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Detailed explanation about the structure and construction of magic squares
Sudoku method (1)
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How to produce a (4x4 pan)magic square from a Sudoku?
 
 A Sudoku mostly consists of 9 rows and 9 columns. Each row and each column (and each nonet) must contain
all the digits from 1 to 9. Using a 4x4 Sudoku (which consists of 4 rows and 4 columns) you can produce a magic
square when you follow the next four steps.
 
(1st) Do not fill in the digits 1, 2, 3 and 4, but fill in the digits 0, 1, 2, 3. Ensure that every row, column and dia-
gonal contains all the digits 0, 1, 2 and 3.
 
(2nd) Produce a second 4x4 Sudoku by rotating the first Sudoku (a quarter turn to the right).
 
(3rd) Take a digit from the first Sudoku multiplied by 4 and add (1x) the digit from the same cell of the second
Sudoku.
 
(4th) Add 1 to each cell.
 
 4x digit                         + 1x digit               =              +1               =   4x4 magic square   
0
1
2
3
 
2
1
3
0
 
2
5
11
12
 
3
6
12
13
3
2
1
0
 
3
0
2
1
 
15
8
6
1
 
16
9
7
2
1
0
3
2
 
0
3
1
2
 
4
3
13
10
 
5
4
14
11
2
3
0
1
 
1
2
0
3
 
9
14
0
7
 
10
15
1
8
 

 
This magic square also just happens to be a 4x4 panmagic square!


See more information on page:  4x4 panmagic square



Most perfect (Franklin pan)magic 8x8 square

It is possible to use the Sudoku patterns of the panmagic 4x4 square to produce a most perfect
(Franklin pan)magic 8x8 square. You need three 8x8 Sudoku patterns.
 
● The first 8x8 Sudoku pattern is a 2x2 carpet of the first 4x4 Sudoku pattern.
 
● To produce the second 8x8 Sudoku pattern you need to split up the second 4x4 Sudoku pattern in
two sub-squares and enter digits from the same sub-square in the empty cells crosswise:
 
 
 split up the (2nd )    4x4 Sudoku pattern:         Enter digits in the empty cells crosswise:
 
1
3
 
 
2
 
 
0
 
 
0
1
3
2
 
2
3
1
0
3
 
 
1
 
 
0
2
 
 
 
3
2
0
1
 
1
0
2
3
0
 
 
2
 
 
3
1
 
 
 
0
1
3
2
 
2
3
1
0
 
2
0
 
 
1
 
 
3
 
 
3
2
0
1
 
1
0
2
3
 
 
Combine the 2 sub-squares and add the same 2 sub-squares to the bottom.
 
● The third 8x8 Sudoku pattern is fixed (= column pattern of the basic pattern method).
 
 
 
4x digit from first pattern     +   1x digit from second pattern  +   16x digit from third pattern  =
0
1
2
3
0
1
2
3
 
 
0
1
3
2
2
3
1
0
 
 
0
3
0
3
0
3
0
3
3
2
1
0
3
2
1
0
 
 
3
2
0
1
1
0
2
3
 
 
0
3
0
3
0
3
0
3
1
0
3
2
1
0
3
2
 
 
0
1
3
2
2
3
1
0
 
 
3
0
3
0
3
0
3
0
2
3
0
1
2
3
0
1
 
 
3
2
0
1
1
0
2
3
 
 
3
0
3
0
3
0
3
0
0
1
2
3
0
1
2
3
 
 
0
1
3
2
2
3
1
0
 
 
1
2
1
2
1
2
1
2
3
2
1
0
3
2
1
0
 
 
3
2
0
1
1
0
2
3
 
 
1
2
1
2
1
2
1
2
1
0
3
2
1
0
3
2
 
 
0
1
3
2
2
3
1
0
 
 
2
1
2
1
2
1
2
1
2
3
0
1
2
3
0
1
 
 
3
2
0
1
1
0
2
3
 
 
2
1
2
1
2
1
2
1
 
 
                         +1                                =      Most perfect (Franklin pan)magic 8x8 square
0
53
11
62
2
55
9
60
 
 
1
54
12
63
3
56
10
61
15
58
4
49
13
56
6
51
 
 
16
59
5
50
14
57
7
52
52
1
63
10
54
3
61
8
 
 
53
2
64
11
55
4
62
9
59
14
48
5
57
12
50
7
 
 
60
15
49
6
58
13
51
8
16
37
27
46
18
39
25
44
 
 
17
38
28
47
19
40
26
45
31
42
20
33
29
40
22
35
 
 
32
43
21
34
30
41
23
36
36
17
47
26
38
19
45
24
 
 
37
18
48
27
39
20
46
25
43
30
32
21
41
28
34
23
 
 
44
31
33
22
42
29
35
24
 
 
 
Note that above mentioned method is the ‘Sudoku version’ of the basic pattern method (see also on this
website).



Most perfect (Franklin pan)magic 16x16 square

It is also possible to use the Sudoku patterns of the panmagic 8x8 square to produce a most perfect
(Franklin pan)magic 16x16 square. You need four 16x16 Sudoku patterns.
 
● The first and second 16x16 Sudoku pattern is a 2x2 carpet of the first respectively second 8x8 Sudoku
pattern.
 
● The third and fourth 16x16 Sudoku pattern are fixed patterns (the third and fourth pattern together =
the column pattern of the basic pattern method).
 
 
 4x digit from first pattern                                            1x digit from second pattern
0
1
2
3
0
1
2
3
0
1
2
3
0
1
2
3
 
 
0
1
3
2
2
3
1
0
0
1
3
2
2
3
1
0
3
2
1
0
3
2
1
0
3
2
1
0
3
2
1
0
 
 
3
2
0
1
1
0
2
3
3
2
0
1
1
0
2
3
1
0
3
2
1
0
3
2
1
0
3
2
1
0
3
2
 
 
0
1
3
2
2
3
1
0
0
1
3
2
2
3
1
0
2
3
0
1
2
3
0
1
2
3
0
1
2
3
0
1
 
 
3
2
0
1
1
0
2
3
3
2
0
1
1
0
2
3
0
1
2
3
0
1
2
3
0
1
2
3
0
1
2
3
 
 
0
1
3
2
2
3
1
0
0
1
3
2
2
3
1
0
3
2
1
0
3
2
1
0
3
2
1
0
3
2
1
0
 
 
3
2
0
1
1
0
2
3
3
2
0
1
1
0
2
3
1
0
3
2
1
0
3
2
1
0
3
2
1
0
3
2
 
 
0
1
3
2
2
3
1
0
0
1
3
2
2
3
1
0
2
3
0
1
2
3
0
1
2
3
0
1
2
3
0
1
 
 
3
2
0
1
1
0
2
3
3
2
0
1
1
0
2
3
0
1
2
3
0
1
2
3
0
1
2
3
0
1
2
3
 
 
0
1
3
2
2
3
1
0
0
1
3
2
2
3
1
0
3
2
1
0
3
2
1
0
3
2
1
0
3
2
1
0
 
 
3
2
0
1
1
0
2
3
3
2
0
1
1
0
2
3
1
0
3
2
1
0
3
2
1
0
3
2
1
0
3
2
 
 
0
1
3
2
2
3
1
0
0
1
3
2
2
3
1
0
2
3
0
1
2
3
0
1
2
3
0
1
2
3
0
1
 
 
3
2
0
1
1
0
2
3
3
2
0
1
1
0
2
3
0
1
2
3
0
1
2
3
0
1
2
3
0
1
2
3
 
 
0
1
3
2
2
3
1
0
0
1
3
2
2
3
1
0
3
2
1
0
3
2
1
0
3
2
1
0
3
2
1
0
 
 
3
2
0
1
1
0
2
3
3
2
0
1
1
0
2
3
1
0
3
2
1
0
3
2
1
0
3
2
1
0
3
2
 
 
0
1
3
2
2
3
1
0
0
1
3
2
2
3
1
0
2
3
0
1
2
3
0
1
2
3
0
1
2
3
0
1
 
 
3
2
0
1
1
0
2
3
3
2
0
1
1
0
2
3
 
 
 16x digit from third (fixed) pattern                             64x digit from fourth (fixed) pattern
2
3
0
1
2
3
0
1
0
1
2
3
0
1
2
3
 
 
0
3
0
3
0
3
0
3
0
3
0
3
0
3
0
3
0
1
2
3
0
1
2
3
2
3
0
1
2
3
0
1
 
 
0
3
0
3
0
3
0
3
0
3
0
3
0
3
0
3
3
2
1
0
3
2
1
0
1
0
3
2
1
0
3
2
 
 
3
0
3
0
3
0
3
0
3
0
3
0
3
0
3
0
1
0
3
2
1
0
3
2
3
2
1
0
3
2
1
0
 
 
3
0
3
0
3
0
3
0
3
0
3
0
3
0
3
0