Problem 24 · 2024 Math Kangaroo
Stretch
Spatial & Visual Reasoning
tiling-tessellation
Tiler Teri wants to cover a square floor with a regular pattern (see diagram) using six-sided and three-sided tiles. She estimates that she will need about 3000 six-sided tiles for the whole floor. About how many three-sided tiles will she need?
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Answer: D — 6000
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Hint 1 of 2
Focus on one hexagon and count how many triangles touch it, then notice each triangle is shared between hexagons.
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Hint 2 of 2
Pick out the small repeating block of the pattern and count hexagons versus triangles inside it to get the fixed ratio.
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Approach: use the fixed triangle-to-hexagon ratio of the repeating pattern
- In this regular pattern, six triangles ring each hexagon, but every triangle is shared by three hexagons, so each hexagon effectively owns 6 ÷ 3 = 2 triangles.
- That means the tiling always has 2 triangles for every hexagon.
- With about 3000 hexagons, the number of triangles is about 2 × 3000 = 6000.
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