An Ant Walks Around a Cube
The problem
An ant sits at one vertex of a solid wooden cube with side length . It wants to reach the vertex diagonally opposite, but it must crawl over the surface of the cube rather than through it. What is the length of the shortest such path?
- The ant stays on the surface at all times.
- The cube has unit side length; the straight-line distance through the interior would be , which is not allowed.
Take a minute
Reveal hint 1
A surface path looks curved in three dimensions, but any single face it crosses is flat. Try unfolding.
Reveal hint 2
To leave a vertex and arrive at the opposite one, the ant must cross exactly two faces; cut the cube open along the right edge and lay those two faces flat.
Solution
Answer units.
Derivation
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Reduce to a flat problem.
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Draw the diagonal.
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Check the crossing point is legitimate.
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Compare with crossing three faces.
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Compare it with the straight-through distance.