There is a (well known) continuous path, made of four straight sections, which passes exactly once through each of 9 points arranged in a square 3x3 array.
Using three of these paths, plus two plane-to-plane straight sections, it is clearly possible to make a continuous path, made of 14 straight sections, which passes exactly once through each of 27 points arranged in a 3x3x3 grid.
However, there is at least one such path made up of only 13 straight sections.
Can you find such a path?
The grid consists of all points (x,y,z) with x, y and z being integers satisfying 0 <= x, y, z <= 2. For a path, give the starting point and all subsequent "target points".