An ant across a cube
An ant walks on the surface of a cube with edges of length 1, from one corner to the diagonally opposite corner. What is the length of the shortest path it can take?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Unfold the cube: lay two neighbouring faces flat, side by side.
On the flat net, the shortest path is a straight line.
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s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. A path on the surface between opposite corners crosses two faces. Unfolding those two faces into one flat rectangle turns the shortest path into a straight line.
Step 1: unfold. Two adjacent faces laid flat form a rectangle, with the two corners at opposite ends of its diagonal.
Step 2: the straight line. The path crosses the shared edge at its midpoint.
Why this is the minimum. Any surface path from one corner to the opposite one must pass through at least two faces, and it can be unfolded onto a flat net of the faces it crosses, where it is at least as long as the straight line. All the nets give or more.
Two traps. is the straight line through the inside of the cube, which the ant cannot use. follows an edge and then a face diagonal, which is longer.