Problem 8 · 2013 Math Kangaroo
Medium
Spatial & Visual Reasoning
tiling-tessellation
Anne has several grey tiles shaped like the one in the picture. What is the greatest number of these tiles she can place on the 5 × 4 rectangle without any overlaps?
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Answer: C — 4
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Hint 1 of 3
First count how many squares the board has and how many each tile covers.
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Hint 2 of 3
Area says at most 5 tiles could fit, but try actually drawing them in.
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Hint 3 of 3
The bumpy T-shape always leaves a few squares stranded, so you can't reach 5.
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Approach: bound by area, then test placement
- The board has \(5 \times 4 = 20\) squares and each grey tile covers 4 squares, so at most \(20 \div 4 = 5\) tiles could fit.
- But when you slot the T-shaped tiles in, they keep leaving small gaps, so 5 is impossible.
- You can fit 4 tiles with no overlap (covering 16 squares), so the most is 4, choice C.
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