--- In [log in to unmask], Sai Emrys <sai@S...> wrote:
> Actually, no - simpler:
> 123
> 4O5
> 678
> There, you have 8 connections, all directly touching the central
> character, all of the same size.
>  - Sai

If you consider all of those nodes to be squares, then, the pairs 
{1,O}, {2,4}, {2,5}, {3,O}, {4,7}, {O,6}, {O,8}, and {5,7}, share 
only a single point -- a corner.  In tiling, tiles aren't considered 
to be neighbors unless they share a non-zero-length portion of 
border.  So at least some of the above pairs can't be 
considered "connected" in the "semantic domain" in which Jeff is 
talking about the "<=6" limitation.


Also, if the connections 
1-2, 2-3, 4-O, O-5, 6-7, 7-8, 
1-4, 4-6, 2-O, O-7, 3-5, and 5-8,
are all the same length as each other, then the connections
1-O, 3-O, O-6, and O-8 cannot all also be that same length.

If they were, then you would have eight non-overlapping equilateral 
triangles all having O as a vertex; triangles 12O, 14O, 23O, 35O, 
46O, 58O, 67O, 78O.

You can't fit more than six (non-overlapping) equilateral triangles 
all around a single shared vertex, without warping the "paper" (the 
writing-surface) so that it is no longer flat.

You could make just 1-O and O-8 be the same length as the other 
eight, leaving 3-O and O-6 to be different; 
or, instead, make 3-O and O-6 to be the same length as the other 
eight, and 1-O and O-8 different.

Or, you could make 1-2, 1-4, 2-3, 3-5, 4-6, 5-8, 6-7, and 7-8 be one 
length, shorter than the lengths of 1-O, 2-O, 3-O, 4-O, O-5, O-6, O-
7, O-8, which could all be equal.

But you can't make all sixteen lengths equal.

Tom H.C. in MI