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9.0   Latin Squares (9 x 9)

A Latin Square of order 9 is a 9 x 9 square filled with 9 different symbols, each occurring only once in each row and only once in each column.

Based on this definition 5.524.751.496.156.892.842.531.225.600 ea order 9 Latin Squares can be found (ref. OEIS A002860).

For the construction of order 9 Magic Squares normally only those Latin Squares are used for which the 9 different symbols occur also only once in each of the main diagonals (Latin Diagonal Squares).

9.1   Latin Diagonal Squares (9 x 9)

Based on the definition formulated above 1.835.082.219.864.832.081.920 Latin Diagonal Squares can be found (ref. OEIS A274806).

Consequently, pairs of suitable Latin Diagonal Squares (A, B) should be constructed separately rather than be selected from this massive amount.

9.2   Magic Squares, Natural Numbers

9.2.1 General

(Simple) Magic Square M of order 9 with the integers 1 ... 81 can be written as M = A + 9 * B + [1] where the squares A and B contain only the integers 0, 1, 2, 3, 4, 5, 6, 7 and 8.

Consequently order 9 (Simple) Magic Squares can be based on pairs of Orthogonal (Latin Diagonal) Squares (A, B).

The construction of miscellaneous types Orthogonal (Latin Diagonal) Squares (A, B) has been discussed in detail in the sections listed below:

A few additional types of Orthogonal Squares (A, B) - and resulting Magic Squares - will be discussed in following sections.

9.2.2 Concentric Magic Squares

Order 9 Concentric Magic Squares M can be constructed based on pairs of Orthogonal Concentric Semi-Latin Squares (A, B), as shown below for the symbols {ai, i = 1 ... 9} and {bj, j = 1 ... 9).

(A, B)
 a9, b5 a2, b9 a3, b9 a1, b1 a4, b1 a6, b1 a7, b9 a8, b9 a5, b1 a1, b8 a8, b5 a2, b8 a3, b2 a4, b8 a6, b8 a7, b2 a5, b2 a9, b2 a1, b7 a8, b7 a7, b5 a3, b3 a4, b7 a6, b7 a5, b3 a2, b3 a9, b3 a9, b6 a2, b6 a3, b4 a6, b5 a4, b4 a5, b6 a7, b6 a8, b4 a1, b4 a9, b4 a2, b4 a7, b3 a4, b6 a5, b5 a6, b4 a3, b7 a8, b6 a1, b6 a9, b1 a8, b3 a3, b6 a5, b4 a6, b6 a4, b5 a7, b4 a2, b7 a1, b9 a1, b3 a2, b2 a5, b7 a7, b7 a6, b3 a4, b3 a3, b5 a8, b8 a9, b7 a1, b2 a5, b8 a8, b2 a7, b8 a6, b2 a4, b2 a3, b8 a2, b5 a9, b8 a5, b9 a8, b1 a7, b1 a9, b9 a6, b9 a4, b9 a3, b1 a2, b1 a1, b5

All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:

A
 8 1 2 0 3 5 6 7 4 0 7 1 2 3 5 6 4 8 0 7 6 2 3 5 4 1 8 8 1 2 5 3 4 6 7 0 8 1 6 3 4 5 2 7 0 8 7 2 4 5 3 6 1 0 0 1 4 6 5 3 2 7 8 0 4 7 6 5 3 2 1 8 4 7 6 8 5 3 2 1 0
B
 4 8 8 0 0 0 8 8 0 7 4 7 1 7 7 1 1 1 6 6 4 2 6 6 2 2 2 5 5 3 4 3 5 5 3 3 3 3 2 5 4 3 6 5 5 0 2 5 3 5 4 3 6 8 2 1 6 6 2 2 4 7 6 1 7 1 7 1 1 7 4 7 8 0 0 8 8 8 0 0 4
M = A + 9 * B + 1
 45 74 75 1 4 6 79 80 5 64 44 65 12 67 69 16 14 18 55 62 43 21 58 60 23 20 27 54 47 30 42 31 50 52 35 28 36 29 25 49 41 33 57 53 46 9 26 48 32 51 40 34 56 73 19 11 59 61 24 22 39 71 63 10 68 17 70 15 13 66 38 72 77 8 7 81 78 76 3 2 37

A pair of order 9 Orthogonal Semi-Latin Borders can be constructed for each pair of order 7 Orthogonal Concentric Semi-Latin Squares (A7, B7), as found in Section 7.2.3.

Each pair of order 9 Orthogonal Semi-Latin Borders corresponds with 8 * (7!)2 = 203212800 pairs.

Consequently 2,69685 1016 Concentric Magic Squares can be constructed based on the method described above.

9.2.3 Bordered Magic Squares

Diamond Inlay Order 3

Order 9 Bordered Magic Squares M can be constructed based on pairs of Orthogonal Bordered Semi-Latin Squares (A, B) for miscellaneous types of Center Squares.

The example shown below is based on Center Squares with order 3 Diamond Inlays as discussed in Section 7.2.4 and the symbols {ai, i = 1 ... 9} and {bj, j = 1 ... 9).

(A, B)
 a9, b5 a2, b9 a3, b9 a1, b1 a4, b1 a6, b1 a7, b9 a8, b9 a5, b1 a1, b8 a8, b5 a2, b8 a3, b2 a4, b8 a6, b8 a7, b2 a5, b2 a9, b2 a1, b7 a8, b7 a7, b5 a3, b3 a5, b6 a6, b7 a4, b4 a2, b3 a9, b3 a9, b6 a2, b6 a3, b4 a4, b5 a5, b7 a6, b3 a7, b6 a8, b4 a1, b4 a9, b4 a2, b4 a6, b4 a7, b3 a5, b5 a3, b7 a4, b6 a8, b6 a1, b6 a9, b1 a8, b3 a3, b6 a4, b7 a5, b3 a6, b5 a7, b4 a2, b7 a1, b9 a1, b3 a2, b2 a6, b6 a7, b7 a5, b4 a4, b3 a3, b5 a8, b8 a9, b7 a1, b2 a5, b8 a8, b2 a7, b8 a6, b2 a4, b2 a3, b8 a2, b5 a9, b8 a5, b9 a8, b1 a7, b1 a9, b9 a6, b9 a4, b9 a3, b1 a2, b1 a1, b5

All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:

A
 8 1 2 0 3 5 6 7 4 0 7 1 2 3 5 6 4 8 0 7 6 2 4 5 3 1 8 8 1 2 3 4 5 6 7 0 8 1 5 6 4 2 3 7 0 8 7 2 3 4 5 6 1 0 0 1 5 6 4 3 2 7 8 0 4 7 6 5 3 2 1 8 4 7 6 8 5 3 2 1 0
B
 4 8 8 0 0 0 8 8 0 7 4 7 1 7 7 1 1 1 6 6 4 2 5 6 3 2 2 5 5 3 4 6 2 5 3 3 3 3 3 2 4 6 5 5 5 0 2 5 6 2 4 3 6 8 2 1 5 6 3 2 4 7 6 1 7 1 7 1 1 7 4 7 8 0 0 8 8 8 0 0 4
M = A + 9 * B + 1
 45 74 75 1 4 6 79 80 5 64 44 65 12 67 69 16 14 18 55 62 43 21 50 60 31 20 27 54 47 30 40 59 24 52 35 28 36 29 33 25 41 57 49 53 46 9 26 48 58 23 42 34 56 73 19 11 51 61 32 22 39 71 63 10 68 17 70 15 13 66 38 72 77 8 7 81 78 76 3 2 37

A pair of order 9 Orthogonal Semi-Latin Borders can be constructed for each pair of order 7 Orthogonal Inlaid Semi-Latin Squares (A7, B7), as found in Section 7.2.4.

Each pair of order 9 Orthogonal Semi-Latin Borders corresponds with 8 * (7!)2 = 203212800 pairs.

Consequently 3,783 1016 Bordered Magic Squares with order 3 Diamond Inlays can be constructed based on the method described above.

Diamond Inlay Order 4

The example shown below is based on Center Squares with order 4 Diamond Inlays as discussed in Section 7.2.7 and the symbols {ai, i = 1 ... 9} and {bj, j = 1 ... 9).

(A, B)
 a9,b5 a2,b9 a3,b9 a1,b1 a4,b1 a6,b1 a7,b9 a8,b9 a5,b1 a1,b8 a8,b4 a7,b7 a5,b4 a4,b8 a3,b2 a2,b2 a6,b8 a9,b2 a1,b7 a2,b8 a7,b2 a5,b2 a4,b6 a6,b3 a8,b7 a3,b7 a9,b3 a9,b6 a2,b7 a3,b4 a4,b3 a5,b7 a7,b4 a8,b5 a6,b5 a1,b4 a9,b4 a8,b6 a6,b6 a7,b5 a5,b5 a3,b5 a4,b4 a2,b4 a1,b6 a9,b1 a4,b5 a2,b5 a3,b6 a5,b3 a6,b7 a7,b6 a8,b3 a1,b9 a1,b3 a7,b3 a2,b3 a4,b7 a6,b4 a5,b8 a3,b8 a8,b2 a9,b7 a1,b2 a4,b2 a8,b8 a7,b8 a6,b2 a5,b6 a3,b3 a2,b6 a9,b8 a5,b9 a8,b1 a7,b1 a9,b9 a6,b9 a4,b9 a3,b1 a2,b1 a1,b5

All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:

A
 8 1 2 0 3 5 6 7 4 0 7 6 4 3 2 1 5 8 0 1 6 4 3 5 7 2 8 8 1 2 3 4 6 7 5 0 8 7 5 6 4 2 3 1 0 8 3 1 2 4 5 6 7 0 0 6 1 3 5 4 2 7 8 0 3 7 6 5 4 2 1 8 4 7 6 8 5 3 2 1 0
B
 4 8 8 0 0 0 8 8 0 7 3 6 3 7 1 1 7 1 6 7 1 1 5 2 6 6 2 5 6 3 2 6 3 4 4 3 3 5 5 4 4 4 3 3 5 0 4 4 5 2 6 5 2 8 2 2 2 6 3 7 7 1 6 1 1 7 7 1 5 2 5 7 8 0 0 8 8 8 0 0 4
M = A + 9 * B + 1
 45 74 75 1 4 6 79 80 5 64 35 61 32 67 12 11 69 18 55 65 16 14 49 24 62 57 27 54 56 30 22 59 34 44 42 28 36 53 51 43 41 39 31 29 46 9 40 38 48 23 60 52 26 73 19 25 20 58 33 68 66 17 63 10 13 71 70 15 50 21 47 72 77 8 7 81 78 76 3 2 37

A pair of order 9 Orthogonal Semi-Latin Borders can be constructed for each pair of order 7 Orthogonal Inlaid Semi-Latin Squares (A7, B7), as found in Section 7.2.7.

Each pair of order 9 Orthogonal Semi-Latin Borders corresponds with 8 * (7!)2 = 203212800 pairs.

Consequently 2 * 224 * 8 * (7!)2 = 9,104 1010 Bordered Magic Squares with order 4 Diamond Inlays can be constructed based on the method described above.

9.2.4 Bordered Magic Squares

Overlapping Sub Squares Order 4

Another order 9 Bordered Magic Squares M, which can be constructed based on pairs of Orthogonal Bordered Semi-Latin Squares (A, B), and the symbols {ai, i = 1 ... 9} and {bj, j = 1 ... 9), is shown below.

The Center Square is Associated and contains 2 ea other overlapping order 4 Simple Magic Squares.

(A, B)
 a9, b5 a2, b9 a3, b9 a1, b1 a4, b1 a6, b1 a7, b9 a8, b9 a5, b1 a1, b8 a7, b7 a2, b7 a6, b3 a5, b3 a8, b6 a4, b2 a3, b7 a9, b2 a1, b7 a7, b2 a6, b6 a4, b8 a3, b4 a2, b5 a5, b8 a8, b2 a9, b3 a9, b6 a3, b6 a8, b4 a2, b2 a7, b8 a5, b4 a6, b5 a4, b6 a1, b4 a9, b4 a3, b5 a4, b3 a8, b7 a5, b5 a2, b3 a6, b7 a7, b5 a1, b6 a9, b1 a6, b4 a4, b5 a5, b6 a3, b2 a8, b8 a2, b6 a7, b4 a1, b9 a1, b3 a2, b8 a5, b2 a8, b5 a7, b6 a6, b2 a4, b4 a3, b8 a9, b7 a1, b2 a7, b3 a6, b8 a2, b4 a5, b7 a4, b7 a8, b3 a3, b3 a9, b8 a5, b9 a8, b1 a7, b1 a9, b9 a6, b9 a4, b9 a3, b1 a2, b1 a1, b5

All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:

A
 8 1 2 0 3 5 6 7 4 0 6 1 5 4 7 3 2 8 0 6 5 3 2 1 4 7 8 8 2 7 1 6 4 5 3 0 8 2 3 7 4 1 5 6 0 8 5 3 4 2 7 1 6 0 0 1 4 7 6 5 3 2 8 0 6 5 1 4 3 7 2 8 4 7 6 8 5 3 2 1 0
B
 4 8 8 0 0 0 8 8 0 7 6 6 2 2 5 1 6 1 6 1 5 7 3 4 7 1 2 5 5 3 1 7 3 4 5 3 3 4 2 6 4 2 6 4 5 0 3 4 5 1 7 5 3 8 2 7 1 4 5 1 3 7 6 1 2 7 3 6 6 2 2 7 8 0 0 8 8 8 0 0 4
M = A + 9 * B + 1
 45 74 75 1 4 6 79 80 5 64 61 56 24 23 53 13 57 18 55 16 51 67 30 38 68 17 27 54 48 35 11 70 32 42 49 28 36 39 22 62 41 20 60 43 46 9 33 40 50 12 71 47 34 73 19 65 14 44 52 15 31 66 63 10 25 69 29 59 58 26 21 72 77 8 7 81 78 76 3 2 37

A pair of order 9 Orthogonal Semi-Latin Borders can be constructed for each pair of order 7 Orthogonal Semi-Latin Squares (A7, B7) with Overlapping Sub Squares, as found in Section 7.2.6.

Each pair of order 9 Orthogonal Semi-Latin Borders corresponds with 8 * (7!)2 = 203212800 pairs.

Consequently 8,89584 1012 Bordered Magic Squares with Overlapping Sub Squares can be constructed based on the method described above.

9.2.5 Composed Magic Squares

Associated Border

Order 9 Magic Squares M, composed out of two order 3 Semi Magic, two order 4 Magic Center Squares and an Associated Border, can be constructed based on pairs of Orthogonal Inlaid Semi-Latin Squares (A, B).

The example shown below is based on order 3 and 4 Latin Magic Sub Squares - as discussed in Section 2.0 and Section 4.2.1 - for the symbols {ai, i = 1 ... 9} and {bj, j = 1 ... 9}.

(A, B)
 a1,b5 a4,b9 a8,b9 a7,b9 a1,b1 a7,b1 a8,b1 a4,b1 a5,b9 a9,b4 a2,b2 a3,b5 a5,b6 a6,b3 a6,b8 a3,b7 a2,b4 a9,b6 a9,b8 a5,b3 a6,b6 a2,b5 a3,b2 a3,b4 a2,b8 a6,b7 a9,b2 a9,b7 a6,b5 a5,b2 a3,b3 a2,b6 a2,b7 a6,b4 a3,b8 a9,b3 a9,b1 a3,b6 a2,b3 a6,b2 a5,b5 a4,b8 a8,b7 a7,b4 a1,b9 a1,b7 a4,b2 a7,b6 a8,b3 a8,b4 a7,b7 a5,b8 a4,b5 a1,b3 a1,b8 a7,b3 a8,b2 a4,b6 a7,b8 a8,b5 a4,b4 a5,b7 a1,b2 a1,b4 a8,b6 a4,b3 a7,b2 a4,b7 a5,b4 a7,b5 a8,b8 a1,b6 a5,b1 a6,b9 a2,b9 a3,b9 a9,b9 a3,b1 a2,b1 a6,b1 a9,b5

All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:

A
 0 3 7 6 0 6 7 3 4 8 1 2 4 5 5 2 1 8 8 4 5 1 2 2 1 5 8 8 5 4 2 1 1 5 2 8 8 2 1 5 4 3 7 6 0 0 3 6 7 7 6 4 3 0 0 6 7 3 6 7 3 4 0 0 7 3 6 3 4 6 7 0 4 5 1 2 8 2 1 5 8
Sa
 12 8 16 20
B
 4 8 8 8 0 0 0 0 8 3 1 4 5 2 7 6 3 5 7 2 5 4 1 3 7 6 1 6 4 1 2 5 6 3 7 2 0 5 2 1 4 7 6 3 8 6 1 5 2 3 6 7 4 2 7 2 1 5 7 4 3 6 1 3 5 2 1 6 3 4 7 5 0 8 8 8 8 0 0 0 4
Sb
 12 16 8 20
M = A + 9 * B + 1
 37 76 80 79 1 7 8 4 77 36 11 39 50 24 69 57 29 54 72 23 51 38 12 30 65 60 18 63 42 14 21 47 56 33 66 27 9 48 20 15 41 67 62 34 73 55 13 52 26 35 61 68 40 19 64 25 17 49 70 44 31 59 10 28 53 22 16 58 32 43 71 46 5 78 74 75 81 3 2 6 45
Sm
 124 155 91 204

The balanced series {0, 1, 2, 3, 4 ... 8} have been split into two unbalanced sub series:

{1, 2, 5}, {3, 4, 6, 7} and a pair {0, 8}

which have been used for the construction of four Magic Sub Squares and the associated border.

Attachment 9.25.1 shows a few suitable sets (4 ea) of order 3/4 sub series for the integers 0 ... 8.

Attachment 9.25.2 shows the resulting order 9 Inlaid Magic Squares with Associated Border.

Based on the method described above 12 * 12 * (3!)2 = 5184 squares can be constructed for each set of order 3/4 series shown in Attachment 9.25.1.

Each square shown in Attachment 9.25.2 corresponds with (12 * 72) * n42 * (3!)2 squares, which can be obtained by permutations within the border and/or selecting other aspects of the order 3 and 4 Sub Squares.

9.2.6 Composed Magic Squares

Pan Magic Sub Squares Order 4 and 5

Order 9 Composed Magic Squares, as discussed in detail in Section 9.7.2, can be constructed based on pairs of Orthogonal Composed Semi-Latin Squares (A, B) for miscellaneous types of Sub Squares.

The example shown below is based on order 4 and 5 Latin Diagonal Pan Magic Sub Squares, as discussed in resp. Section 4.2.2 and Section 5.2.2, and the symbols {ai, i = 1 ... 9} and {bj, j = 1 ... 9).

(A, B)
 a4, b4 a1, b6 a9, b1 a6, b9 a8, b6 a7, b1 a2, b5 a5, b9 a3, b4 a6, b1 a9, b9 a1, b4 a4, b6 a2, b9 a5, b4 a3, b6 a8, b1 a7, b5 a1, b9 a4, b1 a6, b6 a9, b4 a3, b1 a8, b5 a7, b9 a2, b4 a5, b6 a9, b6 a6, b4 a4, b9 a1, b1 a7, b4 a2, b6 a5, b1 a3, b5 a8, b9 a4, b8 a1, b2 a9, b3 a6, b7 a5, b5 a3, b9 a8, b4 a7, b6 a2, b1 a6, b3 a5, b7 a4, b5 a1, b8 a9, b2 a3, b3 a2, b7 a8, b2 a7, b8 a1, b5 a9, b8 a6, b2 a5, b3 a4, b7 a7, b2 a8, b8 a2, b3 a3, b7 a5, b2 a4, b3 a1, b7 a9, b5 a6, b8 a2, b8 a3, b2 a7, b7 a8, b3 a9, b7 a6, b5 a5, b8 a4, b2 a1, b3 a8, b7 a7, b3 a3, b8 a2, b2

All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:

A
 3 0 8 5 7 6 1 4 2 5 8 0 3 1 4 2 7 6 0 3 5 8 2 7 6 1 4 8 5 3 0 6 1 4 2 7 3 0 8 5 4 2 7 6 1 5 4 3 0 8 2 1 7 6 0 8 5 4 3 6 7 1 2 4 3 0 8 5 1 2 6 7 8 5 4 3 0 7 6 2 1
B
 3 5 0 8 5 0 4 8 3 0 8 3 5 8 3 5 0 4 8 0 5 3 0 4 8 3 5 5 3 8 0 3 5 0 4 8 7 1 2 6 4 8 3 5 0 2 6 4 7 1 2 6 1 7 4 7 1 2 6 1 7 2 6 1 2 6 4 7 7 1 6 2 6 4 7 1 2 6 2 7 1
M = A + 9 * B + 1
 31 46 9 78 53 7 38 77 30 6 81 28 49 74 32 48 8 43 73 4 51 36 3 44 79 29 50 54 33 76 1 34 47 5 39 80 67 10 27 60 41 75 35 52 2 24 59 40 64 18 21 56 17 70 37 72 15 23 58 16 71 20 57 14 22 55 45 69 65 12 61 26 63 42 68 13 19 62 25 66 11

Based on the distribution of the integers 0, 1 ... 8 over the four Sub Squares shown above, 589824 (= 162 * 482) Semi Latin Squares, with Latin Rows and - Diagonals can be constructed.

This collection will result in 21.743.271.936 (= 1282 * 11522) Magic Squares composed of Pan Magic Sub Squares, for the integer distribution shown above.

The distribution of the integers 0, 1 ... 8, is defined by the applied Balanced Series {0, 3, 4, 5, 8}, {0, 3, 5, 8}, {1, 2, 4, 6, 7} and {1, 2, 6, 7} for the 4 Latin Diagonal Pan Magic Sub Squares.

Other distributions can be based on the (unique) order 4 and 5 Balanced Series for the integers 0 ... 8, as shown in Attachment 9.2.5.

9.2.7 Associated Magic Squares

Sub Squares Order 4 and 5

Alternatively Order 9 Associated Composed Magic Squares can be constructed, based on pairs of Orthogonal Associated Composed Semi-Latin Squares (A, B) for miscellaneous types of Sub Squares. The example shown below is based on

• 2 ea order 4 Latin Diagonal Simple Magic Sub Squares as discussed in Section 4.2.1 and
• 2 ea order 5 Latin Diagonal Pan    Magic Sub Squares as discussed in Section 5.2.2,

and the symbols {ai, i = 1 ... 9} and {bj, j = 1 ... 9).

(A, B)
 a1, b9 a7, b6 a8, b3 a4, b2 a4, b8 a7, b4 a8, b5 a5, b7 a1, b1 a4, b3 a8, b2 a7, b9 a1, b6 a8, b7 a5, b1 a1, b8 a4, b4 a7, b5 a7, b2 a1, b3 a4, b6 a8, b9 a1, b4 a4, b5 a7, b7 a8, b1 a5, b8 a8, b6 a4, b9 a1, b2 a7, b3 a7, b1 a8, b8 a5, b4 a1, b5 a4, b7 a2, b6 a3, b2 a6, b9 a9, b3 a5, b5 a1, b7 a4, b1 a7, b8 a8, b4 a6, b3 a9, b5 a5, b6 a2, b2 a3, b9 a3, b7 a9, b8 a6, b1 a2, b4 a5, b2 a2, b9 a3, b3 a6, b5 a9, b6 a2, b1 a6, b4 a9, b7 a3, b8 a3, b5 a6, b6 a9, b2 a5, b9 a2, b3 a9, b4 a3, b1 a2, b8 a6, b7 a9, b9 a5, b3 a2, b5 a3, b6 a6, b2 a6, b8 a2, b7 a3, b4 a9, b1

All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:

A
 0 6 7 3 3 6 7 4 0 3 7 6 0 7 4 0 3 6 6 0 3 7 0 3 6 7 4 7 3 0 6 6 7 4 0 3 1 2 5 8 4 0 3 6 7 5 8 4 1 2 2 8 5 1 4 1 2 5 8 1 5 8 2 2 5 8 4 1 8 2 1 5 8 4 1 2 5 5 1 2 8
B
 8 5 2 1 7 3 4 6 0 2 1 8 5 6 0 7 3 4 1 2 5 8 3 4 6 0 7 5 8 1 2 0 7 3 4 6 5 1 8 2 4 6 0 7 3 2 4 5 1 8 6 7 0 3 1 8 2 4 5 0 3 6 7 4 5 1 8 2 3 0 7 6 8 2 4 5 1 7 6 3 0
M = 9 * A + B + 1
 9 60 66 29 35 58 68 43 1 30 65 63 6 70 37 8 31 59 56 3 33 72 4 32 61 64 44 69 36 2 57 55 71 40 5 34 15 20 54 75 41 7 28 62 67 48 77 42 11 27 25 80 46 13 38 18 21 50 78 10 49 79 26 23 51 74 45 12 76 19 17 52 81 39 14 24 47 53 16 22 73

Attachment 9.2.6 contains 1008 ea order 9 Associated Semi-Latin Squares - with Latin Rows and Diagonals - based on the properties mentioned above (ref. CompLat9b).

Based on this collection 508032 (= 2 * 254016) order 9 Associated Magic Squares can be generated (ref. CnstrSqrs9a).

9.2.8 Associated Magic Squares

Diamond Inlays order 4 and 5

Order 9 Associated Magic Squares M, with order 4 and 5 Diamond Inlays, can be constructed based on pairs of Orthogonal Associated Semi-Latin Squares (A, B), as shown below for the symbols {ai, i = 1 ... 9} and {bj, j = 1 ... 9).

(A, B)
 a2,b1 a5,b4 a7,b8 a1,b6 a8,b7 a6,b2 a4,b6 a3,b9 a9,b2 a9,b7 a7,b6 a5,b1 a8,b5 a2,b8 a1,b1 a3,b5 a4,b7 a6,b5 a6,b6 a5,b7 a1,b2 a3,b1 a4,b9 a7,b7 a8,b1 a9,b5 a2,b7 a2,b4 a1,b9 a7,b3 a6,b7 a8,b4 a3,b6 a9,b3 a5,b8 a4,b1 a7,b2 a8,b8 a9,b6 a4,b2 a5,b5 a6,b8 a1,b4 a2,b2 a3,b8 a6,b9 a5,b2 a1,b7 a7,b4 a2,b6 a4,b3 a3,b7 a9,b1 a8,b6 a8,b3 a1,b5 a2,b9 a3,b3 a6,b1 a7,b9 a9,b8 a5,b3 a4,b4 a4,b5 a6,b3 a7,b5 a9,b9 a8,b2 a2,b5 a5,b9 a3,b4 a1,b3 a1,b8 a7,b1 a6,b4 a4,b8 a2,b3 a9,b4 a3,b2 a5,b6 a8,b9

All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:

A
 1 4 6 0 7 5 3 2 8 8 6 4 7 1 0 2 3 5 5 4 0 2 3 6 7 8 1 1 0 6 5 7 2 8 4 3 6 7 8 3 4 5 0 1 2 5 4 0 6 1 3 2 8 7 7 0 1 2 5 6 8 4 3 3 5 6 8 7 1 4 2 0 0 6 5 3 1 8 2 4 7
B = R(A)
 0 3 7 5 6 1 5 8 1 6 5 0 4 7 0 4 6 4 5 6 1 0 8 6 0 4 6 3 8 2 6 3 5 2 7 0 1 7 5 1 4 7 3 1 7 8 1 6 3 5 2 6 0 5 2 4 8 2 0 8 7 2 3 4 2 4 8 1 4 8 3 2 7 0 3 7 2 3 1 5 8
M = 9 * A + B +1
 10 40 62 6 70 47 33 27 74 79 60 37 68 17 1 23 34 50 51 43 2 19 36 61 64 77 16 13 9 57 52 67 24 75 44 28 56 71 78 29 41 53 4 11 26 54 38 7 58 15 30 25 73 69 66 5 18 21 46 63 80 39 31 32 48 59 81 65 14 45 22 3 8 55 49 35 12 76 20 42 72

With procedure SemiLat9b 1090 pairs of Orthogonal Semi-Latin Associated Squares (A, R(A)) with order 4 and 5 Diamond Inlays could be found, of which 76 are shown in Attachment 9.2.9.

Attachment 9.2.10 shows the resulting order 9 Associated Magic Squares with order 4 and 5 Diamond Inlays.

Note:
Although with a comparable routine 1000 Latin Diagonal Associated Squares with order 4 and 5 Diamond Inlays could be found, this collection did not result in any orthogonal pair (A, B).

9.2.9 Composed Magic Squares

Sub Squares Order 3 and 6

Magic squares composed of order 3 (Semi) Magic Sub Squares and order 6 Associated or Pan Magic Sub Squares can be obtained by transformation of Associated Compact Pan Magic Squares (ref. Section 9.6.7).

Order 9 Associated Compact Pan Magic Squares, can be constructed based on pairs of Orthogonal Semi-Latin Squares (A, B) composed of order 3 Latin Sub Squares, based on the symbols {ai, i = 1 ... 9} and {bj, j = 1 ... 9).

(A, B)
 a9, b9 a5, b4 a1, b2 a2, b9 a7, b4 a6, b2 a4, b9 a3, b4 a8, b2 a4, b5 a3, b3 a8, b7 a9, b5 a5, b3 a1, b7 a2, b5 a7, b3 a6, b7 a2, b1 a7, b8 a6, b6 a4, b1 a3, b8 a8, b6 a9, b1 a5, b8 a1, b6 a9, b2 a5, b9 a1, b4 a2, b2 a7, b9 a6, b4 a4, b2 a3, b9 a8, b4 a4, b7 a3, b5 a8, b3 a9, b7 a5, b5 a1, b3 a2, b7 a7, b5 a6, b3 a2, b6 a7, b1 a6, b8 a4, b6 a3, b1 a8, b8 a9, b6 a5, b1 a1, b8 a9, b4 a5, b2 a1, b9 a2, b4 a7, b2 a6, b9 a4, b4 a3, b2 a8, b9 a4, b3 a3, b7 a8, b5 a9, b3 a5, b7 a1, b5 a2, b3 a7, b7 a6, b5 a2, b8 a7, b6 a6, b1 a4, b8 a3, b6 a8, b1 a9, b8 a5, b6 a1, b1

All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:

A
 8 4 0 1 6 5 3 2 7 3 2 7 8 4 0 1 6 5 1 6 5 3 2 7 8 4 0 8 4 0 1 6 5 3 2 7 3 2 7 8 4 0 1 6 5 1 6 5 3 2 7 8 4 0 8 4 0 1 6 5 3 2 7 3 2 7 8 4 0 1 6 5 1 6 5 3 2 7 8 4 0
B = T(A)
 8 3 1 8 3 1 8 3 1 4 2 6 4 2 6 4 2 6 0 7 5 0 7 5 0 7 5 1 8 3 1 8 3 1 8 3 6 4 2 6 4 2 6 4 2 5 0 7 5 0 7 5 0 7 3 1 8 3 1 8 3 1 8 2 6 4 2 6 4 2 6 4 7 5 0 7 5 0 7 5 0
M = A + 9 * B + 1
 81 32 10 74 34 15 76 30 17 40 21 62 45 23 55 38 25 60 2 70 51 4 66 53 9 68 46 18 77 28 11 79 33 13 75 35 58 39 26 63 41 19 56 43 24 47 7 69 49 3 71 54 5 64 36 14 73 29 16 78 31 12 80 22 57 44 27 59 37 20 61 42 65 52 6 67 48 8 72 50 1

The corresponding transformations are illustrated below:

Compact, Associated, Pan Magic
 81 32 10 74 34 15 76 30 17 40 21 62 45 23 55 38 25 60 2 70 51 4 66 53 9 68 46 18 77 28 11 79 33 13 75 35 58 39 26 63 41 19 56 43 24 47 7 69 49 3 71 54 5 64 36 14 73 29 16 78 31 12 80 22 57 44 27 59 37 20 61 42 65 52 6 67 48 8 72 50 1
Composed, Associated Corner
 11 79 33 13 75 35 18 77 28 63 41 19 56 43 24 58 39 26 49 3 71 54 5 64 47 7 69 29 16 78 31 12 80 36 14 73 27 59 37 20 61 42 22 57 44 67 48 8 72 50 1 65 52 6 74 34 15 76 30 17 81 32 10 45 23 55 38 25 60 40 21 62 4 66 53 9 68 46 2 70 51
Composed, Pan Magic Corner
 11 79 33 13 75 35 18 77 28 63 41 19 56 43 24 58 39 26 49 3 71 54 5 64 47 7 69 29 16 78 31 12 80 73 14 36 27 59 37 20 61 42 44 57 22 67 48 8 72 50 1 6 52 65 74 34 15 9 68 46 51 70 2 45 23 55 38 25 60 62 21 40 4 66 53 76 30 17 10 32 81

Attachment 9.2.7 contains 96 ea order 9 (Semi) Latin Associated Compact Pan Magic Squares - with Latin Rows and Diagonals - based on the properties mentioned above (ref. CompLat9c).

Based on this collection 4096 order 9 Associated Compact Pan Magic Squares can be generated, which can be transformed to Composed Magic Squares with orde 6 Associated or Pan Magic Corner Squares (ref. CnstrSqrs9a).

9.2.10 Composed Magic Squares

Overlapping Sub Squares Order 3 and 7

Order 9 Composed Magic Squares, as discussed in detail in Section 9.6.6, can be constructed based on pairs of Orthogonal Composed Semi-Latin Squares (A, B) for miscellaneous types of order 7 Sub Squares.

The example shown below is based on (transformed) order 7 Latin Diagonal Ultra Magic Sub Squares, as discussed in Section 7.2.2, and the symbols {ai, i = 1 ... 9} and {bj, j = 1 ... 9).

(A, B)
 a9, b5 a5, b1 a1, b9 a4, b1 a1, b2 a1, b3 a9, b6 a7, b9 a8, b9 a5, b9 a1, b5 a9, b1 a6, b9 a9, b8 a9, b7 a1, b4 a3, b1 a2, b1 a1, b1 a9, b9 a5, b5 a3, b3 a6, b2 a8, b6 a2, b4 a4, b8 a7, b7 a7, b1 a3, b9 a6, b4 a8, b8 a2, b7 a4, b5 a7, b3 a5, b2 a3, b6 a8, b1 a2, b9 a2, b3 a4, b2 a7, b6 a5, b4 a3, b8 a6, b7 a8, b5 a9, b4 a1, b6 a7, b8 a5, b7 a3, b5 a6, b3 a8, b2 a2, b6 a4, b4 a1, b7 a9, b3 a3, b2 a6, b6 a8, b4 a2, b8 a4, b7 a7, b5 a5, b3 a1, b8 a9, b2 a8, b7 a2, b5 a4, b3 a7, b2 a5, b6 a3, b4 a6, b8 a4, b9 a6, b1 a4, b6 a7, b4 a5, b8 a3, b7 a6, b5 a8, b3 a2, b2

All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:

A
 8 4 0 3 0 0 8 6 7 4 0 8 5 8 8 0 2 1 0 8 4 2 5 7 1 3 6 6 2 5 7 1 3 6 4 2 7 1 1 3 6 4 2 5 7 8 0 6 4 2 5 7 1 3 0 8 2 5 7 1 3 6 4 0 8 7 1 3 6 4 2 5 3 5 3 6 4 2 5 7 1
B
 4 0 8 0 1 2 5 8 8 8 4 0 8 7 6 3 0 0 0 8 4 2 1 5 3 7 6 0 8 3 7 6 4 2 1 5 0 8 2 1 5 3 7 6 4 3 5 7 6 4 2 1 5 3 6 2 1 5 3 7 6 4 2 7 1 6 4 2 1 5 3 7 8 0 5 3 7 6 4 2 1
M =A + 9 * B + 1
 45 5 73 4 10 19 54 79 80 77 37 9 78 72 63 28 3 2 1 81 41 21 15 53 29 67 61 7 75 33 71 56 40 25 14 48 8 74 20 13 52 32 66 60 44 36 46 70 59 39 24 17 47 31 55 27 12 51 35 65 58 43 23 64 18 62 38 22 16 50 30 69 76 6 49 34 68 57 42 26 11

The Frames (A', B'), composed out of a 3 x 3 Semi-Magic Corner Square and two 2 x 6 Symmetric Magic Rectangles, are Non-Latin but Orthogonal.

Attachment 9.2.8 contains 33 (unique) Non-Latin Orthogonal Frames, based on the order 3 Semi Magic Sub Square as used in the example shown above (CompLat9d).

Each Frame corresponds with 8 * 2 * 22 * (6!)2 = 33.177.600 Non-Latin Orthogonal Frames (A', B'), suitable for pairs of order 7 Orthogonal Latin Diagonal Magic Sub Squares (A7, B7).

Taking the limiting condition of the second diagonal (blue) into account 8 * 587.759.616 Composed Magic Squares with Overlapping Sub Squares can be constructed (transformed Ultra Magic Corner Squares).

9.2.11 Evaluation of the Results

Following table compares a few enumeration results for order 9 Magic Squares with the results based on the construction methods described above:

 Type Enumerated Source Constructed Base Borders 7,25063 1011)* 203.212.800 Semi-Latin Comp 4/5, Pm Sub 1,56 1012)** 21.743.271.936 Semi-Latin Comp 4/5, Associated 508.032 Semi-Latin

)*  Based on integers 1 ... 16 and 66 ... 81
)** Based on predefined integers per Pan Magic Sub Square

The constructability by means of Orthogonal (Latin Diagonal) Squares can be considered as an additional property.

9.3   Magic Squares, Prime Numbers

9.3.1 Simple Magic Squares

When the elements {ai, i = 1 ... 9} and {bj, j = 1 ... 9) of a valid pair of Orthogonal Latin Diagonal Squares (A, B) complies with following condition:

• mij = ai + bj = prime for i = 1 ... 9 and j = 1 ... 9 (correlated)

the resulting square M = A + B will be an order 9 Prime Number Simple Magic Square.

Sa = 2543
 1 3 33 79 163 379 453 583 849 79 163 379 453 583 849 1 3 33 453 583 849 1 3 33 79 163 379 33 1 3 379 79 163 849 453 583 379 79 163 849 453 583 33 1 3 849 453 583 33 1 3 379 79 163 3 33 1 163 379 79 583 849 453 163 379 79 583 849 453 3 33 1 583 849 453 3 33 1 163 379 79
Sb = 3708
 4 10 70 190 238 280 598 1030 1288 598 1030 1288 4 10 70 190 238 280 190 238 280 598 1030 1288 4 10 70 10 70 4 238 280 190 1030 1288 598 1030 1288 598 10 70 4 238 280 190 238 280 190 1030 1288 598 10 70 4 70 4 10 280 190 238 1288 598 1030 1288 598 1030 70 4 10 280 190 238 280 190 238 1288 598 1030 70 4 10
Sm = 6251
 5 13 103 269 401 659 1051 1613 2137 677 1193 1667 457 593 919 191 241 313 643 821 1129 599 1033 1321 83 173 449 43 71 7 617 359 353 1879 1741 1181 1409 1367 761 859 523 587 271 281 193 1087 733 773 1063 1289 601 389 149 167 73 37 11 443 569 317 1871 1447 1483 1451 977 1109 653 853 463 283 223 239 863 1039 691 1291 631 1031 233 383 89

Attachment 9.3 contains miscellaneous correlated magic series {ai, i = 1 ... 9} and {bj, j = 1 ... 9).

Attachment 9.3.1 contains the resulting Prime Number Simple Magic Squares for miscellaneous Magic Sums (Sm).

Each square shown corresponds with numerous Prime Number Pan Magic Squares.

9.3.21 Symmetric Magic Squares

Prime Number Symmetric Magic Squares require that the series {ai, i = 1 ... 9} and {bj, j = 1 ... 9) of a valid pair of Orthogonal Latin Diagonal Squares (A, B) comply with following conditions:

• mij = ai + bj = prime for i = 1 ... 9 and j = 1 ... 9 (correlated)
• a1 + a9 = 2 * a5 and b1 + b9 = 2 * b5                 (balanced)
a2 + a8 = 2 * a5 and b2 + b8 = 2 * b5
a3 + a7 = 2 * a5 and b3 + b7 = 2 * b5
a4 + a6 = 2 * a5 and b4 + b6 = 2 * b5

Such order 9 Correlated Balanced Magic Series, resulting in Prime Number Magic Squares, have not yet been found.

9.4   Summary

The obtained results regarding the order 9 Latin - and related Magic Squares, as deducted and discussed in previous sections, are summarized in following table:

 Attachment Subject Subroutine Composed Semi-Latin Associated Magic Squares (Semi) Latin Associated Compact Pan Magic Squares (Unique) Orthogonal Frames, Non-Lattin Semi-Latin Squares, Associated, Diamond inLay order 4 and 5 Magic Squares, Associated, Diamond inLay order 4 and 5 - - - Correlated Magic Series - Prime Number Simple Magic Squares - - -

Comparable methods as described above, can be used to construct order 10 (Semi) Latin - and related (Composed) Magic Squares, which will be described in following sections.