Problem 25 · 2021 Math Kangaroo
Stretch
Geometry & Measurement
symmetry
What is the smallest number of shaded squares that can be added to the diagram to create a design, including the grid, with 4 axes of symmetry?
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Answer: E — 21
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Hint 1 of 2
Four axes of symmetry means both diagonals and both midlines must mirror the shading.
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Hint 2 of 2
Reflect the two shaded cells across all four axes and count every new cell those reflections demand.
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Approach: enforce all four mirror symmetries
- A design with 4 axes of symmetry must look the same under both diagonal flips and both horizontal/vertical flips.
- Reflecting the two already-shaded cells through all those symmetries generates a full set of matching cells.
- The number of extra cells that must be shaded to complete that symmetric set is 21.
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