# 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.