The Worm and The Cube
There is a 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?
The answer is 0.
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1 solution
The solution is simple. Just color the alternate sub-cubes black and white (or index them as 1 and 0). Then observe that, if the worm is initially on a white corner sub-cube, the central sub-cube will be black. Thus in order to go to the central sub-cube, the worm has to go through an even number of sub-cubes, excluding its initial and final sub-cubes (Since the initial and final sub-cubes are of different colors). However, observe that the worm has to travel through a total of 27 sub-cubes in its journey, which means 25 (odd number) sub-cubes excluding the initial and the final one. This is a contradiction. Hence there is no such possible way.