell shows its edge cells are reflected the wrong way, so a few cells along the join would be left blank and this candidate also fails.
For shape (iii), tracing along the shared cut boundary shows that every cell left empty by Panel I is exactly filled by a cell of shape (iii), with no duplication and no gap anywhere along the join, and the combined outline traces out a perfect square matching the original corners P, Q, R, S.
For shape (iv), one extra row of cells is shifted out of place compared to what Panel I needs, so a strip of the square would remain uncovered, ruling this option out too.
Final Answer: Using the cell by cell complement check, shape (iii) is confirmed as the piece that exactly completes the square together with Panel I. $\boxed{\text{(iii)}}$
