What is the smallest number of circles k that pass through all the points of an n x n square grid of points?

geometry/circles.through.grid New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

What is the smallest number of circles k that pass through all the
points of an n x n square grid of points?

[ Editor: problem statement reconstructed from a Web page titled
"Circles Through Points". ]

Show solution

The smallest known number of circles k through the points of an n x n
square grid:

   n    k
   2    1    Trivial.
   3    3    Found and proved by Erich Friedman.
   4    3    Found and proved by Erich Friedman.
   5    5    Found and proved by Erich Friedman.
   6    6    Found by Erich Friedman.
   7    8    Found by Joseph W. DeVincentis.
   8    8    Found by Joseph W. DeVincentis.
   9   11    Found by Joseph W. DeVincentis.
  10   11    Found by Joseph W. DeVincentis.
  11   15    Found by Joseph W. DeVincentis.

[ Editor: the Web page showed each configuration as a picture
(cp2.gif ... cp11.gif); the pictures are not in the collected material.
The page names no author or address. ]