# Symlateral Magic Squares ## Description

These squares have complementary numbers that:

• in each octant are all symmetric about the center of their row or column
• in each octant boundary are all symmetric about the middle

See symlateral.

These can be transformed from/to concentric magic squares by repeating transform 2. Swap all pairs of rows on the top side of the square in increasing sequence, e.g., for orders 9 and 10, swap rows (1 2), (1 3), (1 4), (2 3), (2 4), (3 4).

## Construction

SymlateralSquares makes these squares.

## Odd Order Squares

### Example Squares: Order 5, 7 The center 3x3 square can be any center of an odd concentric magic square of the same order including any aspect of the Lo Shu square.

### Bones of Above Squares Each order n bones is conceptually made from the n-2 bones by:

• expanding the square at the middle row and middle column to make space for two new rows and columns
• filling the new rows and columns with the remaining complementary pairs such that:
• a pair is in each of the main diagonals
• a pair is in each of the center row and center column
• a pair is in each top and bottom row
• a pair is in each left and right column
• the new rows and columns have the magic sum. ## Even Order Squares

### Example Squares: Order 6, 8 The center 4x4 square can be any center of an even concentric magic square of the same order including any aspect of the 880 order 4 squares.

### Bones of Above Squares These are conceptually made as for odd order bones, but reserving two middle rows and columns here, (see octant boundaries).

Note 1: ½ has been omitted from all the bones numbers displayed here. Therefore, to convert these displayed bones numbers to the actual square numbers:

• add n²/2 + 1 to the positive numbers
• add n²/2 to the negative numbers

Note 2: Here the outside cells of the middle rows and middle columns are not middle symmetric.