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Topic: Path through a 3x3x3 grid
Replies: 2   Last Post: Feb 27, 2013 12:48 PM

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The Last Danish Pastry

Posts: 739
Registered: 12/13/04
Re: Path through a 3x3x3 grid
Posted: Feb 27, 2013 12:48 PM
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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 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.

> > 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 line-intersections?  (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 line-intersections 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



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