I have always wondered if 3-orchard could be tackled topologically. For example, suppose primitive A-B-C denotes that distinct points A, B, and C occur on a line in that order. We could formulate some constraints related to betweenness, e.g. not A-B-A not A-B-B if (A-B-C and A-C-D) then A-B-D and planarity(?) and 3-orchard constraints if A-B-C then not B-C-D The object would be to devise a handful of axioms of this sort that would be sufficient to show whether or not a pile of A-B-C statements is consistent with a 3-orchard configuration without having to assign geometric locations to the points.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Ed Pegg Jr Sent: Wednesday, December 06, 2017 7:08 PM To: math-fun Subject: [math-fun] 4-orchard problem
I recently tackled the 3-orchard problem. Nice pictures are at http://oeis.org/A003035 .
The 4-orchard problem is proving harder because the solutions don't seem as strong. In particular, the solutions for 22-25 points seem easily beatable. I've posted values and a picture at https://math.stackexchange.com/questions/2554709/the-4-orchard- problem
If anyone can improve these solutions, please let me know.
--Ed Pegg Jr _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun