There are 21,446 order 7 bent diagonal magic squares. In addition to the rows, columns, and main diagonals, all the bent diagonals add up to the magic constant.
They were made with a FormulaOne Compiler program. See details.
The squares file is in sorted Frénicle form, (program Frenicle).
Francis Gaspalou analysed the properties arising from the magic square sums and bent diagonal sums. See emails.
Miguel Angel Amela of Argentina also analysed the properties and wrote a program in Quick Basic. See emails.
The square center number is always 25. There are many interesting cell patterns, for example:
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These can be centred on any cell in the middle row or middle colum, (13 patterns in all). |
Fourteen of the squares are pandiagonal.
There are 4497 complement pair pattern groups. The biggest 2 groups each have 544 squares. The smallest 1305 groups each have 2, (complementary), squares.
There are 16 group sizes.
The number of squares in the group is shown below each group number.
Pairs that are not in the same row, column, or main diagonal are shown in color.
A total of 4330 squares have 4 cross-corner pairs like the squares of groups 1 and 3.
Only 4 pairs in each of these patterns are not center symmetric.
While not a sufficient confirmation that these are all the bent diagonal squares, there is some comfort from checking the necessary condition that all the complementary squares are present:
Similarly, for the pandiagonal squares, (extract the squares using program CopySquaresByType).
Heinz, H.D. and J.R. Hendricks "Magic Square Lexicon: Illustrated"
http://recmath.org/Magic%20Squares/Downloads/Lexicon_Sample.pdf,
Magic Square Lexicon: Illustrated, Complementary pair patterns, page 24.
K2 Software Corp. "FormulaOne Compiler" http://www.f1compiler.com/
White, S.H. "Order 5 Magic Square Groups" https://budshaw.ca/Groups5.html.