Step 1: Understanding the Question: A square PQRS is filled solid black and is cut into two jigsaw like pieces along an irregular staircase style boundary. Panel I shows one of those two pieces, drawn with corners S, R on top and P, Q on the bottom marking the original square outline. Panel II shows four candidate shapes labelled (i), (ii), (iii) and (iv), and exactly one of them is the true second piece that, when placed together with the Panel I piece, reforms the complete solid black square PQRS with no gaps and no overlaps.
Step 2: Key Formula or Approach: The correct approach for this kind of cut and reassemble figure question is to treat the missing piece as the exact complement of the shown piece within the square boundary. Every notch that cuts into the Panel I shape must be filled by a matching tooth on the correct answer piece, and every tooth that sticks out on the Panel I shape must correspond to a matching notch on the correct answer piece, so that the two pieces interlock perfectly like two halves of a puzzle to recreate a perfect square with straight uniform edges.
Step 3: Detailed Explanation: Checking option (A), shape (i) has a boundary profile that does not line up with the stepped cuts of Panel I. Some of its projecting teeth fall in the same positions as the teeth already present in Panel I instead of filling the gaps, so combining it with Panel I would leave overlapping black regions and empty white regions, meaning the two pieces cannot form a full square.
Checking option (B), shape (ii) is a mirror flipped version of a plausible complement. While it looks visually close to the correct piece, close comparison of its step pattern shows the notches are reversed left to right relative to what Panel I requires, so it again fails to interlock cleanly along the cut line.
Checking option (C), shape (iii) has its projecting and recessed sections placed exactly where Panel I has the opposite feature, meaning every notch in Panel I lines up with a tooth in shape (iii) and every tooth in Panel I lines up with a notch in shape (iii). Fitting shape (iii) against the Panel I piece completes a full, evenly bordered square with the original corners P, Q, R, S restored, confirming that this is the genuine second half of the cut square.
Checking option (D), shape (iv) has an extra offset step compared to the true complement, which means part of its boundary would either leave a gap or cause an overlap once matched against Panel I, so it does not correctly complete the square either.
Step 4: Final Answer: Only the shape in option (iii) has a boundary that is the exact complementary mirror of the cut edge of Panel I, so joining it with Panel I recreates the original solid square PQRS. \[ \boxed{\text{(iii)}} \]
