[seqfan] Looking for help with a sequence: the longest non self - intersecting path through a cube
mcgreg265 at msn.com
Thu Jul 8 03:05:54 CEST 2010
A few years back I proposed the sequence A047838. Dmitry Kamenetsky (Dmitry.Kamenetsky(AT)rsise.anu.edu.au) posted a note within the description of the sequence that spoke to the longest non self-intersecting spiral in an n x n array. In a similar vein, I have been trying to figure out the longest non- self-intersecting path through a cube.
I have attached some preliminary drawings I have made. I can also envision a computer program that could do a brute force exhaustive search of the possibilities:
in a 3x3x3 cube for instance
a. create a permutation 1 through 27 and place the numbers in the cube keeping a sorted track of the position of each number
number level row column
1 3 1 2
2 1 3 3
3 2 2 2
27 3 3 3
b. start at 1 and try to travel to 2, 3, and so on, but check to see if the path has self-intersected
i. a path can only continue if the two consecutive integers are in cells that are adjacent
ii. as the string of visited cells grows, the cells that cannot be visited are created (I tried to show this in the sketches by x-ing out nodes that cannot be visited after the cells on a certain level have been visited)
Unfortunately, I do not have the computer or the software to do such a program.
Any help, or if someone wants to take the ball and run with this, would be appreciated!
spring lake park senior high
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