The Worm and The CubeDiscrete Mathematics Level 5
There is a \(3\times 3\times 3\) cube (i.e., a cube which is made up of 27 sub-cubes). A worm sits on one of its corner-faces. The worm starts digging into the cube according to the following set of rules: i) It can dig parallel to the edges of the cube only and cannot dig diagonally. ii) At each step, the behavior of the worm is somewhat like a Bohr's electron. It can be found inside exactly one of the sub-cubes and can't be found in a state of transition, i.e., halfway from one sub-cube into another. The worm's motive is to dig a tunnel through all 26 sub-cubes finally reach the central sub-cube. In how many ways can it do this?