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Re: Path through a 3x3x3 grid
Posted:
Feb 27, 2013 12:48 PM


On Feb 26, 11:53 pm, James Waldby <n...@valid.invalid> wrote: > On Tue, 26 Feb 2013 13:59:17 0800, clive tooth wrote: > > 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. > > I believe that there are exactly 26 essentially distinct paths through > > the 27 points of the 3x3x3 grid. Here are images of all of them: > > <http://www.flickr.com/photos/lhc_logs/sets/72157632868867524/> > > Which ones are of minimal diameter? Ie which ones minimize the > maximum distance between two lineintersections? (Another > "diameter" is min diameter of a containing sphere, and another > is twice the max distance from center of mass to an intersection, > but maximum distance between two lineintersections is easy to > compute and understand.)
Numbers 10, 12 and 19 all have minimal diameter [in your first sense], sqrt(20).
24 and 25 both have maximal diameter, Route 66. I mean sqrt(66).
 Clive



