I'm still confused by your argument, but rather than focus on specific passages that I don't follow I'd rather take a global approach. What would you say is the general situation for a 2k-by-2k grid? What is the minimal number of grid-squares that don't achieve maximal winding number? Is it k/2+1, or k-1, or something else? Jim On Wednesday, March 23, 2016, Tom Rokicki <rokicki@gmail.com> wrote:
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.)
I mean optimal configuration that does not require connectedness of the surrounding path; that is, the nested rectangle configuration.
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).
All the cells with 1 in them form a border around the cells with 2 in them, which form a border around the cells with 3, which form a border around the cells with 4. A chess king cannot get from a cell with 1 to a cell with 3 without stepping on a 2; the 3's (and 4's) live in a hole in the surface of 2's. Thus, the edges included in the solution that are outside the 2's cannot be connected to the edges that are inside the 2's. If the chess king can get from 2 to 4 without stepping on 3, that means that the edges on the inside of the 2's and those on the outside are connected.
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 regions are those with 1's, 2's, and 3's. The region with 4's does not contain a hole; there are no interior edges that need to be connected to the outside edges.
The only modification we can make to the optimal configuration is to *decrease* the winding number of a cell by one.
All this confusion is my fault for not defining what I meant by "optimal configuration".
-tom _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun