SODLS Construction Methods

Self-orthogonal Diagonal Latin Squares

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Odd

Small Orders

Backtracking

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

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: .xlsx

Centers 5, 7, 9, 11

Made for orders from 17 to 45:

See method and examples for center 4 for even orders.

Example, order 17: .xlsx

Example, order 31: .xlsx

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: .xlsx

x ks+1

Some orders: 17 (4 4s+1), 29 (4 7s+1) (7 4s+1), 49 (4 12s+1) (12 4s+1), etc.

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

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: .xlsx

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: .xlsx

Example, order 33: .xlsx

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.
Example, order 27: .xlsx

x ks+c

For c=4, 8, 10, 12, 14, 16: 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.
Example, order 39: .xlsx

Note: The c values can be increased by two's if needed for bigger orders. For orders up to 25,000 only order 22,623 required a value bigger than 10.

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: .xlsx

Centers 4, 8, 10, 12

Made for orders from 16 to 48:

Make from Latin square: LS.xlsx SODLS.xlsx

Consider the Latin square as divided into:

Fill diagonals in the N-4 sub-square. Four are filled with the biggest numbers;
the rest start with randomly chosen numbers subject to constrained results:

Similarly, fill the columns above and the rows left of the order 4 SODLS.

Convert the Latin square to a SODLS:

The middle 4 rows and/or columns may be permuted as appropriate to make other SODLS. Optionally, the results may be converted to natural \diagonal.

A computer program quickly makes hundreds of thousands for order 18; for order 30, the algorithm has to be tweaked to make a hundred.

Example, order 18: .xlsx nd.xlsx

Example, order 30: .xlsx

Example, order 32: .xlsx

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: .xlsx

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: .xlsx

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: .xlsx

x ks+1

Some orders: 36 (5 7s+1), 56 (5 11s+1), 64 (7 9s+1) (9 7s+1), etc.

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

Singly Even

Center 4

Order 14.

Fill the center with a 4×4 SODLS of the middle numbers.
In each of the corners:

Fill the remaining cells of the first row and use a backtracking program to complete the square.
.xlsx

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.
For order 18, millions are made in a day; for order 30, the algorithm has to be tweaked to make even one. This algorithm will not make SODLS for doubly even orders, because, for example, in the upper left quarter the middle diagonal wraps around to its own transpose position.

Example, order 18: .xlsx

Example, order 30: .xlsx

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: .xlsx

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: .xlsx

Example, order 38: .xlsx

x ks+c

For c=4, 8, 10, 12, 14, 16: 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 note.

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