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
Count the distinct edge types on the four pieces (one curve pair, one jagged edge, one spike) and check which candidate figure shows exactly that many of each, no extra bumps or missing spikes.
A reliable way to solve a piece-assembly question like this is to count distinctive features on the four loose pieces first, a smooth curved bite, a matching curved bump, a zig-zag jagged edge, and a twin-peaked spike, and then check each candidate figure for exactly those same features, no more and no less.
The rounded piece and the dome piece share one smooth curve between them (one is concave, one is convex), so together they should produce one clean, unbroken curve somewhere in the final outline, with no extra bumps added there. The zig-zag piece is the only source of jagged, saw-tooth edging in the picture, so the final outline should show exactly one jagged run, not two. The twin-peaked piece is the only source of a beak-like spike, so the final outline must show exactly one spike sticking out, and the smaller of its two peaks has to be absorbed inside the curved bite rather than showing up as a second bump.
Figure Q fails immediately: it shows no spike at all. Since the twin-peaked piece must contribute a spike somewhere on the boundary, an outline with no spike cannot be built from these four pieces.
Figure R shows the beak, but it also shows extra jagged bumps crowding the base of the beak, more jagged detail than the single zig-zag piece can supply once its edge is already used for the neck step. That is one jagged feature too many.
Figure S shows a star-like double spike at the neck in addition to the beak, again asking for jagged or pointed detail beyond what the four pieces contain between them.
Figure P shows exactly one clean spike (the beak), one smooth curve (head to shoulder), and one jagged step (the neck), which is precisely the feature count carried by the four given pieces: one spike source, one curve pair, one zig-zag source.
Matching feature for feature rules out Q, R and S, leaving Figure P as the only outline whose count and type of edges agrees with the four given pieces.
Let's summarize the feature count check:
One curve pair (the rounded piece and the dome piece) can only ever produce one smooth curved boundary, never a jagged one.
One zig-zag piece can only ever produce one jagged run of edge, so an outline with two separate jagged regions, like R, is impossible.
One twin-peaked piece can only ever produce one visible spike once its smaller peak is tucked away inside the curved bite, so an outline with no spike (Q) or with a double spike (S) cannot be built.
Since Figure P is the only outline whose curve count, jagged-edge count and spike count all match the four given pieces exactly, once, it is the figure that can be assembled from them.
