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Topic: Path through a 3x3x3 grid
Replies: 3   Last Post: Feb 26, 2013 5:01 PM

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The Last Danish Pastry

Posts: 739
Registered: 12/13/04
Re: Path through a 3x3x3 grid
Posted: Feb 23, 2013 3:41 PM
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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.
>
> 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 0-5 and 6-11 are congruent - one being a 180
degree

rotation of the other about the line {x=1, y=1}.

The last two line segments 11-13 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



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