4.0 Latin Squares (4 x 4)
A Latin Square of order 4 is a 4 x 4 square filled with 4 different symbols, each occurring only once in each row and only once in each column.
Attachment 4.0.1 shows the 576 ea order 4 Latin Squares, which can be found based on this definition
(ref. LatSqr4).
For the construction of order 4 Magic Squares normally only those Latin Squares are used for which the 4 different symbols occur also only once in each of the main diagonals (Latin Diagonal Squares).
4.1 Latin Diagonal Squares (4 x 4)
Attachment 4.1.1 shows the 48 ea order 4 Latin Diagonal Squares, which can be found based on the definition formulated above (ref. LatSqr4).
Suitable Euler Squares can be found, by selecting pairs of Latin Diagonal Squares (A, B)
while ensuring that the resulting square M contains 16 distinct pairs.
Attachment 4.1.2 shows the 1152 ea resulting Euler Squares
for the symbols {a_{i}} = {0, 1, 2, 3} and {b_{i}} = {0, 1, 2, 3}
(ref. CnstrSqrs4a).
If the series {a_{i}} and {b_{i}} are replaced by the symbols
{J,
Q,
K,
A} and
{♥,
♦,
♠,
♣}
the first solution can be rewritten as:
A
K 
A 
J 
Q 
J 
Q 
K 
A 
Q 
J 
A 
K 
A 
K 
Q 
J 

B
♦ 
♥ 
♣ 
♠ 
♠ 
♣ 
♥ 
♦ 
♥ 
♦ 
♠ 
♣ 
♣ 
♠ 
♦ 
♥ 

(A, B)
K♦ 
A♥ 
J♣ 
Q♠ 
J♠ 
Q♣ 
K♥ 
A♦ 
Q♥ 
J♦ 
A♠ 
K♣ 
A♣ 
K♠ 
Q♦ 
J♥ 

which is one of the solutions  found by Jacques Ozanam (1725)  of an old but comparable playing cards puzzle.
The puzzle was later studied by Rouse Ball, Martin Gardner and finally by Kathleen Ollerenshaw,
who did find the correct number of possible solution (8 x 144 = 1152).
4.2 Magic Squares, Natural Numbers
4.2.1 Simple Magic Squares
Simple Magic Square M of order 4 with the integers 1 ... 16 can be written as
M = A + 4 * B + [1]
where the squares A and B contain only the integers 0, 1, 2 and 3.
Consequently order 4 Simple Magic Squares
can be based on pairs of Orthogonal Latin Diagonal Squares (A, B)
which is illustrated below for a Simple Magic Square M.
A
2 
3 
0 
1 
0 
1 
2 
3 
1 
0 
3 
2 
3 
2 
1 
0 

B
1 
0 
3 
2 
2 
3 
0 
1 
0 
1 
2 
3 
3 
2 
1 
0 

M = A + 4 * B + 1
7 
4 
13 
10 
9 
14 
3 
8 
2 
5 
12 
15 
16 
11 
6 
1 

Attachment 4.2.1 shows the 1152 ea Simple Magic Squares which can be constructed based on the
1152 ea Euler Squares found in Section 4.1 above.
Note: Based on the complete collection of 576 Latin Squares (ref. Attachment 4.0.1) 1536 order 4 Simple Magic Squares can be constructed.
4.2.2 Pan Magic Squares
Attachment 4.1.3, page 1
shows the 16 order 4 Latin Diagonal Squares for which the Main  and Broken Diagonals sum to 6.
Attachment 4.1.3, page 2
shows the 32 order 4 Latin Squares for which the Main  and Broken Diagonals sum to 6.
Attachment 4.1.3, page 3
shows 16 order 4 SemiLatin Squares with Latin Main  and Broken Diagonals (summing to 6).
Order 4 Pan Magic Squares M
can be based on pairs of Orthogonal (Latin Diagonal) Squares (A, B)
out of these collections, which is illustrated below:
A
1 
0 
3 
2 
2 
3 
0 
1 
0 
1 
2 
3 
3 
2 
1 
0 

B
1 
2 
0 
3 
0 
3 
1 
2 
3 
0 
2 
1 
2 
1 
3 
0 

M = A + 4 * B + 1
6 
9 
4 
15 
3 
16 
5 
10 
13 
2 
11 
8 
12 
7 
14 
1 

Attachment 4.2.2, page 1
shows the 128 ea Pan Magic Squares which can be constructed based on the 16 ea Latin Diagonal Squares found above.
Attachment 4.2.2, page 2
shows the 256 ea Pan Magic Squares which can be constructed based on the 32 ea Latin Squares found above.
The 384 Pan Magic Squares as shown in Attachment 1, can be constructed
based on page 2 and 3 of Attachment 4.1.3.
4.2.3 Associated Magic Squares
Attachment 4.1.4, page 1
shows the 16 possible order 4 Associated Latin Diagonal Squares.
Attachment 4.1.4, page 2
shows the 32 possible order 4 Associated Latin Squares.
Attachment 4.1.4, page 3
shows 16 order 4 Associated SemiLatin Squares.
Order 4 Associated Magic Squares M
can be based on pairs of Orthogonal (Latin Diagonal) Squares (A, B)
out of these collections,
as shown below for a set of SemiLatin Squares and the symbols
{a_{i}, i = 1 ... 4}
and
{b_{j}, j = 1 ... 4).
A
a4 
a3 
a2 
a1 
a1 
a2 
a3 
a4 
a1 
a2 
a3 
a4 
a4 
a3 
a2 
a1 

B
b4 
b1 
b1 
b4 
b3 
b2 
b2 
b3 
b2 
b3 
b3 
b2 
b1 
b4 
b4 
b1 

(A, B)
a4, b4 
a3, b1 
a2, b1 
a1, b4 
a1, b3 
a2, b2 
a3, b2 
a4, b3 
a1, b2 
a2, b3 
a3, b3 
a4, b2 
a4, b1 
a3, b4 
a2, b4 
a1, b1 

All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:
A
3 
2 
1 
0 
0 
1 
2 
3 
0 
1 
2 
3 
3 
2 
1 
0 

B
3 
0 
0 
3 
2 
1 
1 
2 
1 
2 
2 
1 
0 
3 
3 
0 

M = A + 4 * B + 1
16 
3 
2 
13 
9 
6 
7 
12 
5 
10 
11 
8 
4 
15 
14 
1 

Attachment 4.2.3, page 1
shows the 128 ea Associated Magic Squares which can be constructed based on the 16 ea Associated Latin Diagonal Squares found above.
Attachment 4.2.3, page 2
shows the 256 ea Associated Magic Squares which can be constructed based on the 32 ea Associated Latin Squares found above.
The 384 Associated Magic Squares as shown in Attachment 2.5, can be constructed
based on page 2 and 3 of Attachment 4.1.4.
4.2.4 Evaluation of the Results
Following table compares the enumeration results for order 4 Magic Squares
(ref. Section 2.7)
with the results based on the construction methods described above:
Type

Enumerated

Constructed

Base

Simple (All)

7040

1536

Latin

1252

Latin Diagonal

Pan Magic

384

256

Latin

128

Latin Diagonal

Associated

384

256

Latin

128

Latin Diagonal

The constructability by means of Orthogonal (Latin Diagonal) Squares can be considered as an additional property.
4.3 Magic Squares, Prime Numbers
4.3.1 Simple Magic Squares
When the elements
{a_{i}, i = 1 ... 4}
and
{b_{j}, j = 1 ... 4)
of a valid pair of Orthogonal Latin Diagonal Squares (A, B)
 as applied in
Section 4.2.1 above  complies with following condition:

m_{ij} = a_{i} + b_{j} = prime
for i = 1 ... 4 and j = 1 ... 4 (correlated)
the resulting square M = A + B will be an order 4 Prime Number Simple Magic Square:
Sa = 82
3 
1 
57 
21 
21 
57 
1 
3 
1 
3 
21 
57 
57 
21 
3 
1 

Sb = 68
40 
2 
16 
10 
16 
10 
40 
2 
10 
16 
2 
40 
2 
40 
10 
16 

Sm = 150
43 
3 
73 
31 
37 
67 
41 
5 
11 
19 
23 
97 
59 
61 
13 
17 

Attachment 4.3, page 1 contains miscellaneous correlated unbalanced series
{a_{i}, i = 1 ... 4}
and
{b_{j}, j = 1 ... 4).
Attachment 4.3.1 contains the resulting Prime Number Simple Magic Squares and related Magic Sums (Sm).
Each square shown corresponds with 1152 Prime Number Simple Magic Squares.
4.3.2 Pan Magic Squares
When the elements
{a_{i}, i = 1 ... 4}
and
{b_{j}, j = 1 ... 4)
of a valid pair of Orthogonal Latin Diagonal Squares (A, B)
 as applied in
Section 4.2.2 above  comply with following conditions:

m_{ij} = a_{i} + b_{j} = prime
for i = 1 ... 4 and j = 1 ... 4 (correlated)

a_{1} + a_{4} =
a_{2} + a_{3} and
b_{1} + b_{4} =
b_{2} + b_{3} (balanced)
the resulting square M = A + B will be an order 4 Prime Number Pan Magic Square:
Sa = 36
7 
1 
17 
11 
11 
17 
1 
7 
1 
7 
11 
17 
17 
11 
7 
1 

Sb = 204
96 
6 
72 
30 
72 
30 
96 
6 
30 
72 
6 
96 
6 
96 
30 
72 

Sm = 240
103 
7 
89 
41 
83 
47 
97 
13 
31 
79 
17 
113 
23 
107 
37 
73 

Attachment 4.3, page 2 contains miscellaneous correlated balanced series
{a_{i}, i = 1 ... 4}
and
{b_{j}, j = 1 ... 4).
Attachment 4.3.2 contains the resulting Prime Number Pan Magic Squares and related Magic Sums (Sm).
Each square shown corresponds with 384 Prime Number Pan Magic Squares.
4.3.3 Associated Magic Squares
When the elements
{a_{i}, i = 1 ... 4}
and
{b_{j}, j = 1 ... 4)
of a valid pair of Orthogonal Squares (A, B)
 as applied in
Section 4.2.3 above  comply with following conditions:

m_{ij} = a_{i} + b_{j} = prime
for i = 1 ... 4 and j = 1 ... 4 (correlated)

a_{1} + a_{4} =
a_{2} + a_{3} and
b_{1} + b_{4} =
b_{2} + b_{3} (balanced)
the resulting square M = A + B will be an order 4 Prime Number Associated Magic Square:
Sa = 36
17 
11 
7 
1 
1 
7 
11 
17 
1 
7 
11 
17 
17 
11 
7 
1 

Sb = 204
96 
6 
6 
96 
72 
30 
30 
72 
30 
72 
72 
30 
6 
96 
96 
6 

Sm = 240
113 
17 
13 
97 
73 
37 
41 
89 
31 
79 
83 
47 
23 
107 
103 
7 

Attachment 4.3, page 2 contains miscellaneous correlated balanced series
{a_{i}, i = 1 ... 4}
and
{b_{j}, j = 1 ... 4).
Attachment 4.3.3 contains the resulting Prime Number Associated Magic Squares and related Magic Sums (Sm).
Each square shown corresponds with 384 Prime Number Associated Magic Squares.
4.4 Summary
The obtained results regarding the order 4 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 5 Latin  and related (Pan) Magic Squares, which will be described in following sections.
