The following table gives you a feeling of the density distribution for the markers 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 markers of the 2x541 different configurations are summed up and these values are decreased by the expected number of markers (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 -20So negative values, like in the corner, indicate densities lower than the average and positive values, like in the middle of the sides, are higher than the average of 54.1 .
shows the whole grid
where these data are maped into grayscale values with high densities are bright
and low densities are dark represented. The rotational symmetry around the center is evident. So the density distribution
is roughly a function of the euclidean distance to the grid center. Here are
the according pictures for n=38 
and n=36 
.
shows for n=42 the density 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 rot 4 further. So he could generate for all 10441 configurations in symmetry class rot4 of the 56 x 56 grid this
corresponding frequency map aka. "heatmap" in April 2026.
Achim Flammenkamp