
COMMENTS

A generalization of the wellknown "Nine Dots Problem", where the regular axisaligned bounding box (RAABB:=[0, n] X [0, n] X [0, n]) has been declared.
In particular, if we loosen the constraint on the allowed AABB, covering paths characterized by a shorter linklength can be found, such as 5 <= a(2) <= 6, where the aforementioned upper bound has been discovered by Koki Goma in August 2021, providing the selfcrossing covering path (0,0,0)(2,2,0)(1/2,1/2,3/2)(2,1,0)(0,1,0)(0,1,1)(0,0,1).
Moreover, the above pattern suggests different uncrossing covering paths of the same linklength, such as (1,0,0)(0,0,0)(2,2,2)(1/2,1,1/2)(1/2,1,3/2)(1,1,0)(1,1,0) and also the (selfcrossing) covering path (1,0,0)(0,0,0)(0,1,0)(3/2,1,3/2)(1/2,1,1/2)(1/2,1,3/2)(1,1,0) which is entirely contained inside a box of 1.5 X 2 X 2 units^3 but which does not match the RAABB. (End)
