Friedrich-Schiller-Universität Jena
Fakultät für Mathematik und Informatik
Institut für Angewandte Mathematik

Frequency Distributions of Points of No-Three-in-Line Configurations

The following table gives you a feeling of the frequency distribution for the selected points of large no-three-in-line configurations. Here you see the upper left quarter of the grid of all solutions for n=20 in symmetry class rot4. In each of the 20x20 locations of the quarter the points of the 2x541 different configurations are summed up and these values are decreased by the expected number of these points (54.1) for each location.

 -54 -54 -45 -40 -42 -38 -20 -29 -31   4  -6   1  32 -41  45  23  65  65  99  68

 -54 -54 -48 -44 -44 -41 -29 -34 -19 -17  22  23  27  49  57  31  37  23  38  79

 -45 -48 -34 -39 -31 -25 -16 -16   4  -1   9  20  60  42  14  32   8  35  19  14

 -40 -44 -39 -40 -26 -14  20  23  10  32  34  17  10   6 -34  28   7  -3  25  30

 -42 -44 -31 -26 -40  -2  33  12  17   7  29   6  23  20  29  25   3 -13 -10   6

 -38 -41 -25 -14  -2 -30  32  23  21  19  18  16   2  -7   9  14  -3 -10  -7  25

 -20 -29 -16  20  33  32 -34  23   2  32  12  11  23  -7  -7 -31  -7 -24   4 -15

 -29 -34 -16  23  12  23  23  -8  26  26 -13 -16  -4   0   6  13  -6 -21   2  -5

 -31 -19   4  10  17  21   2  26 -34  11  -5   4  -2  13 -19   2  -6   9   9 -10

   4 -17  -1  32   7  19  32  26  11 -30  -5  -5 -23   4  -2 -24 -21   7  -1 -11

  -6  22   9  34  29  18  12 -13  -5  -5 -40   1 -16  10  -8  -7  11  -7 -21 -16

   1  23  20  17   6  16  11 -16   4  -5   1 -50   1   9  -7   7 -11   1 -16 -10

  32  27  60  10  23   2  23  -4  -2 -23 -16   1 -14   0 -17 -28 -15 -19 -28 -10

 -41  49  42   6  20  -7  -7   0  13   4  10   9   0 -48  -9   2  -5 -16  -6 -14

  45  57  14 -34  29   9  -7   6 -19  -2  -8  -7 -17  -9 -30  -6   1   3  -8 -15

  23  31  32  28  25  14 -31  13   2 -24  -7   7 -28   2  -6 -22  -8 -16 -16 -17

  65  37   8   7   3  -3  -7  -6  -6 -21  11 -11 -15  -5   1  -8  -8 -15 -24  -1

  65  23  35  -3 -13 -10 -24 -21   9   7  -7   1 -19 -16   3 -16 -15  56 -18 -35

  99  38  19  25 -10  -7   4   2   9  -1 -21 -16 -28  -6  -8 -16 -24 -18   2 -41

  68  79  14  30   6  25 -15  -5 -10 -11 -16 -10 -10 -14 -15 -17  -1 -35 -41 -20

So negative values, like in the corner, indicate frequencies lower than the average and positive values, like in the middle of the sides, are higher than the average of 54.1 .
This picture grayscale pict

shows the whole grid where these data are maped into grayscale values with high frequencies/densities are bright and low densities are dark represented. The rotational symmetry around the center is evident. So the frequency/density distribution is roughly a function of the euclidean distance to the grid center. Here are the according pictures for n=38 grayscale pict

and n=36

.
And

shows for n=42 the frequencies calculated of those 746 configurations.

Everything above was done and known until 1997-01-02.
Since 2026 Thomas Prellberg, Queen Mary University of London, UK, has achieved much progress in discovering previously unknown no-three-in-line solutions/configurations and pushed the limits for grid sizes e.g. in symmetry class rot4 further. So he could generate for all 10441 configurations in symmetry class rot4 of the 56 x 56 grid this RGB-color picture corresponding frequency map aka. "heatmap" in April 2026. And finally for n=57 in symmetry class rct4 he constructed this 2 dimensional frequency distribution from all 833 configurations on 31th May 2026.
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Achim Flammenkamp
2026-06-12