Question

let us consider the ant at one corner of cube.if the ant needs to go to the other corner.what is the shortest distance covered by the ant

Answer 1

Proof : First of all, note that the optimal path will lie on 2 faces of the cube only. If the path lay on more than two faces, it would have to leave a face and return to it (This is because the cube has only 6 faces, 3 touching the initial vertex and 3 touching the final vertex). Hence we can obtain a shorter path by replacing the trajectory between the two approaches to a face by a straight line.

With this information, the proof becomes trivial. The two faces on which the trajectory lies are adjacent, and we can flatten them out to obtain a 2a×a2a×a rectangle. we need to find the shortest path lying on this rectangle between two opposite corners, which is just the length of the diagonal between them.
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