# Order 7 Bent Diagonal Magic Squares ## Bent Diagonal

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). ### Analysis

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. ### Properties

The square center number is always 25. There are many interesting cell patterns, for example:   ``` These can be centred on any cell in the middle row or middle colum, (13 patterns in all). ```

Fourteen of the squares are pandiagonal. ### Groups

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. #### Groups 1 to 4 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.

#### Almost Associative Groups Only 4 pairs is each of these patterns are not center symmetric. ### Complement Check

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:

• make the complementary squares, (program Complement)
• put the original and the complementary squares in sorted Frénicle form, (program Frenicle)
• confirm that the two Frénicle files are the same, e.g., with Command Prompt, command comp or fc

Similarly, for the pandiagonal squares, (extract the squares using program CopySquaresByType). ##### REFERENCES

Heinz, H.D. and J.R. Hendricks "Magic Square Lexicon: Illustrated"