Problem 13 · 2013 Math Kangaroo
Medium
Spatial & Visual Reasoning
tiling-tessellation
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Answer: E
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Hint 1 of 3
A round carpet colours every tile it touches, so the grey region must be a single rounded, bulging blob.
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Hint 2 of 3
A disk has two axes of symmetry, so the set of tiles it touches must look the same when flipped left-right and top-bottom.
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Hint 3 of 3
Check each picture for that mirror symmetry and for any one-tile 'bump' a smooth circle could not reach.
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Approach: use the symmetry a disk's grey set must have
- The tiles a disk touches always form one solid, convex-looking blob that is symmetric about both the horizontal and the vertical line through the disk's centre.
- Patterns (A)–(D) each have that double mirror symmetry, so a suitably placed circle can produce them.
- Pattern (E) has an off-centre tile that breaks the symmetry — no single circle can touch exactly those tiles, so the impossible one is E.
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