The ant crawling across a box

Question

A rectangular box measures 12 by 3 by 4. An ant sits at one bottom corner and wants to reach the opposite corner (the far top corner, diagonally across the whole box). The ant can only crawl along the surface, it cannot fly through the inside or dig. What is the length of the shortest path along the surface?

Accepted Answer

The shortest surface path is \sqrt{193} \approx \mathbf{13.89}. The trick: unfold the box flat. A path over two adjacent faces becomes a straight line once you unfold those faces into a single plane. The shortest surface route is the straight line across the best unfolding. Between opposite corners of an a 	imes b 	imes c box, the path crosses two faces, and there are three ways to unfold, giving three candidate straight-line distances. With a=12, b=3, c=4: - Combine the two "side" dimensions with the long one: \sqrt{12^2 + (3+4)^2} = \sqrt{144 + 49} = \sqrt{193} \approx 13.89. - \sqrt{3^2 …

The ant crawling across a box

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