Question

The shown figure is first cloned and flipped on the axis PP, and the resulting image is cloned and flipped on the axis QQ. How many triangles does the resulting image have? 

Hint:

In complex counting problems involving reflections, a reliable method is the "component and join" strategy. Count the figures in the base component, multiply by the number of components in the final image, and then carefully count only the new figures formed at the joins between components.

Answer 1

Step 1: Start from the base figure.
The original figure (before any flip) has 16 triangles.

Step 2: Apply the first flip, across PP.
Joining the figure to its mirror image across PP simply doubles the triangle count, since no new triangle is formed exactly on this particular seam:
\[ 16 + 16 = 32 \]

Step 3: Apply the second flip, across QQ.
Joining this 32-triangle shape to its own mirror image across QQ again doubles the count, and this seam does create new triangles, where the tips of the mirrored halves meet:
\[ 32 + 32 = 64 \]
plus 4 new bridging triangles formed exactly along QQ.

Step 4: Add the new triangles.
\[ 64 + 4 = 68 \]
\[ \boxed{68} \]

Answer 2

A third way to organise the same count is to separate, at each stage of assembly, the triangles carried over unchanged from the two source copies from the triangles newly created purely by the mirror symmetry.

At the first join (flip across PP), every triangle in the two source copies survives unchanged, so this stage contributes \( 16 + 16 = 32 \) carried-over triangles, and no new triangle is created at this particular seam.

At the second join (flip across QQ), the same carry-over rule applies to the 32-triangle shape, giving \( 32 + 32 = 64 \) carried-over triangles. This time the join also produces new triangles at the seam, since the outer tips of the mirrored halves line up to complete extra triangular outlines that did not exist in either half on its own. Counting these, 4 new triangles appear along QQ.

Total triangles = carried-over triangles + newly created triangles = 64 + 4 = 68. So the correct answer is 68.