Naturally, I wondered about the same problem on an NxN-grid torus. In this case, I can make sense of the winding number of a closed path about a grid square only if it can be shrunk to a point on the torus (i.e., is "null-homotopic"). Or is there a satisfactory definition in other cases? But there are still many null-homotopic closed paths on the torus that would not exist on the square. Is the answer the same, say, in the 8x8 case? —Dan
On Mar 23, 2016, at 2:48 PM, James Propp <jamespropp@gmail.com> wrote:
It might be best if in continuing our discussion of the 8-by-8 case we switched over to the general 2k-by-2k case. That might clarify some aspects of the discussion.
Jim
On Wednesday, March 23, 2016, James Propp <jamespropp@gmail.com> wrote:
Tom,
Let me try again, then.
Start with the "optimal" configuration; it is unique.
Alan's solution is unique only up to symmetry. Or are you referring to the disconnected "path" consisting of several square loops? (That would explain the scare-quotes.)
Consider each region of like-marked cells.
Sure; there are four of them.
If it contains a hole
(considering connectivity by N/S/E/W and not by diagonal), then the circuit is not closed (the used edges are not connected).
I don't understand what you mean here.
Since there are three such regions (with holes), we need to modify at least one cell in each of these three regions to eliminate the hole.
I see four, not three, such regions.
The only modification we can make to the optimal configuration is to *decrease* the winding number of a cell by one.
I still don't follow this. Does anyone else?
120-3=117.
Okay, *that* I understand. :-)
Jim
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