Cubical Box | AMC-10A, 2010 | Problem 20

Given that "A fly trapped inside a cubical box with side length \(1\) meter decides to relieve its boredom by visiting each corner of the box. It will begin and end in the same corner and visit each of the other corners exactly once".................................Therefore we may say that from a corner to any other corner the straight path will be \(A \to G \to B \to H \to C \to E \to D \to F \to A\)

can you finish the problem........

The distance of an interior diagonal in this cube is \(\sqrt 3\) ( i.e \(HB\)) and the distance of a diagonal on one of the square faces is \(\sqrt 2\) ( i.e \(HA\))

can you finish the problem........

Now the fly visits each corner exactly once, it cannot traverse such a line segment twice. Also, the cube has exactly four such diagonals, so the path of the fly can contain at most four segments of length.Therefore  the maximum distance traveled is \(4\sqrt 3+4\sqrt 2\)