Graphic charts for the world record Q27

Distribution of solutions, by queen on first row

The first two charts show the number of solutions, depending on the position of the queen on the first row; on the left the values for Q26 (world record of 2009), on the right side the new values of 2017. Note that the Q26 values use a different scale.

Distribution by queen on first row, for Q26 and Q27.

Discussion of the similarity of distribution, for different sizes

The two charts look very similar. This changes when we combine them into one chart, using the same scale for both board sizes. In order to focus on the similarity, I made another chart with relative values in a double sense: horizontally the relative position of the first queen, scaled to 0 to 300 (inclusive), and vertically such that the maximum (mid value) is the reference point 1.0 (or 100 percent). This gives curves which are nearly identical; I added a line for Q21, but also this curve has almost no deviations.

I found a simple formula for the curve; it is:

r(x) = 0.2 + 0.4 * (1 + sin (x / n - 0.72))

One can use it to estimate the number of solutions for a given first queen. I added this formulal also in the chart.

On the left side, an absolute comparision of Q26 and Q27; on the right, double relative: horizontally the relative position of the first queen, vertically the ration of the actual number to the maximum for the same board size which is in the mid. Additional, values for Q21 and the curve of the formula.

As the deviation is so small, I show an amplification. It contains the delta of the values to the formula, for all three board sizes. To avoid negative numbers, the values are shifted by 0.01; i.e. value 0.01 means: no deviation.

Distribution again, by the pair of queens on first and second row

More details are contained in the next images: the number of solutions, by the first pair of queens. The left chart for Q26 (of 2009), the chart on the right side for the new Q27 numbers of 2017.

The gaps on the left side are caused by the fact that cells (1,n-1), (1,n), (1,n+1) are forbidden for a queen, if there is a queen on (0,n). This is obvious; but I do not know what causes the crimpy lines, with regular up and down; it would be interesting to get an explanation!

A hint for the connection to the first charts above (nr by first queen): each line here corresponds to a bar there, and the size of the bar there is the area under the line here.

The charts are not symmetric, as only the lines for the left queens on the first row, i.e. between 0 and n/2, are contained. If we add the other lines, we get a chart which contains the mirror line for every line. They meet on the middle axis, x=13.