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?

Question

Pointing to a photograph, a man says, "I have no brothers or sisters, but the father of the person in the photograph is my father’s son." Who is the person in the photograph?

Answer 1

Step 1: Count the original figure's triangles.
The given figure has 16 triangles in total (8 smallest, 4 made of 2 units, 4 made of 4 units).

Step 2: Apply the first flip, across PP.
Flipping across PP places a mirrored copy next to the original. Since the two halves meet edge to edge without new triangles forming across that particular fold, the figure now has \(2 \times 16 = 32\) triangles.

Step 3: Apply the second flip, across QQ.
Flipping the whole 32-triangle figure across QQ doubles it again to \(2 \times 32 = 64\) triangles, and this second fold is where the halves line up so that 4 new, larger triangles are formed spanning across the fold line.

Step 4: Final Answer.
\[ 64 + 4 = \boxed{68} \]

Answer 2

Another way to see this is to treat the final image as 4 quadrants meeting at a shared centre point, and to work out separately what happens inside each quadrant and what happens only where quadrants touch.

Each of the 4 quadrants is an exact copy of the original figure (arranged so that reflecting across PP swaps left and right, and reflecting the result across QQ swaps top and bottom), and each copy on its own contains 16 triangles, none of which reach outside that quadrant's own boundary. \[ 4 \times 16 = 64 \text{ triangles fully inside the 4 quadrants} \]
Now look only at what happens where two quadrants meet. Along the horizontal fold (QQ), the top and bottom quadrants line up so that parts of one quadrant's triangle combine with parts of the neighbouring quadrant's triangle to outline a new, bigger triangle that did not exist in either quadrant alone. This happens twice on the left pair of quadrants and twice on the right pair, giving 4 new triangles that only exist because the quadrants were joined.
No further new triangles form across the vertical fold (PP) in this arrangement, since the shapes meeting there do not line up into a bigger triangular outline the way they do across QQ.
Adding the within-quadrant total to the newly formed boundary triangles gives the count for the whole image. \[ 64 + 4 = 68 \]

So the correct answer is 68.

Answer 3

Another way to see this is to treat the final image as 4 quadrants meeting at a shared centre point, and to work out separately what happens inside each quadrant and what happens only where quadrants touch.

Each of the 4 quadrants is an exact copy of the original figure (arranged so that reflecting across PP swaps left and right, and reflecting the result across QQ swaps top and bottom), and each copy on its own contains 16 triangles, none of which reach outside that quadrant's own boundary. \[ 4 \times 16 = 64 \text{ triangles fully inside the 4 quadrants} \]
Now look only at what happens where two quadrants meet. Along the horizontal fold (QQ), the top and bottom quadrants line up so that parts of one quadrant's triangle combine with parts of the neighbouring quadrant's triangle to outline a new, bigger triangle that did not exist in either quadrant alone. This happens twice on the left pair of quadrants and twice on the right pair, giving 4 new triangles that only exist because the quadrants were joined.
No further new triangles form across the vertical fold (PP) in this arrangement, since the shapes meeting there do not line up into a bigger triangular outline the way they do across QQ.
Adding the within-quadrant total to the newly formed boundary triangles gives the count for the whole image. \[ 64 + 4 = 68 \]

So the correct answer is 68.

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?

Show Hint

Correct Answer: 68

Approach Solution - 1

Approach Solution -2

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