SODLS Construction Methods

Odd

Small Orders

Backtracking

Orders 5, 7, 9, 11, 13, 15 SODLS for associative magic squares can be made by a bactracking program.
Examples:

Composite

Orders 25 (5 5s), 35 (5 7s) (7 5s), 45 (5 9s) (9 5s), 49 (7 7s), 55 (5 11s) (11 5s), etc.
For some orders, the magic squares can be associative, pandiagonal, or ultramagic.

Example, order 25 (5 5s) associative:

Odd - Not Multiple of 3

Shift Left 2

Orders 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, etc.

Let N be the order of the SODLS Q.

Make other associative by swapping rows/columns of ultramagic.

Example, orders 5, 7, 11, 13 ultramagic:

Odd - Multiple of 3

x 5s+4

Orders 39 (7 5s+4), 69 (13 5s+4), 129 (25 5s+4), 159 (31 5s+4), 249 (49 5s+4), etc.

Let N be the order of the SODLS Q, nB = x, nA = 5, n = N-4.

Example: order 39 (7 5s+4), N = 39, nB = 7, n = N-4 = 35.

x ks+1

Orders 21 (5 4s+1), 33 (4 8s+1), 51 (5 10s+1), 57 (4 14s+1), 69 (4 17s+1), 93 (4 23s+1), etc.

Let N be the order of the SODLS Q, nB = x, nA = k, n = N-1.

Example: order 21, N = 21, nB = 5, nA = 4, n = N-1 = 20.

x ks+k/2

Orders 27 (5 5s+2), 57 (11 5s+2), 87 (17 5s+2), 93 (5 17s+8), 573 (15 37s+18), etc.

See method for x ks+k/2 for singly even orders.

x ks+c

For c=4, 8, 10: orders 39 (5 7s+4), 69 (5 13s+4), 87 (11 7s+10), 93 (5 17s+ 8), etc.

See method and example for x ks+c for order 24.

Even

Composite

Orders 16 (4 4s), 20 (4 5s) (5 4s), 28 (4 7s) (7 4s), 32 (4 8s) (8 4s), 36 (4 9s) (9 4s),
40 (4 10s) (5 8s) (8 5s) (10 4s), 50 (5 10s) (10 5s), etc.
For some orders, the magic squares can be associative, pandiagonal, or ultramagic.

Example, order 16 (4 4s) pandiagonal:

Doubly Even

Small Orders

Backtracking

Orders 4, 8, 12, 16 SODLS for associative magic squares can be made by a bactracking program. SODLS for pandiagonal magic squares can be made from some of these by Planck's "A-D method".
Example for orders 4, 8, 12:

Order 24

x 5s+4

General method: Let N be the order of the SODLS Q, nB = x, n = N-4.

Example order 24: N = 24, nB = 4, n = N-4 = 20.

x ks+c

General method: Let N be the order of the SODLS Q, nB = x, nA = k, n = N-c.

Example order 24: N = 24, nB = 5, nA = 4, c = 4, n = N-4 = 20.

Singly Even

diagonal, center 4

Orders 18, 22, 26, 30.

Fill the center with a 4×4 SODLS of the middle numbers.
Consider the rest of the SODLS as divided into:

In each of the corners:

Fill the rest of the diagonals with numbers either all smaller or all bigger than the middle numbers.

A backtracking program was used to make SODLS with this specification.

x 5s+4

Orders 54 (10 5s+4), 74 (14 5s+4), 94 (18 5s+4), 114 (22 5s+4), 134 (26 5s+4), etc.

See method and example for x 5s+4 for order 24.

x 8s+2

Orders 34 (4 8s+2), 42 (5 8s+2), 58 (7 8s+2), 66 (8 8s+2), 82 (10 8s+2), 90 (11 8s+2), etc.

Let N be the order of the SODLS Q, nB = x, n = N-2.

Example: order 34, (4 8s+2), N = 34, nB = 4, n = N-2 = 32.

x ks+1

Some orders: 46 (5 9s+1), 50 (7 7s+1), 66 (5 13s+1), 78 (7 11s+1), 86 (5 17s+1), etc.

See method and examples for x ks+1 for odd multiple of 3.

x ks+k/2

Orders 22 (4 5s+2), 38 (5 7s+3), 62 (12 5s+2), 66 (9 7s+3), 82 (16 5s+2), 94 (13 7s+3), etc.

Let N be the order of the SODLS Q, nB = x, nA = k, n = N-k/2

Example: order 22, (4 5s+2), N = 22, nB = 4, nA = 5, n = N-5/2 = 22-2 = 20.

x ks+c

For c=4, 8, 10: orders 54 (5 10s+4) (11 4s+10), 62 (13 4s+10), 74 (5 14s+4), 78 (17 4s+10), 86 (19 4s+10), 94 (5 18s+4), 98 (11 8s+10), 250 (11 22s+8), etc.

See method and example for x ks+c for order 24.