Progress in the No-Three-in-Line Problem, II

Abstract

First solutions to the no-three-in-line problem for n=33, 37, 39, 41, 43, 45, 48, 50, 52 and for certain symmetry classes for n=26, 42, 44 are presented. All configurations with n 16 have been generated. Further, the significance of a new symmetry class for configurations which are almost as symmetric as those in class rot4 is demonstrated.1998 Academic Press, Inc.Key words and phrases: no-three-in-line; lattice configurations; branch-and-bound technique.We consider a square n_n grid in the Euclidean plane. The task is to mark as many of the intersection points as possible under the restriction that no three of the marked points lie in a straight line. One can obviously mark at most 2n points. The problem of finding for which n this value is reached is known as the no-three-in-line problem. For a short concise overview of the historical development, see [1], [2].Usually the solutions are partitioned into the following eight symmetry classes: These having full symmetry (abbreviation, full), having only rotational symmetry (half rotation, rot2; quarter rotation, rot4), having only diagonal reflection symmetry (in one long-diagonal, dia1; in both longdiagonals, dia2), those having only orthogonal reflection symmetry (in one mid-perpendicular, ort1; in both mid-perpendiculars, ort2) and those having no symmetry (abbreviated, iden).

A note on the no-three-in-line problem on a torus

“…The obvious upper bound is 2n since one can put at most two points in each row. This bound is attained for many small cases, for details see [4] and [5]. In [7] the authors give a probabilistic argument to support the conjecture that for a large n this limit is unattainable.…”

Arcs in $$\mathbb Z^2_{2p}$$ Z 2 p 2

“…This question has been widely studied (see Erdös 1951;Flammenkamp 1992Flammenkamp , 1998Guy and Kelly 1968;Hall et al 1975), but is still not resolved.…”