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

Typical Density Distributions of Configurations

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 -20

So 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 .
This picture grayscale pict

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 grayscale pict

and n=36

.
And

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

Achim Flammenkamp
97-01-02 17:00