Problem 17 · 2018 Math Kangaroo
Hard
Spatial & Visual Reasoning
transformationssymmetry
A regular pentagon is cut out of a page of lined paper. Step by step the pentagon is then rotated 21° counter-clockwise about its midpoint. The result after step one is shown in the diagram. Which of the diagrams shows the situation when the pentagon fills the hole entirely again for the first time?
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Answer: B
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Hint 1 of 3
The pentagon outline refits its hole only when the total turn is a whole multiple of its rotational-symmetry angle.
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Hint 2 of 3
Find the smallest number of 21° steps that is a whole multiple of 72°, then see how far the drawn lines have turned.
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Hint 3 of 3
The lines printed on the pentagon turn by the full accumulated angle, reduced modulo 360°.
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Approach: rotational symmetry: the hole is refilled when the total rotation is a multiple of 72°
- A regular pentagon looks the same after a turn of 72°, so its outline fits the hole when the accumulated rotation is a multiple of 72°.
- Stepping 21° at a time first lands on a multiple of 72° after 24 steps, since 24·21° = 504° = 7·72°.
- The outline is back in place, but the lines printed across it have turned 504° ≡ 144°, so they are no longer horizontal — they appear tilted.
- The diagram with the outline refit and the lines at that tilt is option (B).
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