As already mentioned in other chapters there are 136 244 essentially different bimagic 8x8-squares (SQ).
Here we want to demonstrate how many SQ are there for the each magic series appearing in the row with number 64.
But each of the 136 244 SQ consists of two bimagic series with number 64 inside.
Dependent on the normalization concept one of these series occurs as row and the other as column with 64.
To avoid the normalization influence we have to give rows and columns the same importance.
Therefore we here consider each square in the normalized aspect and in its reflection with respect to the main diagonal.
We speak of the 'main aspects' of SQ. The sets with main aspects are twice as large as the sets with SQ.
The vertical scale of the diagram is logarithmic.
Row64 is the row with number 64 inside. LN is the lowest integer in row64. All data of the distributions of SQ and SM (XLSX)
Number of main aspects of bimagic 8x8-squares per row64 (Click on the image to enlarge it.)
Questions:
- Why has the number of SQ a vertical gap from 158 to 853?
- What is special about the 30 rows with more than 853 SQ (hot spots)?
- Why is there a lower and a higher horizontal branch in the diagram (except of the 30 hot spots)?
- Why are there more SQ (except of the 30 hot spots) if LN = 1, 4, 7, 10, 13 (= 1 modulo 3)?
A first analysis of the 30 hot spot series was done by Francis Gaspalou.
In our statistics there are 32288 squares which have got a hot spot series.
There are 14840 SQ where the row and the column with number 64 is a hot spot series.
See XLSX file of Francis Gaspalou.
There are 17448 SQ where the row or the column with number 64 is a hot spot series.
The 'hot spot' question: Why are there so many SQ with hot spot series?
The 'two branches"-problem and the '1 modulo 3'-problem are still unsolved.
The two horizontal branches can be seen more clearly in the distribution of semi-bimagic 8x8-squares.
Number of semi-bimagic squares (SM) per row64 (Click on the image to enlarge it.)
Image of the distribution of all SQ with respect to the indices of row64 and col64 (Click on the image to enlarge it.)