Problem 8 · 2020 Math Kangaroo
Medium
Geometry & Measurement
perimeterspatial-reasoning
Gaspar has these seven different pieces, each made of equal little squares. He uses all the pieces to build rectangles with different shapes (and so different perimeters). How many different perimeters can he get?
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Answer: B — 3
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Hint 1 of 2
All the pieces together cover a fixed number of small squares — that's the area of every rectangle he makes.
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Hint 2 of 2
List the rectangle shapes with that area and count their different perimeters.
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Approach: fix the total area, then count distinct rectangle shapes
- The pieces have 1, 2, 3, 4, 5, 6 and 7 squares, so together they always cover 1+2+3+4+5+6+7 = 28 squares.
- A rectangle of 28 squares can be 1×28, 2×14 or 4×7, which give 3 different perimeters.
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