Office Applications and Entertainment, Latin Squares

7.0   Latin Squares (7 x 7)

A Latin Square of order 7 is a 7 x 7 square filled with 7 different symbols, each occurring only once in each row and only once in each column.

Based on this definition 61.479.419.904.000 ea order 7 Latin Squares can be found (ref. OEIS A002860).

For the construction of order 7 Magic Squares normally only those Latin Squares are used for which the 7 different symbols occur also only once in each of the main diagonals (Latin Diagonal Squares).

7.1   Latin Diagonal Squares (7 x 7)

Based on the definition formulated above 862.848.000 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.

7.2   Magic Squares, Natural Numbers

7.2.1 Pan Magic Squares

(Pan) Magic Square M of order 7 with the integers 1 ... 49 can be written as M = A + 7 * B + [1] where the squares A and B contain only the integers 0, 1, 2, 3, 4, 5 and 6.

Consequently order 7 (Pan) Magic Squares can be based on pairs of Orthogonal Latin Diagonal Squares (A, B).

The required Orthogonal Latin Diagonal Squares (A, B) for Pan Magic Squares can be constructed as follows:

1. Fill the first row of square A and square B with the numbers 0, 1, 2, 3, 4, 5 and 6.
   While starting with 0 there are 6! = 720 possible combinations for each square.
   
   
2. Complete square A and B by copying the first row into the following rows of the applicable square,
   according to one of the following schemes:
   
   
   1. A: shift 2 columns to the left / B: shift 2 columns to the right
   2. A: shift 2 columns to the left / B: shift 3 columns to the right
   3. A: shift 2 columns to the left / B: shift 3 columns to the left
   4. A: shift 3 columns to the left / B: shift 2 columns to the right
   5. A: shift 3 columns to the left / B: shift 3 columns to the right
   6. A: shift 3 columns to the left / B: shift 2 columns to the left
      
      

An example of such a pair (A, B) and the resulting Pan Magic Square M is shown below:

A 0 1 2 3 4 5 6 2 3 4 5 6 0 1 4 5 6 0 1 2 3 6 0 1 2 3 4 5 1 2 3 4 5 6 0 3 4 5 6 0 1 2 5 6 0 1 2 3 4

B 0 1 2 3 4 5 6 5 6 0 1 2 3 4 3 4 5 6 0 1 2 1 2 3 4 5 6 0 6 0 1 2 3 4 5 4 5 6 0 1 2 3 2 3 4 5 6 0 1

M = A + 7 * B + 1 1 9 17 25 33 41 49 38 46 5 13 21 22 30 26 34 42 43 2 10 18 14 15 23 31 39 47 6 44 3 11 19 27 35 36 32 40 48 7 8 16 24 20 28 29 37 45 4 12

The Latin Diagonal Squares described above, generated with routine SudSqr7a in 150 seconds (4 x 37,5), are shown in Attachment 7.3.4.

The possible combinations of square A and B described above will result in 6 * 720 * 720/4 = 777.600 unique solutions.

Each of these 777.600 Pan Magic Squares will result in a Class C_n and finally in 777.600 * 49 * 8 = 304.819.200 possible Pan Magic Squares of the 7^th order.

Collections of Pan Magic Squares, based on Latin Diagonal Squares, can be generated very fast with routine CnstrSqrs7a.

7.2.2 Ultra Magic Squares

Attachment 7.4.3 shows the 192 ea order 7 Symmetric Latin Diagonal Squares, with the Broken Diagonals summing to 21 which can be generated with routine SudSqr7b.

Based on the pairs of Orthogonal Symmetric Latin Diagonal Squares (A, B) out of this collection 27648 (= 192 * 144) order 7 Ultra Magic Squares can be constructed (ref. CnstrSqrs7c).

7.2.3 Concentric Magic Squares

Order 7 Concentric Magic Squares M can be constructed based on pairs of Orthogonal Concentric Semi-Latin Squares (A, B), as shown below for the symbols {a_i, i = 1 ... 7} and {b_j, j = 1 ... 7).

(A, B)

a7, b4	a1, b7	a2, b1	a3, b7	a5, b7	a6, b1	a4, b1
a7, b6	a6, b4	a2, b2	a3, b6	a5, b6	a4, b2	a1, b2
a1, b5	a2, b3	a5, b4	a3, b3	a4, b5	a6, b5	a7, b3
a1, b3	a6, b2	a3, b5	a4, b4	a5, b3	a2, b6	a7, b5
a7, b2	a2, b5	a4, b3	a5, b5	a3, b4	a6, b3	a1, b6
a1, b1	a4, b6	a6, b6	a5, b2	a3, b2	a2, b4	a7, b7
a4, b7	a7, b1	a6, b7	a5, b1	a3, b1	a2, b7	a1, b4

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

A 6 0 1 2 4 5 3 6 5 1 2 4 3 0 0 1 4 2 3 5 6 0 5 2 3 4 1 6 6 1 3 4 2 5 0 0 3 5 4 2 1 6 3 6 5 4 2 1 0

B 3 6 0 6 6 0 0 5 3 1 5 5 1 1 4 2 3 2 4 4 2 2 1 4 3 2 5 4 1 4 2 4 3 2 5 0 5 5 1 1 3 6 6 0 6 0 0 6 3

M = A + 7 * B + 1 28 43 2 45 47 6 4 42 27 9 38 40 11 8 29 16 26 17 32 34 21 15 13 31 25 19 37 35 14 30 18 33 24 20 36 1 39 41 12 10 23 49 46 7 48 5 3 44 22

A pair of order 7 Orthogonal Semi-Latin Borders can be constructed for each pair of order 5 Orthogonal Concentric Semi-Latin Squares (A5, B5), as found in Section 5.2.5.

Each pair of order 7 Orthogonal Semi-Latin Borders corresponds with 8 * (5!)² = 115200 pairs.

Consequently 132.710.400 Concentric Magic Squares can be constructed based on the method described above.

7.2.4 Bordered Magic Squares

Inlaid Center Square, Diamond Inlay

Order 7 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 5.2.6 - and the symbols {a_i, i = 1 ... 7} and {b_j, j = 1 ... 7).

(A, B)

a7, b4	a1, b7	a2, b1	a3, b7	a5, b7	a6, b1	a4, b1
a7, b6	a6, b4	a2, b2	a4, b5	a5, b6	a3, b3	a1, b2
a1, b5	a2, b3	a3, b4	a4, b6	a5, b2	a6, b5	a7, b3
a1, b3	a5, b3	a6, b2	a4, b4	a2, b6	a3, b5	a7, b5
a7, b2	a2, b5	a3, b6	a4, b2	a5, b4	a6, b3	a1, b6
a1, b1	a5, b5	a6, b6	a4, b3	a3, b2	a2, b4	a7, b7
a4, b7	a7, b1	a6, b7	a5, b1	a3, b1	a2, b7	a1, b4

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

A 6 0 1 2 4 5 3 6 5 1 3 4 2 0 0 1 2 3 4 5 6 0 4 5 3 1 2 6 6 1 2 3 4 5 0 0 4 5 3 2 1 6 3 6 5 4 2 1 0

B 3 6 0 6 6 0 0 5 3 1 4 5 2 1 4 2 3 5 1 4 2 2 2 1 3 5 4 4 1 4 5 1 3 2 5 0 4 5 2 1 3 6 6 0 6 0 0 6 3

M = A + 7 * B + 1 28 43 2 45 47 6 4 42 27 9 32 40 17 8 29 16 24 39 12 34 21 15 19 13 25 37 31 35 14 30 38 11 26 20 36 1 33 41 18 10 23 49 46 7 48 5 3 44 22

A pair of order 7 Orthogonal Semi-Latin Borders can be constructed for each pair of order 5 Orthogonal Inlaid Semi-Latin Squares (A5, B5), as found in Section 5.2.6.

Each pair of order 7 Orthogonal Semi-Latin Borders corresponds with 8 * (5!)² = 115200 pairs.

Consequently 186.163.200 Bordered Magic Squares with Diamond Inlays can be constructed based on the method described above.

7.2.5 Composed Magic Squares

Overlapping Sub Squares order 3 and 5

Order 7 Composed Magic Squares M 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 3 and 5 each other Overlapping Sub Squares and the symbols {a_i, i = 1 ... 7} and {b_j, j = 1 ... 7).

(A, B)

a7, b4	a4, b1	a1, b7	a2, b1	a3, b7	a5, b7	a6, b1
a4, b7	a1, b4	a7, b1	a6, b7	a5, b1	a3, b1	a2, b7
a1, b1	a7, b7	a4, b4	a2, b3	a3, b6	a5, b2	a6, b5
a7, b6	a1, b2	a6, b2	a3, b5	a5, b3	a4, b6	a2, b4
a7, b2	a1, b6	a5, b5	a6, b6	a2, b2	a3, b4	a4, b3
a1, b3	a7, b5	a2, b6	a5, b4	a4, b5	a6, b3	a3, b2
a1, b5	a7, b3	a3, b3	a4, b2	a6, b4	a2, b5	a5, b6

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

A 6 3 0 1 2 4 5 3 0 6 5 4 2 1 0 6 3 1 2 4 5 6 0 5 2 4 3 1 6 0 4 5 1 2 3 0 6 1 4 3 5 2 0 6 2 3 5 1 4

B 3 0 6 0 6 6 0 6 3 0 6 0 0 6 0 6 3 2 5 1 4 5 1 1 4 2 5 3 1 5 4 5 1 3 2 2 4 5 3 4 2 1 4 2 2 1 3 4 5

M = A + 7 * B + 1 28 4 43 2 45 47 6 46 22 7 48 5 3 44 1 49 25 16 38 12 34 42 8 13 31 19 39 23 14 36 33 41 9 24 18 15 35 37 26 32 20 10 29 21 17 11 27 30 40

Pairs of order 7 Orthogonal Semi-Latin Composed Squares with Overlapping Sub Squares can be constructed for the majority of the 2304 (= 57600/25) pairs of order 5 Orthogonal Simple Latin Diagonal Squares (A5, B5), as found in Section 5.1.

Taking the limiting condition of the second diagonal (top/right to bottom/left) into account 8 * 44032 = 352.256 Composed Magic Squares with Overlapping Sub Squares can be constructed (ref. CmbSqrs7).

7.2.6 Associated Magic Squares

Overlapping Sub Squares order 4

Order 7 Associated Magic Squares M, with order 4 Overlapping Sub Squares, can be constructed based on pairs of Orthogonal Associated Semi-Latin Squares (A, B), as shown below for the symbols {a_i, i = 1 ... 7} and {b_j, j = 1 ... 7).

(A, B)

a6, b6	a1, b6	a5, b2	a4, b2	a7, b5	a3, b1	a2, b6
a6, b1	a5, b5	a3, b7	a2, b3	a1, b4	a4, b7	a7, b1
a2, b5	a7, b3	a1, b1	a6, b7	a4, b3	a5, b4	a3, b5
a2, b4	a3, b2	a7, b6	a4, b4	a1, b2	a5, b6	a6, b4
a5, b3	a3, b4	a4, b5	a2, b1	a7, b7	a1, b5	a6, b3
a1, b7	a4, b1	a7, b4	a6, b5	a5, b1	a3, b3	a2, b7
a6, b2	a5, b7	a1, b3	a4, b6	a3, b6	a7, b2	a2, b2

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

A 5 0 4 3 6 2 1 5 4 2 1 0 3 6 1 6 0 5 3 4 2 1 2 6 3 0 4 5 4 2 3 1 6 0 5 0 3 6 5 4 2 1 5 4 0 3 2 6 1

B 5 5 1 1 4 0 5 0 4 6 2 3 6 0 4 2 0 6 2 3 4 3 1 5 3 1 5 3 2 3 4 0 6 4 2 6 0 3 4 0 2 6 1 6 2 5 5 1 1

M = A + 7 * B + 1 41 36 12 11 35 3 37 6 33 45 16 22 46 7 30 21 1 48 18 26 31 23 10 42 25 8 40 27 19 24 32 2 49 29 20 43 4 28 34 5 17 44 13 47 15 39 38 14 9

Attachment 7.2.6 shows 480 ea order 7 Associated Semi-Latin Squares, with order 4 Overlapping Sub Squares, which can be found with procedure SemiLat7a.

Based on this collection 43.776 order 7 Associated Magic Squares can be constructed.

7.2.7 Associated Magic Squares

Diamond Inlay order 4

Order 7 Associated Magic Squares M, with an order 4 Diamond Inlay, can be constructed based on pairs of Orthogonal Associated Semi-Latin Squares (A, B), as shown below for the symbols {a_i, i = 1 ... 7} and {b_j, j = 1 ... 7).

(A, B)

a7, b3	a6, b6	a4, b3	a3, b7	a2, b1	a1, b1	a5, b7
a1, b7	a6, b1	a4, b1	a3, b5	a5, b2	a7, b6	a2, b6
a1, b6	a2, b3	a3, b2	a4, b6	a6, b3	a7, b4	a5, b4
a7, b5	a5, b5	a6, b4	a4, b4	a2, b4	a3, b3	a1, b3
a3, b4	a1, b4	a2, b5	a4, b2	a5, b6	a6, b5	a7, b2
a6, b2	a1, b2	a3, b6	a5, b3	a4, b7	a2, b7	a7, b1
a3, b1	a7, b7	a6, b7	a5, b1	a4, b5	a2, b2	a1, b5

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

A 6 5 3 2 1 0 4 0 5 3 2 4 6 1 0 1 2 3 5 6 4 6 4 5 3 1 2 0 2 0 1 3 4 5 6 5 0 2 4 3 1 6 2 6 5 4 3 1 0

B 2 5 2 6 0 0 6 6 0 0 4 1 5 5 5 2 1 5 2 3 3 4 4 3 3 3 2 2 3 3 4 1 5 4 1 1 1 5 2 6 6 0 0 6 6 0 4 1 4

M = A + 7 * B + 1 21 41 18 45 2 1 47 43 6 4 31 12 42 37 36 16 10 39 20 28 26 35 33 27 25 23 17 15 24 22 30 11 40 34 14 13 8 38 19 46 44 7 3 49 48 5 32 9 29

Procedure SemiLat7b generated 5072 Associated Semi-Latin Squares with order 4 Diamond Inlays (Latin Rows and Latin Diagonals).

Based on this collection {A} and the collection {B} = {T(A)} 224 pairs of Orthogonal Semi-Latin Squares (A, B) could be found, which are shown in Attachment 7.2.9.

Attachment 7.2.10 shows the resulting order 7 Associated Magic Squares with order 4 Diamond Inlays.

Note:
It can be proven that Associated Magic Squares with order 3 and 4 Diamond Inlays can't be constructed based on Latin or Semi-Latin (Diagonal) Squares.

7.2.8 Associated Magic Squares

Based on the defining equations for Associated Magic Squares 135.168 Associated Latin Diagonal Squares can be found (ref. MgcSqr7j2).

Although pairs of suitable Associated Latin Diagonal Squares (A, B) can be selected from this massive amount, more feasible methods will be discussed below.

Based on Self Orthogonal Latin Squares

A more controllable collection of Associated Magic Squares can be obtained by means of Self Orthogonal Associated Latin Squares, as illustrated below:

A 6 5 4 2 0 3 1 2 4 3 1 6 0 5 3 0 1 6 4 5 2 0 2 5 3 1 4 6 4 1 2 0 5 6 3 1 6 0 5 3 2 4 5 3 6 4 2 1 0

B = T(A) 6 2 3 0 4 1 5 5 4 0 2 1 6 3 4 3 1 5 2 0 6 2 1 6 3 0 5 4 0 6 4 1 5 3 2 3 0 5 4 6 2 1 1 5 2 6 3 4 0

M = A + 7 * B + 1 49 20 26 3 29 11 37 38 33 4 16 14 43 27 32 22 9 42 19 6 45 15 10 48 25 2 40 35 5 44 31 8 41 28 18 23 7 36 34 46 17 12 13 39 21 47 24 30 1

Square A is an Associated Latin Diagonal Square. The Associated Latin Diagonal Square B is the transposed square of A (rows and columns exchanged).

Based on this principle 3072 Associated Magic Squares can be constructed of which 192 Ultra Magic (ref. CnstrSqrs7a2).

Based on Ultra Magic Latin Squares

The Ultra Magic Latin Diagonal Squares, as discussed in Section 7.2.3 above, can be used as a starting point for the construction of Associated Magic Squares based on Latin Diagonal Squares.

Each of the 192 order 7 Ultra Magic Latin Diagonal Squares as shown in Attachment 7.4.3 correspond with 24, occasionally Pan Magic, Associated Latin Diagonal Squares.

Subject Latin Diagonal Squares can be obtained by means of following classical transformations:

• Any line n can be interchanged with line (8 - n). The possible number of transformations is 2³ = 8.
  It should be noted that for each square the 180^o rotated aspect is included in this collection.
• Any permutation can be applied to the lines 1, 2, 3 provided that the same permutation is applied to the lines 7, 6, 5. The possible number of transformations is 3! = 6.
• The resulting number of transformations, excluding the 180^o rotated aspects, is 8/2 * 6 = 24.

As the 192 Ultra Magic Latin Diagonal Squares are not essential different, the total collection results in 1536 different Associated Latin Diagonal Squares, which are shown in Attachment 7.2.7.

Based on these 1536 Associated Latin Diagonal Squares, 221.184 Associated Squares can be constructed, which is eight times the number of Ultra Magic Squares found above (= 8 * 27648).

Based on LDR Base Squares

Walter Trump and Holger Danielsson have developed an elegant method to execute the full enumeration of Associated Magic Squares based on Latin Diagonal Squares, which can be summarized as follows:

• Determine the number (n7) of essential different (LDR-format) Associated Magic Squares which can be constructed based on Latin Diagonal Squares;
• Determine the total number by application of the classical row and column permutations, resulting in 8 * 24 * n7 Associated Magic Squares (Latin Diagonal Square based).

The (essential different) LDR Squares can be constructed based on symmetrical transformations applied on Latin Squares A of LDR Base Squares (A, B) as defined below:

A 0 6 5 4 3 2 1 2 1 0 6 5 4 3 4 3 2 1 0 6 5 6 5 4 3 2 1 0 1 0 6 5 4 3 2 3 2 1 0 6 5 4 5 4 3 2 1 0 6

B 0 5 3 1 6 4 2 3 1 6 4 2 0 5 6 4 2 0 5 3 1 2 0 5 3 1 6 4 5 3 1 6 4 2 0 1 6 4 2 0 5 3 4 2 0 5 3 1 6

LDR Base (A + 7 * B + 1) 1 42 27 12 46 31 16 24 9 43 35 20 5 39 47 32 17 2 36 28 13 21 6 40 25 10 44 29 37 22 14 48 33 18 3 11 45 30 15 7 41 26 34 19 4 38 23 8 49

The Latin Diagonal Squares A and B with main diagonal 0, 1, 2, 3, 4, 5, 6 (highlighted) can be generated with routine MgcSqr7j2, which generated 2816 of subject Latin Diagonal Squares.

Based on this collection 128 LDR Base Squares can be constructed, which are shown in Attachment 7.2.8.

Based on the 48 possible symmetric transformations 6144 Associated Magic Squares can be constructed, of which however only 3072 essential different (ref. CnstrLdr7).

The 3072 essential different Squares result finally in 8 * 24 * 3072 = 589.824 Associated Magic Squares (73.728 unique).

7.2.9 Evaluation of the Results

Following table compares a few enumeration results for order 7 Magic Squares with the results based on the construction methods described above:

Type	Enumerated	Source	Constructed	Base
Pan Magic	8 * 1,21 * 1017	Walter Trump	304.819.200	Latin Diagonal
Ultra Magic	8 * 20.190.684	Walter Trump	27.648	Latin Diagonal
Concentric	4,91 * 1011	-	132.710.400	Semi-Latin
Inlaid	1,76 * 1011	-	186.163.200	Semi-Latin
Composed 3/5	368.640)*	-	352.256	Semi-Latin
Composed 4/4, Symm	4.156.416	-	43.776	Semi-Latin

)*  Pan Magic Corner Squares

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

Note:
The order 5 Orthogonal Semi-Latin and Latin Diagonal Squares (A5, B5), as used in Section 7.2.3 thru 7.2.5 above have been based on the Balanced Series {1, 2, 3, 4, 5}.

Alternatively the Balanced Series {0, 2, 3, 4, 6} or {0, 1, 3, 5, 6} can be used, which will return for each option the same amount of additional Orthogonal - and resulting Magic Squares.

7.3   Magic Squares, Prime Numbers

7.3.1 Simple Magic Squares

When the elements {a_i, i = 1 ... 7} and {b_j, j = 1 ... 7) of a valid pair of Orthogonal Latin Diagonal Squares (A, B) comply with following condition:

• m_ij = a_i + b_j = prime for i = 1 ... 7 and j = 1 ... 7 (correlated)
  

the resulting square M = A + B will be an order 7 Prime Number Simple or Pan Magic Square.

Due to the vast amount of Orthogonal pairs (A, B) returning Pan Magic Squares, only Prime Number Pan Magic Squares will be considered.

7.3.2 Pan Magic Squares

Attachment 7.3, page 1 contains miscellaneous correlated unbalanced series {a_i, i = 1 ... 7} and {b_j, j = 1 ... 7).

Based on Orthogonal Squares (A,B) and correlated (unbalanced) series, the square M = A + B will be an order 7 Prime Number Pan Magic Square.

Sa = 763 1 3 13 31 141 253 321 13 31 141 253 321 1 3 141 253 321 1 3 13 31 321 1 3 13 31 141 253 3 13 31 141 253 321 1 31 141 253 321 1 3 13 253 321 1 3 13 31 141

Sb = 976 10 16 58 100 136 226 430 226 430 10 16 58 100 136 100 136 226 430 10 16 58 16 58 100 136 226 430 10 430 10 16 58 100 136 226 136 226 430 10 16 58 100 58 100 136 226 430 10 16

Sm = 1739 11 19 71 131 277 479 751 239 461 151 269 379 101 139 241 389 547 431 13 29 89 337 59 103 149 257 571 263 433 23 47 199 353 457 227 167 367 683 331 17 61 113 311 421 137 229 443 41 157

Attachment 7.3.2 contains the resulting Prime Number Pan Magic Squares for miscellaneous Magic Sums (Sm).

Each square shown corresponds with 304.819.200 Prime Number Pan Magic Squares.

7.3.3 Ultra Magic Squares

When the elements {a_i, i = 1 ... 7} and {b_j, j = 1 ... 7) of a valid pair of Orthogonal Latin Diagonal Squares (A, B) comply with following conditions:

• m_ij = a_i + b_j = prime for i = 1 ... 7 and j = 1 ... 7 (correlated)
  
• a₁ + a₇ = 2 * a₄ and b₁ + b₇ = 2 * b₄                 (balanced)
  a₂ + a₆ = 2 * a₄ and b₂ + b₆ = 2 * b₄
  a₃ + a₅ = 2 * a₄ and b₃ + b₅ = 2 * b₄

The resulting square M = A + B will be an order 7 Prime Number Ultra Magic Square.

Sa = 54271 9649 8713 7753 6793 5857 7789 7717 7753 6793 5857 7789 7717 9649 8713 5857 7789 7717 9649 8713 7753 6793 7717 9649 8713 7753 6793 5857 7789 8713 7753 6793 5857 7789 7717 9649 6793 5857 7789 7717 9649 8713 7753 7789 7717 9649 8713 7753 6793 5857	Sb = 127050 36300 30330 5970 0 24780 18150 11520 0 24780 18150 11520 36300 30330 5970 11520 36300 30330 5970 0 24780 18150 5970 0 24780 18150 11520 36300 30330 18150 11520 36300 30330 5970 0 24780 30330 5970 0 24780 18150 11520 36300 24780 18150 11520 36300 30330 5970 0
	Sm = 181321 45949 39043 13723 6793 30637 25939 19237 7753 31573 24007 19309 44017 39979 14683 17377 44089 38047 15619 8713 32533 24943 13687 9649 33493 25903 18313 42157 38119 26863 19273 43093 36187 13759 7717 34429 37123 11827 7789 32497 27799 20233 44053 32569 25867 21169 45013 38083 12763 5857

Attachment 7.3, page 2 contains miscellaneous correlated balanced series {a_i, i = 1 ... 7} and {b_j, j = 1 ... 7).

Attachment 7.3.3 contains the resulting Prime Number Ultra Magic Squares for miscellaneous Magic Sums (Sm).

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

7.3.4 Concentric Magic Squares

Based on Orthogonal Semi-Latin Squares (A,B) as applied in Section 7.2.3 and correlated balanced series, the square M = A + B will be an order 7 Prime Number Concentric Magic Square.

Sa = 54271 9649 5857 6793 7717 7789 8713 7753 9649 8713 6793 7717 7789 7753 5857 5857 6793 7789 7717 7753 8713 9649 5857 8713 7717 7753 7789 6793 9649 9649 6793 7753 7789 7717 8713 5857 5857 7753 8713 7789 7717 6793 9649 7753 9649 8713 7789 7717 6793 5857	Sb = 127050 18150 36300 0 36300 36300 0 0 30330 18150 5970 30330 30330 5970 5970 24780 11520 18150 11520 24780 24780 11520 11520 5970 24780 18150 11520 30330 24780 5970 24780 11520 24780 18150 11520 30330 0 30330 30330 5970 5970 18150 36300 36300 0 36300 0 0 36300 18150
	Sm = 181321 27799 42157 6793 44017 44089 8713 7753 39979 26863 12763 38047 38119 13723 11827 30637 18313 25939 19237 32533 33493 21169 17377 14683 32497 25903 19309 37123 34429 15619 31573 19273 32569 25867 20233 36187 5857 38083 39043 13759 13687 24943 45949 44053 9649 45013 7789 7717 43093 24007

Attachment 7.3.4 contains the resulting Prime Number Concentric Magic Squares for miscellaneous Magic Sums (Sm).

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

7.4   Summary

The obtained results regarding the order 7 Latin - and related Magic Squares, as deducted and discussed in previous sections, are summarized in following table:

Attachment	Subject	Subroutine
Attachment 7.3.4	Latin Diagonal Squares, Pan Magic	SudSqr7a
Attachment 7.4.3	Latin Diagonal Squares, Ultra Magic	SudSqr7b
Attachment 7.2.7	Latin Diagonal Squares, Associated	-
Attachment 7.2.6	Semi-Latin Squares, Overlapping Sub Squares	SemiLat7a
Attachment 7.2.9	Semi-Latin Squares, Associated, Diamond inLay order 4	SemiLat7b
Attachment 7.2.10	Magic Squares, Associated, Diamond inLay order 4	CnstrSqrs7c
Attachment 7.2.8	Base LDR    Squares, Associated	CnstrSqrs7c
-	-	-
Attachment 7.3	Correlated Magic Series	-
Attachment 7.3.2	Prime Number Pan Magic Squares	CnstrSqrs7b
Attachment 7.3.3	Prime Number Ultra Magic Squares
Attachment 7.3.4	Prime Number Concentric Magic Squares

Comparable methods as described above, can be used to construct order 8 (Semi) Latin - and related (Pan) Magic Squares, which will be described in following sections.