
Re: Path through a 3x3x3 grid
Posted:
Feb 23, 2013 3:41 PM


On Feb 20, 3:46 pm, Clive Tooth <cli...@gmail.com> wrote:
> 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 planetoplane 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? > > Suggested notation: > > 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". > > So a path might begin: > (0, 0, 2) > (2, 2, 0) > ...
I think there are a few dozen solutions  here are a couple:
http://www.youtube.com/watch?v=oxlJsUBOEjM Solution 2, steps=13
0: ( 0, 0, 0) 1: ( 2, 0, 2) 2: ( 2, 0, 1) 3: ( 2, 3, 2) 4: (1, 0, 2) 5: ( 1, 2, 0) 6: ( 1, 0, 0) 7: ( 3, 2, 2) 8: ( 0, 1, 2) 9: ( 0, 2, 1) 10: ( 0, 2, 2) 11: ( 2, 2, 0) 12: ( 0, 0, 2) 13: ( 2, 2, 2)
In this path the sections 05 and 611 are congruent  one being a 180 degree
rotation of the other about the line {x=1, y=1}.
The last two line segments 1113 lie in the plane x=y.
===================================== http://www.youtube.com/watch?v=Na8LHCM82tM Solution 21, steps=13
0: ( 0, 0, 1) 1: ( 0, 2, 1) 2: ( 3, 1, 2) 3: ( 0, 2, 2) 4: ( 0, 1, 1) 5: ( 3, 2, 2) 6: ( 1, 0, 2) 7: ( 1, 2, 0) 8: (1, 2, 0) 9: ( 2, 2, 3) 10: ( 2, 2, 0) 11: (2, 0, 4) 12: ( 3, 0, 1) 13: ( 1, 2, 1)
 Clive

