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Size of maximal subset of points of n X n grid such that no two points are at the same distance.
8

%I #54 Jul 16 2022 03:46:14

%S 1,2,3,4,5,6,7,7,8,9,10,10,11,11,12,13,13

%N Size of maximal subset of points of n X n grid such that no two points are at the same distance.

%C Inverse function to A193838, which is the main entry for this problem.

%D R. K. Guy, Unsolved Problems in Number Theory, Third Edition, Springer New York, 2004, F2, 367-368.

%D Keith F. Lynch, Posting to Math Fun Mailing List, Apr 02 2016.

%H P. Erdős and R. K. Guy, <a href="http://dx.doi.org/10.5169/seals-27359">Distinct distances between lattice points</a>, Elemente der Mathematik 25 (1970), 121-123.

%H Wolfram Demonstration Project, <a href="http://demonstrations.wolfram.com/NoRepeatedDistances/">No Repeated Distances</a>

%H A. Zimmermann, <a href="http://azspcs.com/Contest/PointPacking/FinalReport">Al Zimmermann's Programming Contests: Point Packing.</a> (Oct 10, 2009).

%e From _Ehit Dinesh Agarwal_, May 28 2020: (Start)

%e An 11 X 11 grid has only two subsets of size 10, barring symmetry: {(0,0), (0,2), (0,3), (0,7), (1,10), (5,4), (6,0), (8,7), (9,8), (10, 10)} and {(0,0), (0,6), (0,7), (1,2), (4,10), (7,8), (7,10), (9,2), (9,6), (10,5)}.

%e A 12 x 13 grid has only four subsets of size 11, barring symmetry: {(0,0), (0,1), (0,9), (0,12), (2,0), (5,3), (6,12), (7,0), (8,4), (10,10), (11,11)}. (End)

%Y Cf. A193838, A335232 (number of solutions).

%K nonn,hard,more,nice

%O 1,2

%A _N. J. A. Sloane_, Apr 14 2016

%E a(11)-a(13) corrected and extended by _Ehit Dinesh Agarwal_, May 28 2020

%E a(14)-a(16) from _Bert Dobbelaere_, Sep 20 2020

%E a(17) from _Fausto A. C. Cariboni_, Jul 16 2022