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

Examples:

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:

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

Let N be the order of the SODLS Q.

- Fill row 0: for pandiagonal, with natural order; for ultramagic, with "(N-column) % N".
- For other rows, start with the last 2 numbers of the previous row.

Make other associative by swapping rows/columns of ultramagic.

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

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.

- Make a composite SODLS of nB subsquares A, (ultramagic order 5), where nB × 5 = n.

- Add 4 rows and 4 columns after the middle A.
- Let u be the number of the first added row; increase SODLS numbers ≥ u by 4.
- On one diagonal parallel to the main
forward diagonal,
replace numbers on 2 diagonals of subsquare A by u+1, u.

On another parallel diagonal, replace numbers on 2 diagonals of subsquare A by u+3, u+2. - Put the displaced numbers into the corresponding new rows and columns.
- Put an order 4 SODLS with u added to all its numbers at Q[u][u].
- Move the first 2 new rows up 5 rows and move the first 2 new columns left 5 columns.

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

- Make a composite SODLS of 7 subsquares of order 5, (7 × 5 = 35).
- Add 4 new rows at row 20 and 4 new columns at column 20.
- Increase SODLS numbers ≥ 20 by 4.
- On one diagonal parallel to the main
forward diagonal,
replace numbers on 2 diagonals of subsquare A by 21 20.

On another parallel diagonal, replace numbers on 2 diagonals of subsquare A by 23 22. - Put the displaced numbers into the corresponding new rows and columns.

For example,12 14 and 7 9 in the new rows at column 0; 35 36 and 16 17 in the new columns at row 0. - Put an order 4 SODLS with 20 added to all its numbers at Q[20][20].

Note that this order 4 SODLS is split into 4 2×2 pieces by the next step. - Move the new rows 20,21 up to 15,16 and move the new columns 20,21 left to 15,16.

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.

- Make a composite SODLS of nB subsquares A of order nA, where nB x nA = n.

Use natural order main back diagonal for the nA and nB. - Add 1 row and 1 column in the middle for even nB; after the middle A for odd nB.
- Let u be the number of the added row; increase SODLS numbers ≥ u by 1.
- On the main back diagonal, replace each nA subsquare A by numbers from an order nA+1 SODLS.
- In A, replace the number nA by u; add Q[R][R], where R is the first row of A to the other numbers.

Put the row nA numbers in the new row; put the column nA numbers in the new column. - Put number u at the intersection of the new row and new column, Q[u][u].

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

- Make a composite SODLS of 5 subsquares A of order 4, (5 × 4 = 20).
- Add 1 row and 1 column after the third A.
- The number of the added row, (column), u = 12; increase SODLS numbers ≥ 12 by 1.
- On the main back diagonal, replace each 4×4 subsquare A by numbers from an order 5 SODLS.
- In A, replace the number 4 by 12; add Q[R][R], where R is the first row of A to the other numbers.

Add 0 to numbers of the first A, 4 to numbers of the second A, etc.

Put the row 4 numbers in row 12, (3 2 1 0 for the first A);

put the column 4 numbers in column 12, (1 3 0 2 for the first A). - Put number 12 at Q[12][12].

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.

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.

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:

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:

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

- Make a composite SODLS of nB subsquares A, (ultramagic order 5), where nB × 5 = n.
- Add 4 rows and 4 columns in the middle.
- Let u be the number of the first added row; increase SODLS numbers ≥ u by 4.
- On the main
back diagonal
replace numbers on 2 diagonals of each subsquare by u+1, u;

on the main forward diagonal replace numbers on 2 diagonals of each subsquare by u+3, u+2;

The 2 diagonals to use are shown in the example below.

(Note the difference for the forward diagonal versus the back diagonal,

and for the bottom half of the forward diagonal versus the top.) - Move the displaced numbers into the new rows and columns.
- Put an order 4 SODLS with u added to all its numbers at Q[u][u].

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

- Make a composite SODLS of 4 subsquares of order 5, (4 × 5 = 20).
- Add 4 rows and 4 columns in the middle.
- The first added row u = 10; increase SODLS numbers ≥ 10 by 4.
- On the main
back diagonal
replace numbers on 2 diagonals of each subsquare by 11 10.

On the main forward diagonal replace numbers on 2 diagonals of each subsquare by 13 12. - Move the displaced numbers into the new rows and columns.

For example, these are numbers 2 4 14 16 in column 0 of the new rows 10..13;

numbers 1 2 7 8 in row 0 of new columns 10..13. - Put an order 4 SODLS with 10 added to all its numbers at Q[10][10].

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

- Make a composite SODLS of nB subsquares A of order nA, where nB × nA = n.

Use natural order main back diagonal for A; use odd, not multiple of 3, ultramagic for order nB. - Add c rows and c columns after the middle A.
- Let u be the number of the first added row; increase SODLS numbers ≥ u by c.
- On each of c diagonals parallel to the main
forward diagonal,

replace each subsquare A by numbers from an order nA+1 SODLS. - Let U be u for one such diagonal, u+1 for another, etc.
- In A, replace the number nA by U; add Q[R][C], where R is the first row and C is the first column
of A, to the other numbers.

Put the row nA numbers in the new row; put the column nA numbers in the new column. - Put an order c SODLS with u added to all its numbers at Q[u][u].
- Move the first c/2 rows up nA rows and move the first c/2 columns left nA columns.

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

- Make a composite SODLS of 5 subsquares of order 4, (5 × 4 = 20).
- Add 4 rows and 4 columns after the middle 4 starting at Q[12][12].

(Note: 2 of these rows will be moved up and 2 columns moved left later.) - The number of the first added row is 12; increase SODLS numbers ≥ 12 by 4.
- On the each of 4 diagonals parallel to the main
forward diagonal,

replace each 4×4 subsquare A by numbers from an order 5 SODLS. - Let U have the values 12, 13, 14, and 15 for the 4 diagonals.
- In A on the first diagonal, replace the number 4 by 12.

Add Q[0][0] = 0 to the other numbers of the first A, add Q[4][20] = 16 to the other numbers of the second A, and so on.

Put the row 4 numbers in the new row, (3 2 1 0 for the A at Q[0][0]).

Put the column 4 numbers in the new column, (1 3 0 2 for the A at Q[0][0]).

Don't put number A[4][4] anywhere, (it will be in the added order 4 SODLS).

Proceed similarly for the second, third, and fourth diagonals. - Put an order 4 SODLS with 12 added to all its numbers at Q[12][12].

Note that this order 4 SODLS is split into 4 2×2 pieces by the next step. - Move the new rows 12,13 up to 8,9 and move the new columns 12,13 left to 8,9.

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:

- rows left of the 4×4; rows right of the 4×4
- columns above the 4×4; columns below the 4×4
- top left corner, wrap-around diagonals, parallel to the main back diagonal
- bottom right corner, wrap-around diagonals, parallel to the main back diagonal
- top right corner, wrap-around diagonals, parallel to the main forward diagonal
- bottom left corner, wrap-around diagonals, parallel to the main forward diagonal

In each of the corners:

- Fill the main diagonals as shown below.
- Fill each of two diagonals with a middle number, as shown below.

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

- Put the numbers in natural order, increasing or decreasing, with wrap-around.
- In top columns and top left and right diagonals, put small numbers in increasing order from the top; put big numbers in decreasing order.
- In bottom columns and bottom left and right diagonals, put small numbers in decreasing order from the middle; put big numbers in increasing order.
- In left rows, put small numbers in increasing order from the left; put big numbers in decreasing order.
- In right rows, put small numbers in decreasing order from the left; put big numbers in increasing order.
- In top columns, put sequence small, big, small, small.
- In left rows, put sequence small, big, big, small.

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

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.

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.

- Make a composite SODLS of nB subsquares A of order 8, where nB × 8 = n.

Use natural order main back diagonal for A and the order nB. - Add 2 rows and columns in the middle for even nB; after the middle A for odd nB.
- Let u be the number of the first added row; increase SODLS numbers ≥ u by 2.
- Let A' be the subsquare on the main back diagonal immediately following the added rows, columns.
- In the other subsquares with the same numbers as A', replace the first 2 numbers with the new
row numbers;

replace the matching numbers in all the rows of these subsquares. - Move the replaced numbers to the subsquare on the main back diagonal as shown below.

Finally, put the numbers displaced from that subsquare into the new rows, columns. - Replace subsquare A' on the main back diagonal with an order 8+2=10 SODLS, (adding A[0][0]
to all numbers).

Note that the numbers of the order 10 included in the program have been specially permuted to place the correct numbers on the main forward diagonal. - For odd nB, move row u up 8 rows and move column u left 8 columns.

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

- Make a composite SODLS of 4 subsquares of order 8.
- Add 2 rows and columns in the middle.
- The first added row is 16; increase SODLS numbers ≥ 16 by 2.
- Let A' be the subsquare on the main
back diagonal at Q[18][18].

(Note: this subsquare will be replaced below.) - In the other subsquares with the same numbers as A', replace 18 21 with the the new row numbers, 16 17.
- Move the replaced numbers 18 and 21
to the subsquare on the main back diagonal.

Finally, put the displaced numbers into the new rows, columns as shown.

For example, these are numbers 5 4 in column 0 of the new rows 16, 17;

numbers 3 6 in row 0 of new columns 16, 17. - Replace subsquare A' on the main back diagonal with the order 10 SODLS, (adding 16 to all numbers).

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.

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

- Make a composite SODLS of nB subsquares A of order nA, where nB × nA = n.

Use odd, not multiple of 3, ultramagic for A; use natural order main back diagonal for order nB. - Add k/2 rows and columns in the middle for even nB; after the middle A for odd nB.
- Let u be the number of the first added row; increase SODLS numbers ≥ u by k/2.
- Let A' be the subsquare on the main back diagonal immediately preceding the added rows, columns.
- In the other subsquares with the same numbers as A', replace the first k/2 numbers with the new
row numbers;

replace the matching numbers in all the rows of these subsquares. - Move the replaced numbers to the subsquare on the main back diagonal as shown below.

Finally, put the numbers displaced from that subsquare into the new rows, columns. - Replace subsquare A' on the main back diagonal with an order nA+k/2 SODLS, (adding A[0][0] to all numbers).

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

- Make a composite SODLS of 4 subsquares of order 5, (4 × 5 = 20).
- Add 2 rows and columns in the middle.
- The first new row is 10; increase SODLS numbers ≥ 10 by 2.
- Let A' be the subsquare on the main back diagonal at Q[5][5].
- In the other subsquares with the same numbers as A', replace numbers 5, 9 with the new row numbers, 10 11.
- Move the replaced numbers 9 5 to the subsquare on the main back diagonal.

Finally, put the numbers displaced from that subsquare into the new rows, columns.

For example, these are numbers 4 2 in column 0 ot the new rows 10 and 11; numbers 2 1 in row 0 of new columns 10 and 11. - Replace subsquare A' on the main back diagonal with an order 7 SODLS, (adding 5 to all numbers).

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.