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:

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.

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:

A: shift 2 columns to the left / B: shift 2 columns to the right

A: shift 2 columns to the left / B: shift 3 columns to the right

A: shift 2 columns to the left / B: shift 3 columns to the left

A: shift 3 columns to the left / B: shift 2 columns to the right

A: shift 3 columns to the left / B: shift 3 columns to the right

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
SemiLatin 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 SemiLatin Borders can be constructed
for each pair of order 5 Orthogonal Concentric SemiLatin Squares
(A5, B5),
as found in Section 5.2.5.
Each pair of order 7 Orthogonal SemiLatin 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
SemiLatin 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 SemiLatin Borders can be constructed
for each pair of order 5 Orthogonal Inlaid SemiLatin Squares
(A5, B5),
as found in Section 5.2.6.
Each pair of order 7 Orthogonal SemiLatin 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
SemiLatin 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 SemiLatin
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 SemiLatin 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 SemiLatin 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 SemiLatin 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 SemiLatin 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 SemiLatin 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 SemiLatin (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 2^{3} = 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 (LDRformat) 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 * 10^{17}

Walter Trump

304.819.200

Latin Diagonal

Ultra Magic

8 * 20.190.684

Walter Trump

27.648

Latin Diagonal

Concentric

4,91 * 10^{11}



132.710.400

SemiLatin

Inlaid

1,76 * 10^{11}



186.163.200

SemiLatin

Composed 3/5

368.640)*



352.256

SemiLatin

Composed 4/4, Symm

4.156.416



43.776

SemiLatin

)* 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 SemiLatin 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)

a_{1} + a_{7} = 2 * a_{4} and
b_{1} + b_{7} = 2 * b_{4} (balanced)
a_{2} + a_{6} = 2 * a_{4} and
b_{2} + b_{6} = 2 * b_{4}
a_{3} + a_{5} = 2 * a_{4} and
b_{3} + b_{5} = 2 * b_{4}
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 SemiLatin 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:
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.
