The four pieces of a puzzle are shown in the figure below. Which one of the figures labelled as P, Q, R, and S can be constructed by using each of the four pieces only once without overlaps?
Show Hint
Match each piece's jagged edges and peaks to the candidate outline that uses all four pieces exactly once, with no gaps or overlaps.
Step 1: Understanding the Question.
We are given four separate, oddly shaped puzzle pieces and four candidate figures P, Q, R, S. We must find the one candidate that can be built by placing all four pieces together, using each piece exactly once, with no overlaps and no gaps.
Step 2: Key Approach.
For this kind of spatial assembly question, first note that the combined area of the four pieces must exactly equal the area enclosed by the correct candidate figure, no more and no less. Then trace each piece's distinctive edges, sharp points, notches, or scalloped curves, and check where each such edge must sit on the boundary of the assembled figure, since two pieces can only join along edges of matching length and matching shape (a concave notch on one piece must meet a convex bump on another).
Step 3: Detailed Explanation.
Each of the four pieces has its own distinct boundary: a scalloped edge, a slanted arch-like top, a reversed jagged edge, and a twin-peak (mountain-like) edge. Placed together correctly, these edges interlock into the outline of a single bird-like silhouette, with the twin-peak piece forming the crest on top, the scalloped piece forming the beak and front of the head, and the remaining two pieces closing up the body and tail.
Checking each candidate: in (B)/Q and (C)/R, the beak or crest comes out a shade shorter than what the four given pieces actually produce, so fitting the pieces there leaves either a small gap or a small overlap where two edges fail to match in length, which the question rules out.
In (D)/S, the crest shows an extra spike that the four pieces cannot generate between them, since the twin-peak piece only ever supplies two peaks, not three, so this candidate needs an edge that is not actually present on any piece.
Only candidate (A)/P reproduces the exact outline the four pieces interlock into: the scalloped piece forms the beak with no leftover notch, the twin-peak piece sits exactly as the crest, and the remaining two pieces close the body and tail with edges of matching length and shape, using every piece exactly once with no overlap.
Step 4: Final Answer.
The figure that can be constructed from the four pieces is P.
\[ \boxed{\text{P}} \]
