Problem 10 · AMC 8 Stretch
Core
Geometry & Measurement
visual-representationreduce-and-expand
A bug sits at corner \(B\) of a closed box and wants to crawl along the outside surface to the opposite corner \(H\). It must stay on the faces (it cannot fly or tunnel through). What is the shortest route along the faces?
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Answer: Unfold the box flat; the shortest route is the straight segment \(\overline{BH}\) in the unfolding (the unfolding that makes it shortest)
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Hint 1 of 3
A path that bends around a corner of the box is hard to measure. Change the picture: unfold the box flat, like opening a cereal box, so the two faces the bug crosses lie in one flat plane.
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Hint 2 of 3
On a flat surface, the shortest path between two points is a straight line. So once the box is unfolded, just draw the straight segment from \(B\) to \(H\).
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Hint 3 of 3
Fold the box back up: that straight line becomes the bug's route over the faces. Since the box can be unfolded in more than one way, check the different unfoldings and pick the one giving the shortest straight line.
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Approach: Visual representation — unfold the box flat
- Crawling around a 3-D corner is confusing, so change the representation: unfold the box so the faces the bug walks across become one flat sheet.
- On a flat sheet, the shortest path between two points is a straight line. So draw the straight segment from \(B\) to \(H\) in the unfolded picture — that is the shortest possible crawl. When you fold the box back up, that straight line wraps over the faces and shows the bug's actual route.
- There is more than one way to unfold the box (you could open it across different pairs of faces). Each unfolding gives a straight segment of a different length, so try the unfoldings and choose the one with the shortest segment \(\overline{BH}\).
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