Problem 25 · 2024 Math Kangaroo
Stretch
Spatial & Visual Reasoning
net-foldingpath-tracing
The diagram shows an object composed of 7 cubes with edge length 2. How long is the shortest path from M to N on the surface of the object?
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Answer: A — 10
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Hint 1 of 3
A shortest path over a surface becomes a straight line once you unfold the faces it crosses into one flat plane.
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Hint 2 of 3
Unfold the faces between M and N, then the distance is \(\sqrt{\text{horizontal}^2+\text{vertical}^2}\); look for a 6-8-10 right triangle.
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Hint 3 of 3
Each cube edge is 2, so legs come in multiples of 2 — try the unfolding whose legs are 8 and 6.
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Approach: unfold the crossed faces and measure the straight segment
- Flatten the faces the path crosses into a single plane; each cube edge has length 2.
- On the best unfolding M and N are the ends of a straight segment whose legs are \(8\) (four edges) and \(6\) (three edges).
- Its length is \(\sqrt{8^2+6^2}=\sqrt{100}=10\), shorter than every other unfolding's value.
- So the shortest path is 10 (answer A).
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