A black square PQRS has been cut into two parts. One part of it is shown in Panel I. Which one of the shapes labelled (i), (ii), (iii), and (iv) in Panel II is the other part of the same square?
Show Hint
The missing piece must be the exact mirror complement of Panel I along the cut line, so the two pieces rejoin into the full square PQRS.
Step 1: Understand what the two parts must satisfy. The square \(PQRS\) is completely black, and it has been cut along a single jagged line into two separate pieces. Panel I shows one of these two pieces. Since the two pieces together make up the whole square with no gaps and no overlap, the second piece must be the exact geometric complement of the piece shown in Panel I.
Step 2: Use the boundary of the square as a guide. Every edge and corner of the square must be accounted for by exactly one of the two pieces. Wherever Panel I's piece steps inward, the missing piece must step outward by the same amount along the same cut line, so that when the two are placed back together, they form a perfect square with straight outer edges and no overlap between the black regions.
Step 3: Compare each option in Panel II against this requirement. Shapes (i) and (ii) in Panel II look similar to each other but are mirror images, and only one orientation can slot against the cut edge of Panel I's piece without leaving a gap or causing an overlap. Shape (iv) also fails to match the notches and projections along the cut line of Panel I when tested against it. Only shape (iii) has notches and projections that are the exact mirror complement of Panel I's cut edge, so that fitting it against Panel I reconstructs the original square exactly.
Step 4: Final answer. The piece that completes the square together with Panel I is shape (iii). \[ \boxed{\text{(iii)}} \]
