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Transformation method
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How to transform a square with sequential digits to a most-perfect magic square!
 
 
 
Introduction
On the website of Harvey Heinz is shown on page www.magic-squares.net/most-perfect.htm that a 4x4 square with sequential
digits can be transformed to a panmagic 4x4 square.
 
Firstly I present step by step the transformation to the 3 basic squares of the panmagic 4x4 square (see page ‘panmagic 4x4
square’).
 
Secondly I present step by step the transformation of a 8x8, 12x12 and 16x16 square with sequential digits to a most-perfect
8x8, 12x12 and 16x16 square.
 
 
 
Transformation to the 3 basic squares of the panmagic 4x4 square
Transform a 4x4 square with sequential digits to a panmagic 4x4 square in 5 steps by swapping each time ‘yellow’ and ‘red’;
see below:
 
 
1
5
9
13
 
1
5
13
9
 
1
5
13
9
 
1
8
13
12
 
1
8
13
12
 
1
8
13
12
2
6
10
14
 
2
6
14
10
 
2
6
14
10
 
2
6
14
10
 
14
10
2
6
 
15
10
3
6
3
7
11
15
 
3
7
15
11
 
4
8
16
12
 
4
5
16
9
 
4
5
16
12
 
4
5
16
9
4
8
12
16
 
4
8
16
12
 
3
7
15
11
 
3
7
15
11
 
15
11
3
7
 
14
11
2
7
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
3
9
11
 
1
3
11
9
 
1
3
11
9
 
1
8
11
14
 
1
8
11
14
 
1
8
11
14
2
4
10
12
 
2
4
12
10
 
2
4
12
10
 
2
4
12
10
 
12
10
2
4
 
15
10
5
4
5
7
13
15
 
5
7
15
13
 
6
8
16
14
 
6
3
16
9
 
6
3
16
9
 
6
3
16
9
6
8
14
16
 
6
8
16
14
 
5
7
15
13
 
5
7
15
13
 
15
13
5
7
 
12
13
2
7
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
2
9
10
 
1
2
10
9
 
1
2
10
9
 
1
8
10
15
 
1
8
10
15
 
1
8
10
15
3
4
11
12
 
3
4
12
11
 
3
4
12
11
 
3
4
12
11
 
12
11
3
4
 
14
11
5
4
5
6
13
14
 
5
6
14
13
 
7
8
16
15
 
7
2
16
9
 
7
2
16
9
 
7
2
16
9
7
8
15
16
 
7
8
16
15
 
5
6
14
13
 
5
6
14
13
 
14
13
5
6
 
12
13
3
6
 
 
 
Transformation to a Franklin panmagic 8x8 square
Transform a 8x8 square with sequential digits to a Franklin panmagic 8x8 square in 5 steps by swapping each time ‘yellow’ and
‘red’; see below:
 
 
 
 
#
*
 
 
*
#
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
9
17
25
33
41
49
57
 
 
1
9
57
49
33
41
25
17
 
 
1
9
57
49
33
41
25
17
2
10
18
26
34
42
50
58
 
 
2
10
58
50
34
42
26
18
 
 
2
10
58
50
34
42
26
18
3
11
19
27
35
43
51
59
 
#
3
11
59
51
35
43
27
19
 
 
8
16
64
56
40
48
32
24
4
12
20
28
36
44
52
60
 
*
4
12
60
52
36
44
28
20
 
 
7
15
63
55
39
47
31
23
5
13
21
29
37
45
53
61
 
 
5
13
61
53
37
45
29
21
 
 
5
13
61
53
37
45
29
21
6
14
22
30
38
46
54
62
 
 
6
14
62
54
38
46
30
22
 
 
6
14
62
54
38
46
30
22
7
15
23
31
39
47
55
63
 
*
7
15
63
55
39
47
31
23
 
 
4
12
60
52
36
44
28
20
8
16
24
32
40
48
56
64
 
#
8
16
64
56
40
48
32
24
 
 
3
11
59
51
35
43
27
19
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
16
57
56
33
48
25
24
 
 
1
16
57
56
33
48
25
24
 
 
1
16
57
56
33
48
25
24
2
10
58
50
34
42
26
18
 
 
58
50
2
10
26
18
34
42
 
 
63
50
7
10
31
18
39
42
8
9
64
49
40
41
32
17
 
 
8
9
64
49
40
41
32
17
 
 
8
9
64
49
40
41
32
17
7
15
63
55
39
47
31
23
 
 
63
55
7
15
31
23
39
47
 
 
58
55
2
15
26
23
34
47
5
12
61
52
37
44
29
20
 
 
5
12
61
52
37
44
29
20
 
 
5
12
61
52
37
44
29
20
6
14
62
54
38
46
30
22
 
 
62
54
6
14
30
22
38
46
 
 
59
54
3
14
27
22
35
46
4
13
60
53
36
45
28
21
 
 
4
13
60
53
36
45
28
21
 
 
4
13
60
53
36
45
28
21
3
11
59
51
35
43
27
19
 
 
59
51
3
11
27
19
35
43
 
 
62
51
6
11
30
19
38
43
 
 
 
Transformation to a most-perfect 12x12 magic square
Transform a 12x12 square with sequential digits to a most-perfect 12x12 magic square in 5 steps by swapping each time ‘yellow’
and ‘red’; see below:
 
 
 
 
#
*
 
 
@
@
 
 
*
#
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
13
25
37
49
61
73
85
97
109
121
133
 
 
1
13
133
121
49
61
85
73
97
109
37
25
2
14
26
38
50
62
74
86
98
110
122
134
 
 
2
14
134
122
50
62
86
74
98
110
38
26
3
15
27
39
51
63
75
87
99
111
123
135
 
#
3
15
135
123
51
63
87
75
99
111
39
27
4
16
28
40
52
64
76
88
100
112
124
136
 
*
4
16
136
124
52
64
88
76
100
112
40
28
5
17
29
41
53
65
77
89
101
113
125
137
 
 
5
17
137
125
53
65
89
77
101
113
41
29
6
18
30
42
54
66
78
90
102
114
126
138
 
 
6
18
138
126
54
66
90
78
102
114
42
30
7
19
31
43
55
67
79
91
103
115
127
139
 
@
7
19
139
127
55
67
91
79
103
115
43
31
8
20
32
44
56
68
80
92
104
116
128
140
 
@
8
20
140
128
56
68
92
80
104
116
44
32
9
21
33
45
57
69
81
93
105
117
129
141