Topzle Topzle

Magic square

Updated: Wikipedia source

Magic square

In mathematics, especially historical and recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The order of the magic square is the number of integers along one side (n), and the constant sum is called the magic constant. If the array includes just the positive integers 1 , 2 , . . . , n 2 {\displaystyle 1,2,...,n^{2}} , the magic square is said to be normal. Many authors take magic square to mean normal magic square. Magic squares that include repeated entries do not fall under this definition and are referred to as trivial. Some well-known examples, including the Sagrada Família magic square are trivial in this sense. When all the rows and columns but not both diagonals sum to the magic constant, this gives a semimagic square (sometimes called orthomagic square). The mathematical study of magic squares typically deals with its construction, classification, and enumeration. Although completely general methods for producing all the magic squares of all orders do not exist, historically three general techniques have been discovered: by bordering, by making composite magic squares, and by adding two preliminary squares. There are also more specific strategies like the continuous enumeration method that reproduces specific patterns. Magic squares are generally classified according to their order n as: odd if n is odd, evenly even (also referred to as "doubly even") if n is a multiple of 4, oddly even (also known as "singly even") if n is any other even number. This classification is based on different techniques required to construct odd, evenly even, and oddly even squares. Beside this, depending on further properties, magic squares are also classified as associative magic squares, pandiagonal magic squares, most-perfect magic squares, and so on. More challengingly, attempts have also been made to classify all the magic squares of a given order as transformations of a smaller set of squares. Except for n ≤ 5, the enumeration of higher-order magic squares is still an open challenge. The enumeration of most-perfect magic squares of any order was only accomplished in the late 20th century. Magic squares have a long history, dating back to at least 190 BCE in China. At various times they have acquired occult or mythical significance, and have appeared as symbols in works of art. In modern times they have been generalized a number of ways, including using extra or different constraints, multiplying instead of adding cells, using alternate shapes or more than two dimensions, and replacing numbers with shapes and addition with geometric operations.

Tables

3
3
4
3
9
5
2
7
8
8
4
8
9
1
2
6
4
9
2
3
5
7
8
1
6
11
11
2
11
16
5
13
8
3
10
7
7
2
7
16
9
13
12
3
6
14
14
2
14
16
4
13
1
3
15
2
16
13
3
11
5
8
10
7
9
12
6
14
4
1
15
15
15
1
15
23
14
16
7
4
18
21
11
24
24
1
24
23
17
16
13
4
9
21
2
20
20
1
20
23
8
16
19
4
12
21
6
5
5
1
5
23
3
16
10
4
22
21
25
1
23
16
4
21
15
14
7
18
11
24
17
13
9
2
20
8
19
12
6
5
3
10
22
25
31
31
13
31
22
4
18
36
27
9
11
29
20
2
12
12
13
12
22
21
18
14
27
23
11
16
20
25
30
30
13
30
22
3
18
5
27
32
11
34
20
7
17
17
13
17
22
26
18
10
27
19
11
15
20
24
8
8
13
8
22
35
18
28
27
1
11
6
20
33
13
22
18
27
11
20
31
4
36
9
29
2
12
21
14
23
16
25
30
3
5
32
34
7
17
26
10
19
15
24
8
35
28
1
6
33
3
3
46
3
8
40
16
35
20
36
29
18
7
41
49
2
44
44
46
44
8
12
16
33
20
23
29
19
7
38
49
6
28
28
46
28
8
26
16
11
20
25
29
39
7
24
49
22
5
5
46
5
8
37
16
31
20
27
29
17
7
13
49
45
48
48
46
48
8
9
16
15
20
14
29
32
7
10
49
47
1
1
46
1
8
43
16
34
20
30
29
21
7
42
49
4
46
8
16
20
29
7
49
3
40
35
36
18
41
2
44
12
33
23
19
38
6
28
26
11
25
39
24
22
5
37
31
27
17
13
45
48
9
15
14
32
10
47
1
43
34
30
21
42
4
12
12
61
12
3
54
2
55
64
9
57
16
7
50
6
51
60
13
20
20
61
20
3
46
2
47
64
17
57
24
7
42
6
43
60
21
37
37
61
37
3
27
2
26
64
40
57
33
7
31
6
30
60
36
29
29
61
29
3
35
2
34
64
32
57
25
7
39
6
38
60
28
44
44
61
44
3
22
2
23
64
41
57
48
7
18
6
19
60
45
52
52
61
52
3
14
2
15
64
49
57
56
7
10
6
11
60
53
5
5
61
5
3
59
2
58
64
8
57
1
7
63
6
62
60
4
61
3
2
64
57
7
6
60
12
54
55
9
16
50
51
13
20
46
47
17
24
42
43
21
37
27
26
40
33
31
30
36
29
35
34
32
25
39
38
28
44
22
23
41
48
18
19
45
52
14
15
49
56
10
11
53
5
59
58
8
1
63
62
4
22
22
31
22
76
40
13
58
36
27
81
45
18
63
29
20
74
38
11
56
67
67
31
67
76
4
13
49
36
72
81
9
18
54
29
65
74
2
11
47
30
30
31
30
76
75
13
12
36
32
81
77
18
14
29
34
74
79
11
16
21
21
31
21
76
39
13
57
36
23
81
41
18
59
29
25
74
43
11
61
66
66
31
66
76
3
13
48
36
68
81
5
18
50
29
70
74
7
11
52
35
35
31
35
76
80
13
17
36
28
81
73
18
10
29
33
74
78
11
15
26
26
31
26
76
44
13
62
36
19
81
37
18
55
29
24
74
42
11
60
71
71
31
71
76
8
13
53
36
64
81
1
18
46
29
69
74
6
11
51
31
76
13
36
81
18
29
74
11
22
40
58
27
45
63
20
38
56
67
4
49
72
9
54
65
2
47
30
75
12
32
77
14
34
79
16
21
39
57
23
41
59
25
43
61
66
3
48
68
5
50
70
7
52
35
80
17
28
73
10
33
78
15
26
44
62
19
37
55
24
42
60
71
8
53
64
1
46
69
6
51
5
5
2
5
3
8
5
2
8
3
4
4
2
4
3
1
5
7
8
6
7
7
2
7
3
6
5
4
8
1
2
3
5
8
5
8
2
3
4
1
7
6
7
6
4
1
5
5
10
5
3
16
13
2
8
11
4
4
10
4
3
9
13
7
8
14
15
15
10
15
3
6
13
12
8
1
10
3
13
8
5
16
2
11
4
9
7
14
15
6
12
1
10
10
30
10
16
44
18
22
36
24
32
32
30
32
16
14
18
20
36
34
28
28
30
28
16
26
18
40
36
6
30
16
18
36
10
44
22
24
32
14
20
34
28
26
40
6
2
2
7
2
1
8
4
5
6
3
5
5
7
5
1
3
4
2
6
8
4
4
7
4
1
6
4
7
6
1
7
1
4
6
2
8
5
3
5
3
2
8
4
6
7
1
3
3
8
3
1
5
6
7
4
4
8
4
1
9
6
2
8
1
6
3
5
7
4
9
2
8
8
1
8
14
11
4
5
15
10
13
13
1
13
14
2
4
16
15
3
12
12
1
12
14
7
4
9
15
6
1
14
4
15
8
11
5
10
13
2
16
3
12
7
9
6
24
24
16
24
14
17
7
10
30
8
23
31
32
32
16
32
14
25
7
18
30
11
23
4
5
5
16
5
14
28
7
26
30
19
23
12
13
13
16
13
14
6
7
29
30
22
23
20
16
14
7
30
23
24
17
10
8
31
32
25
18
11
4
5
28
26
19
12
13
6
29
22
20
25
25
1
25
35
11
4
9
33
28
32
8
6
30
24
24
1
24
35
14
4
18
33
16
32
17
6
22
13
13
1
13
35
23
4
19
33
21
32
20
6
15
12
12
1
12
35
26
4
27
33
10
32
29
6
7
36
36
1
36
35
2
4
34
33
3
32
5
6
31
1
35
4
33
32
6
25
11
9
28
8
30
24
14
18
16
17
22
13
23
19
21
20
15
12
26
27
10
29
7
36
2
34
3
5
31
46
46
35
46
26
37
17
21
1
12
62
3
53
64
44
55
57
57
35
57
26
41
17
32
1
23
62
14
53
5
44
66
61
61
35
61
26
52
17
43
1
34
62
25
53
16
44
7
2
2
35
2
26
63
17
54
1
45
62
36
53
27
44
11
13
13
35
13
26
4
17
65
1
56
62
47
53
31
44
22
24
24
35
24
26
15
17
6
1
67
62
51
53
42
44
33
35
26
17
1
62
53
44
46
37
21
12
3
64
55
57
41
32
23
14
5
66
61
52
43
34
25
16
7
2
63
54
45
36
27
11
13
4
65
56
47
31
22
24
15
6
67
51
42
33
3
3
60
3
53
14
44
19
37
30
4
59
13
54
20
43
29
38
58
58
60
58
53
55
44
42
37
39
4
2
13
15
20
18
29
31
1
1
60
1
53
16
44
17
37
32
4
57
13
56
20
41
29
40
61
61
60
61
53
52
44
45
37
36
4
5
13
12
20
21
29
28
6
6
60
6
53
11
44
22
37
27
4
62
13
51
20
46
29
35
63
63
60
63
53
50
44
47
37
34
4
7
13
10
20
23
29
26
8
8
60
8
53
9
44
24
37
25
4
64
13
49
20
48
29
33
60
53
44
37
4
13
20
29
3
14
19
30
59
54
43
38
58
55
42
39
2
15
18
31
1
16
17
32
57
56
41
40
61
52
45
36
5
12
21
28
6
11
22
27
62
51
46
35
63
50
47
34
7
10
23
26
8
9
24
25
64
49
48
33
9
9
2
9
7
5
6
1
4
4
2
4
7
3
6
8
2
7
6
9
5
1
4
3
8
9
9
4
9
14
7
15
6
1
12
5
5
4
5
14
11
15
10
1
8
16
16
4
16
14
2
15
3
1
13
4
14
15
1
9
7
6
12
5
11
10
8
16
2
3
13
15
15
21
15
3
17
4
6
12
19
25
8
10
10
21
10
3
24
4
13
12
2
25
16
18
18
21
18
3
7
4
20
12
9
25
11
1
1
21
1
3
14
4
22
12
23
25
5
21
3
4
12
25
15
17
6
19
8
10
24
13
2
16
18
7
20
9
11
1
14
22
23
5
25
25
11
25
22
16
32
7
5
30
23
13
18
20
27
27
11
27
22
6
32
35
5
36
23
4
18
3
10
10
11
10
22
31
32
1
5
2
23
33
18
34
14
14
11
14
22
19
32
8
5
29
23
26
18
15
24
24
11
24
22
17
32
28
5
9
23
12
18
21
11
22
32
5
23
18
25
16
7
30
13
20
27
6
35
36
4
3
10
31
1
2
33
34
14
19
8
29
26
15
24
17
28
9
12
21
4
4
47
4
11
37
8
20
9
17
6
16
45
35
49
46
2
2
47
2
11
18
8
26
9
21
6
28
45
32
49
48
43
43
47
43
11
19
8
27
9
25
6
23
45
31
49
7
38
38
47
38
11
36
8
22
9
29
6
24
45
14
49
12
40
40
47
40
11
15
8
30
9
33
6
34
45
13
49
10
1
1
47
1
11
39
8
42
9
41
6
44
45
5
49
3
47
11
8
9
6
45
49
4
37
20
17
16
35
46
2
18
26
21
28
32
48
43
19
27
25
23
31
7
38
36
22
29
24
14
12
40
15
30
33
34
13
10
1
39
42
41
44
5
3
Saturn=15
3
3
4
3
9
5
2
7
8
8
4
8
9
1
2
6
4
9
2
3
5
7
8
1
6
Jupiter=34
9
9
4
9
14
7
15
6
1
12
5
5
4
5
14
11
15
10
1
8
16
16
4
16
14
2
15
3
1
13
4
14
15
1
9
7
6
12
5
11
10
8
16
2
3
13
Mars=65
4
4
11
4
24
12
7
25
20
8
3
16
17
17
11
17
24
5
7
13
20
21
3
9
10
10
11
10
24
18
7
1
20
14
3
22
23
23
11
23
24
6
7
19
20
2
3
15
11
24
7
20
3
4
12
25
8
16
17
5
13
21
9
10
18
1
14
22
23
6
19
2
15
Sol=111
7
7
6
7
32
11
3
27
34
28
35
8
1
30
19
19
6
19
32
14
3
16
34
15
35
23
1
24
18
18
6
18
32
20
3
22
34
21
35
17
1
13
25
25
6
25
32
29
3
10
34
9
35
26
1
12
36
36
6
36
32
5
3
33
34
4
35
2
1
31
6
32
3
34
35
1
7
11
27
28
8
30
19
14
16
15
23
24
18
20
22
21
17
13
25
29
10
9
26
12
36
5
33
4
2
31
Venus=175
5
5
22
5
47
23
16
48
41
17
10
42
35
11
4
29
30
30
22
30
47
6
16
24
41
49
10
18
35
36
4
12
13
13
22
13
47
31
16
7
41
25
10
43
35
19
4
37
38
38
22
38
47
14
16
32
41
1
10
26
35
44
4
20
21
21
22
21
47
39
16
8
41
33
10
2
35
27
4
45
46
46
22
46
47
15
16
40
41
9
10
34
35
3
4
28
22
47
16
41
10
35
4
5
23
48
17
42
11
29
30
6
24
49
18
36
12
13
31
7
25
43
19
37
38
14
32
1
26
44
20
21
39
8
33
2
27
45
46
15
40
9
34
3
28
Mercury=260
49
49
8
49
58
15
59
14
5
52
4
53
62
11
63
10
1
56
41
41
8
41
58
23
59
22
5
44
4
45
62
19
63
18
1
48
32
32
8
32
58
34
59
35
5
29
4
28
62
38
63
39
1
25
40
40
8
40
58
26
59
27
5
37
4
36
62
30
63
31
1
33
17
17
8
17
58
47
59
46
5
20
4
21
62
43
63
42
1
24
9
9
8
9
58
55
59
54
5
12
4
13
62
51
63
50
1
16
64
64
8
64
58
2
59
3
5
61
4
60
62
6
63
7
1
57
8
58
59
5
4
62
63
1
49
15
14
52
53
11
10
56
41
23
22
44
45
19
18
48
32
34
35
29
28
38
39
25
40
26
27
37
36
30
31
33
17
47
46
20
21
43
42
24
9
55
54
12
13
51
50
16
64
2
3
61
60
6
7
57
Luna=369
6
6
37
6
78
38
29
79
70
30
21
71
62
22
13
63
54
14
5
46
47
47
37
47
78
7
29
39
70
80
21
31
62
72
13
23
54
55
5
15
16
16
37
16
78
48
29
8
70
40
21
81
62
32
13
64
54
24
5
56
57
57
37
57
78
17
29
49
70
9
21
41
62
73
13
33
54
65
5
25
26
26
37
26
78
58
29
18
70
50
21
1
62
42
13
74
54
34
5
66
67
67
37
67
78
27
29
59
70
10
21
51
62
2
13
43
54
75
5
35
36
36
37
36
78
68
29
19
70
60
21
11
62
52
13
3
54
44
5
76
77
77
37
77
78
28
29
69
70
20
21
61
62
12
13
53
54
4
5
45
37
78
29
70
21
62
13
54
5
6
38
79
30
71
22
63
14
46
47
7
39
80
31
72
23
55
15
16
48
8
40
81
32
64
24
56
57
17
49
9
41
73
33
65
25
26
58
18
50
1
42
74
34
66
67
27
59
10
51
2
43
75
35
36
68
19
60
11
52
3
44
76
77
28
69
20
61
12
53
4
45
· Some famous magic squares › Magic square in Parshavnath temple
2
2
7
2
12
13
1
8
14
11
16
16
7
16
12
3
1
10
14
5
9
9
7
9
12
6
1
15
14
4
7
12
1
14
2
13
8
11
16
3
10
5
9
6
15
4
· Some famous magic squares › Albrecht Dürer's magic square
5
5
16
5
3
10
2
11
13
8
9
9
16
9
3
6
2
7
13
12
4
4
16
4
3
15
2
14
13
1
16
3
2
13
5
10
11
8
9
6
7
12
4
15
14
1
· Some famous magic squares › Sagrada Família magic square
11
11
1
11
14
7
14
6
4
9
8
8
1
8
14
10
14
10
4
5
13
13
1
13
14
2
14
3
4
15
1
14
14
4
11
7
6
9
8
10
10
5
13
2
3
15
5
5
10
5
3
16
13
2
8
11
4
4
10
4
3
9
13
7
8
14
15
15
10
15
3
6
13
12
8
1
10
3
13
8
5
16
2
11
4
9
7
14
15
6
12
1
12
12
7
12
14
1
4
15
9
6
13
13
7
13
14
8
4
10
9
3
2
2
7
2
14
11
4
5
9
16
7
14
4
9
12
1
15
6
13
8
10
3
2
11
5
16
12
12
1
12
15
6
14
7
4
9
8
8
1
8
15
10
14
11
4
5
13
13
1
13
15
3
14
2
4
16
1
15
14
4
12
6
7
9
8
10
11
5
13
3
2
16
14
14
1
14
17
6
16
7
4
11
8
8
1
8
17
12
16
13
4
5
15
15
1
15
17
3
16
2
4
18
1
17
16
4
14
6
7
11
8
12
13
5
15
3
2
18
18
18
11
18
10
12
4
6
23
5
17
24
25
25
11
25
10
19
4
13
23
7
17
1
2
2
11
2
10
21
4
20
23
14
17
8
9
9
11
9
10
3
4
22
23
16
17
15
11
10
4
23
17
18
12
6
5
24
25
19
13
7
1
2
21
20
14
8
9
3
22
16
15
24
24
16
24
14
17
7
10
30
8
23
31
32
32
16
32
14
25
7
18
30
11
23
4
5
5
16
5
14
28
7
26
30
19
23
12
13
13
16
13
14
6
7
29
30
22
23
20
16
14
7
30
23
24
17
10
8
31
32
25
18
11
4
5
28
26
19
12
13
6
29
22
20
3
3
8
3
1
5
6
7
4
4
8
4
1
9
6
2
8
1
6
3
5
7
4
9
2
7
7
6
7
1
5
8
3
2
2
6
2
1
9
8
4
6
1
8
7
5
3
2
9
4
9
9
2
9
7
5
6
1
4
4
2
4
7
3
6
8
2
7
6
9
5
1
4
3
8
9
9
4
9
3
5
8
1
2
2
4
2
3
7
8
6
4
3
8
9
5
1
2
7
6
7
7
2
7
9
5
4
3
6
6
2
6
9
1
4
8
2
9
4
7
5
3
6
1
8
3
3
4
3
9
5
2
7
8
8
4
8
9
1
2
6
4
9
2
3
5
7
8
1
6
1
1
8
1
3
5
4
9
6
6
8
6
3
7
4
2
8
3
4
1
5
9
6
7
2
1
1
6
1
7
5
2
9
8
8
6
8
7
3
2
4
6
7
2
1
5
9
8
3
4
12
12
1
12
15
6
14
7
4
9
8
8
1
8
15
10
14
11
4
5
13
13
1
13
15
3
14
2
4
16
1
15
14
4
12
6
7
9
8
10
11
5
13
3
2
16
9
9
16
9
3
6
2
7
13
12
5
5
16
5
3
10
2
11
13
8
4
4
16
4
3
15
2
14
13
1
16
3
2
13
9
6
7
12
5
10
11
8
4
15
14
1
12
12
1
12
15
6
14
7
4
9
8
8
1
8
15
10
14
11
4
5
13
13
1
13
15
3
14
2
4
16
1
15
14
4
12
6
7
9
8
10
11
5
13
3
2
16
1
1
12
1
6
15
7
14
9
4
13
13
12
13
6
3
7
2
9
16
8
8
12
8
6
10
7
11
9
5
12
6
7
9
1
15
14
4
13
3
2
16
8
10
11
5
15
15
6
15
12
1
9
4
7
14
3
3
6
3
12
13
9
16
7
2
10
10
6
10
12
8
9
5
7
11
6
12
9
7
15
1
4
14
3
13
16
2
10
8
5
11
12
12
1
12
15
6
14
7
4
9
8
8
1
8
15
10
14
11
4
5
13
13
1
13
15
3
14
2
4
16
1
15
14
4
12
6
7
9
8
10
11
5
13
3
2
16
13
13
8
13
10
3
11
2
5
16
1
1
8
1
10
15
11
14
5
4
12
12
8
12
10
6
11
7
5
9
8
10
11
5
13
3
2
16
1
15
14
4
12
6
7
9
2
2
11
2
5
16
8
13
10
3
14
14
11
14
5
4
8
1
10
15
7
7
11
7
5
9
8
12
10
6
11
5
8
10
2
16
13
3
14
4
1
15
7
9
12
6
12
12
1
12
15
6
14
7
4
9
8
8
1
8
15
10
14
11
4
5
13
13
1
13
15
3
14
2
4
16
1
15
14
4
12
6
7
9
8
10
11
5
13
3
2
16
2
2
11
2
5
16
8
13
10
3
14
14
11
14
5
4
8
1
10
15
7
7
11
7
5
9
8
12
10
6
11
5
8
10
2
16
13
3
14
4
1
15
7
9
12
6
23
23
17
23
24
5
1
7
8
14
15
16
4
4
17
4
24
6
1
13
8
20
15
22
10
10
17
10
24
12
1
19
8
21
15
3
11
11
17
11
24
18
1
25
8
2
15
9
17
24
1
8
15
23
5
7
14
16
4
6
13
20
22
10
12
19
21
3
11
18
25
2
9
2
2
21
2
3
9
19
25
10
11
12
18
20
20
21
20
3
22
19
13
10
4
12
6
8
8
21
8
3
15
19
1
10
17
12
24
14
14
21
14
3
16
19
7
10
23
12
5
21
3
19
10
12
2
9
25
11
18
20
22
13
4
6
8
15
1
17
24
14
16
7
23
5
9
9
16
9
3
6
2
7
13
12
5
5
16
5
3
10
2
11
13
8
4
4
16
4
3
15
2
14
13
1
16
3
2
13
9
6
7
12
5
10
11
8
4
15
14
1
5
5
16
5
3
10
2
11
13
8
9
9
16
9
3
6
2
7
13
12
4
4
16
4
3
15
2
14
13
1
16
3
2
13
5
10
11
8
9
6
7
12
4
15
14
1
12
12
1
12
15
6
14
7
4
9
8
8
1
8
15
10
14
11
4
5
13
13
1
13
15
3
14
2
4
16
1
15
14
4
12
6
7
9
8
10
11
5
13
3
2
16
1
1
12
1
6
15
7
14
9
4
13
13
12
13
6
3
7
2
9
16
8
8
12
8
6
10
7
11
9
5
12
6
7
9
1
15
14
4
13
3
2
16
8
10
11
5
5
5
10
5
3
16
13
2
8
11
4
4
10
4
3
9
13
7
8
14
15
15
10
15
3
6
13
12
8
1
10
3
13
8
5
16
2
11
4
9
7
14
15
6
12
1
4
4
5
4
16
9
2
7
11
14
15
15
5
15
16
6
2
12
11
1
10
10
5
10
16
3
2
13
11
8
5
16
2
11
4
9
7
14
15
6
12
1
10
3
13
8
7
7
2
7
11
14
5
4
16
9
12
12
2
12
11
1
5
15
16
6
13
13
2
13
11
8
5
10
16
3
2
11
5
16
7
14
4
9
12
1
15
6
13
8
10
3
15
15
1
15
23
14
16
7
4
18
21
11
24
24
1
24
23
17
16
13
4
9
21
2
20
20
1
20
23
8
16
19
4
12
21
6
5
5
1
5
23
3
16
10
4
22
21
25
1
23
16
4
21
15
14
7
18
11
24
17
13
9
2
20
8
19
12
6
5
3
10
22
25
15
15
1
15
16
14
23
7
4
18
21
11
24
24
1
24
16
17
23
13
4
9
21
2
20
20
1
20
16
8
23
19
4
12
21
6
5
5
1
5
16
10
23
3
4
22
21
25
1
16
23
4
21
15
14
7
18
11
24
17
13
9
2
20
8
19
12
6
5
10
3
22
25
15
15
1
15
23
14
16
7
4
18
21
11
24
24
1
24
23
17
16
13
4
9
21
2
20
20
1
20
23
8
16
19
4
12
21
6
5
5
1
5
23
3
16
10
4
22
21
25
1
23
16
4
21
15
14
7
18
11
24
17
13
9
2
20
8
19
12
6
5
3
10
22
25
4
4
21
4
11
14
2
7
6
18
25
22
16
16
21
16
11
17
2
13
6
9
25
10
23
23
21
23
11
8
2
19
6
12
25
3
1
1
21
1
11
15
2
24
6
20
25
5
21
11
2
6
25
4
14
7
18
22
16
17
13
9
10
23
8
19
12
3
1
15
24
20
5
· Special methods of construction › A method for constructing a magic square of order 3
c − (a − b)
c − (a − b)
c − b
c − (a − b)
c + (a + b)
c
c − a
c + (a − b)
c + a
c + a
c − b
c + a
c + (a + b)
c − (a + b)
c − a
c + b
c − b
c + (a + b)
c − a
c − (a − b)
c
c + (a − b)
c + a
c − (a + b)
c + b
step 1
1
step 2
1
2
step 3
3
3
Col 1
3
1
3
2
step 4
3
3
Col 1
3
4
4
Col 1
4
Col 3
2
1
3
4
2
step 5
3
3
Col 1
3
1
5
4
4
Col 1
4
Col 3
2
1
3
5
4
2
step 6
3
3
Col 1
3
1
5
4
4
Col 1
4
6
2
1
6
3
5
4
2
step 7
3
3
Col 1
3
1
5
6
7
4
4
Col 1
4
6
2
1
6
3
5
7
4
2
step 8
3
3
8
3
1
5
6
7
4
4
8
4
6
2
8
1
6
3
5
7
4
2
step 9
3
3
8
3
1
5
6
7
4
4
8
4
1
9
6
2
8
1
6
3
5
7
4
9
2
Order 3
3
3
8
3
1
5
6
7
4
4
8
4
1
9
6
2
8
1
6
3
5
7
4
9
2
Order 5
23
23
17
23
24
5
1
7
8
14
15
16
4
4
17
4
24
6
1
13
8
20
15
22
10
10
17
10
24
12
1
19
8
21
15
3
11
11
17
11
24
18
1
25
8
2
15
9
17
24
1
8
15
23
5
7
14
16
4
6
13
20
22
10
12
19
21
3
11
18
25
2
9
Order 9
57
57
47
57
58
68
69
79
80
9
1
11
12
22
23
33
34
44
45
46
67
67
47
67
58
78
69
8
80
10
1
21
12
32
23
43
34
54
45
56
77
77
47
77
58
7
69
18
80
20
1
31
12
42
23
53
34
55
45
66
6
6
47
6
58
17
69
19
80
30
1
41
12
52
23
63
34
65
45
76
16
16
47
16
58
27
69
29
80
40
1
51
12
62
23
64
34
75
45
5
26
26
47
26
58
28
69
39
80
50
1
61
12
72
23
74
34
4
45
15
36
36
47
36
58
38
69
49
80
60
1
71
12
73
23
3
34
14
45
25
37
37
47
37
58
48
69
59
80
70
1
81
12
2
23
13
34
24
45
35
47
58
69
80
1
12
23
34
45
57
68
79
9
11
22
33
44
46
67
78
8
10
21
32
43
54
56
77
7
18
20
31
42
53
55
66
6
17
19
30
41
52
63
65
76
16
27
29
40
51
62
64
75
5
26
28
39
50
61
72
74
4
15
36
38
49
60
71
73
3
14
25
37
48
59
70
81
2
13
24
35
M = Order 4
13
13
1
13
4
16
1
4
6
7
10
11
13
16
M = Order 4
12
12
1
12
15
6
14
7
4
9
8
8
1
8
15
10
14
11
4
5
13
13
1
13
15
3
14
2
4
16
1
15
14
4
12
6
7
9
8
10
11
5
13
3
2
16
M = Order 4
5
5
1
5
2
6
3
7
4
8
9
9
1
9
2
10
3
11
4
12
13
13
1
13
2
14
3
15
4
16
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
M = Order 4
12
12
16
12
15
11
14
10
13
9
8
8
16
8
15
7
14
6
13
5
4
4
16
4
15
3
14
2
13
1
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
M = Order 4
0
0
1
0
0
1
0
1
1
0
0
0
1
0
0
1
0
1
1
0
1
1
1
1
0
0
0
0
1
1
1
0
0
1
0
1
1
0
0
1
1
0
1
0
0
1
M = Order 4
12
12
1
12
15
6
14
7
4
9
8
8
1
8
15
10
14
11
4
5
13
13
1
13
15
3
14
2
4
16
1
15
14
4
12
6
7
9
8
10
11
5
13
3
2
16
M = Order 8
0
0
1
0
0
1
0
1
1
0
1
0
0
1
0
1
1
0
0
0
1
0
0
1
0
1
1
0
1
0
0
1
0
1
1
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
0
0
1
0
0
1
0
1
1
0
1
0
0
1
0
1
1
0
0
0
1
0
0
1
0
1
1
0
1
0
0
1
0
1
1
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
0
0
1
1
0
0
1
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
1
0
0
1
1
0
0
1
M = Order 8
25
25
1
25
4
28
5
29
8
32
33
33
1
33
4
36
5
37
8
40
57
57
1
57
4
60
5
61
8
64
1
4
5
8
10
11
14
15
18
19
22
23
25
28
29
32
33
36
37
40
42
43
46
47
50
51
54
55
57
60
61
64
M = Order 8
56
56
1
56
63
10
62
11
4
53
5
52
59
14
58
15
8
49
48
48
1
48
63
18
62
19
4
45
5
44
59
22
58
23
8
41
25
25
1
25
63
39
62
38
4
28
5
29
59
35
58
34
8
32
33
33
1
33
63
31
62
30
4
36
5
37
59
27
58
26
8
40
24
24
1
24
63
42
62
43
4
21
5
20
59
46
58
47
8
17
16
16
1
16
63
50
62
51
4
13
5
12
59
54
58
55
8
9
57
57
1
57
63
7
62
6
4
60
5
61
59
3
58
2
8
64
1
63
62
4
5
59
58
8
56
10
11
53
52
14
15
49
48
18
19
45
44
22
23
41
25
39
38
28
29
35
34
32
33
31
30
36
37
27
26
40
24
42
43
21
20
46
47
17
16
50
51
13
12
54
55
9
57
7
6
60
61
3
2
64
4
4
1
4
2
5
3
6
7
7
1
7
2
8
3
9
1
2
3
4
5
6
7
8
9
βa
βa
αa
βa
αb
βb
αc
βc
γa
γa
αa
γa
αb
γb
αc
γc
αa
αb
αc
βa
βb
βc
γa
γb
γc
β
β
α
β
α
β
α
β
γ
γ
α
γ
α
γ
α
γ
α
α
α
β
β
β
γ
γ
γ
a
a
a
a
b
b
c
c
a
a
a
a
b
b
c
c
a
b
c
a
b
c
a
b
c
γ
γ
β
γ
α
β
γ
α
α
α
β
α
α
γ
γ
β
β
α
γ
γ
β
α
α
γ
β
a
a
c
a
a
b
b
c
b
b
c
b
a
c
b
a
c
a
b
a
b
c
b
c
a
γa
γa
βc
γa
αa
βb
γb
αc
αb
αb
βc
αb
αa
γc
γb
βa
βc
αa
γb
γa
βb
αc
αb
γc
βa
7
7
6
7
1
5
8
3
2
2
6
2
1
9
8
4
6
1
8
7
5
3
2
9
4
15
15
10
15
5
10
10
5
10
20
10
0
10
15
10
0
0
10
0
Col 3
10
15
15
10
15
Col 3
0
5
10
10
0
5
10
20
0
0
10
0
10
15
0
10
5
5
10
5
15
10
20
15
0
20
5
0
0
0
10
0
15
5
20
10
0
15
5
20
20
20
10
20
15
0
20
5
0
10
5
15
15
15
10
15
15
20
20
0
0
5
5
10
10
15
20
0
5
5
10
15
20
0
0
5
10
15
20
20
0
5
10
15
15
20
0
5
10
1
1
2
1
1
5
5
4
4
3
3
2
5
5
2
5
1
4
5
3
4
2
3
1
4
4
2
4
1
3
5
2
4
1
3
5
3
3
2
3
1
2
5
1
4
5
3
4
2
1
5
4
3
1
5
4
3
2
5
4
3
2
1
4
3
2
1
5
3
2
1
5
4
6
6
12
6
16
15
25
19
4
23
8
2
5
5
12
5
16
9
25
13
4
17
8
21
24
24
12
24
16
3
25
7
4
11
8
20
Col 6
24
18
18
12
18
16
22
25
1
4
10
8
14
Col 6
18
Col 7
22
12
16
25
4
8
6
15
19
23
2
5
9
13
17
21
24
3
7
11
20
24
18
22
1
10
14
18
22
4
8
12
16
2
6
0
0
10
0
20
10
10
20
10
15
10
5
10
0
10
20
20
10
20
0
10
15
0
5
15
20
5
5
5
10
5
0
20
15
10
5
0
20
15
15
15
10
15
0
5
15
20
5
10
20
0
0
0
10
0
0
15
15
5
5
20
20
10
10
0
15
5
20
20
10
0
15
5
5
20
10
0
15
15
5
20
10
0
0
15
5
20
10
4
4
1
4
4
2
2
5
5
3
3
1
2
2
1
2
4
5
2
3
5
1
3
4
5
5
1
5
4
3
2
1
5
4
3
2
3
3
1
3
4
1
2
4
5
2
3
5
1
4
2
5
3
4
2
5
3
1
2
5
3
1
4
5
3
1
4
2
3
1
4
2
5
24
24
11
24
4
12
17
5
10
18
23
6
7
7
11
7
4
25
17
13
10
1
23
19
20
20
11
20
4
8
17
21
10
14
23
2
3
3
11
3
4
16
17
9
10
22
23
15
11
4
17
10
23
24
12
5
18
6
7
25
13
1
19
20
8
21
14
2
3
16
9
22
15
0
0
10
0
15
5
20
10
0
15
5
20
15
15
10
15
15
20
20
0
0
5
5
10
5
5
10
5
15
10
20
15
0
20
5
0
20
20
10
20
15
0
20
5
0
10
5
15
10
15
20
0
5
0
5
10
15
20
15
20
0
5
10
5
10
15
20
0
20
0
5
10
15
4
4
3
4
1
2
4
5
2
3
5
1
5
5
3
5
1
3
4
1
2
4
5
2
1
1
3
1
1
4
4
2
2
5
5
3
2
2
3
2
1
5
4
3
2
1
5
4
3
1
4
2
5
4
2
5
3
1
5
3
1
4
2
1
4
2
5
3
2
5
3
1
4
4
4
13
4
16
7
24
15
2
18
10
21
20
20
13
20
16
23
24
1
2
9
10
12
6
6
13
6
16
14
24
17
2
25
10
3
22
22
13
22
16
5
24
8
2
11
10
19
13
16
24
2
10
4
7
15
18
21
20
23
1
9
12
6
14
17
25
3
22
5
8
11
19
20
20
15
20
20
0
0
5
5
10
10
15
0
0
15
0
20
5
0
10
5
15
10
20
5
5
15
5
20
10
0
15
5
20
10
0
10
10
15
10
20
15
0
20
5
0
10
5
15
20
0
5
10
20
0
5
10
15
0
5
10
15
20
5
10
15
20
0
10
15
20
0
5
3
3
2
3
4
5
1
2
3
4
5
1
4
4
2
4
4
1
1
3
3
5
5
2
5
5
2
5
4
2
1
4
3
1
5
3
1
1
2
1
4
3
1
5
3
2
5
4
2
4
1
3
5
3
5
2
4
1
4
1
3
5
2
5
2
4
1
3
1
3
5
2
4
23
23
17
23
24
5
1
7
8
14
15
16
4
4
17
4
24
6
1
13
8
20
15
22
10
10
17
10
24
12
1
19
8
21
15
3
11
11
17
11
24
18
1
25
8
2
15
9
17
24
1
8
15
23
5
7
14
16
4
6
13
20
22
10
12
19
21
3
11
18
25
2
9
β
β
α
β
δ
γ
α
δ
δ
α
γ
β
β
γ
γ
γ
α
γ
β
δ
γ
α
δ
β
δ
δ
α
δ
β
γ
γ
β
δ
α
β
β
α
β
β
α
γ
δ
δ
γ
α
β
γ
δ
γ
δ
α
β
δ
γ
β
α
β
α
δ
γ
d
d
a
d
b
c
c
b
d
a
b
b
a
b
b
a
c
d
d
c
c
c
a
c
b
d
c
a
d
b
a
b
c
d
d
c
b
a
b
a
d
c
c
d
a
b
γd
γd
αa
γd
βb
δc
γc
αb
δd
βa
δb
δb
αa
δb
βb
γa
γc
βd
δd
αc
βc
βc
αa
βc
βb
αd
γc
δa
δd
γb
αa
βb
γc
δd
γd
δc
αb
βa
δb
γa
βd
αc
βc
αd
δa
γb
12
12
1
12
6
15
11
2
16
5
14
14
1
14
6
9
11
8
16
3
7
7
1
7
6
4
11
13
16
10
1
6
11
16
12
15
2
5
14
9
8
3
7
4
13
10
24
24
0
24
8
16
16
8
24
0
32
56
40
48
48
40
56
32
48
48
0
48
8
56
16
32
24
40
32
16
40
24
48
0
56
8
40
40
0
40
8
32
16
56
24
48
32
8
40
0
48
24
56
16
56
56
0
56
8
48
16
40
24
32
32
24
40
16
48
8
56
0
32
32
0
32
8
40
16
48
24
56
32
0
40
8
48
16
56
24
8
8
0
8
8
0
16
24
24
16
32
40
40
32
48
56
56
48
16
16
0
16
8
24
16
0
24
8
32
48
40
56
48
32
56
40
0
8
16
24
32
40
48
56
24
16
8
0
56
48
40
32
48
56
32
40
16
24
0
8
40
32
56
48
8
0
24
16
56
48
40
32
24
16
8
0
32
40
48
56
0
8
16
24
8
0
24
16
40
32
56
48
16
24
0
8
48
56
32
40
3
3
1
3
2
4
3
1
4
2
5
7
6
8
7
5
8
6
5
5
1
5
2
6
3
7
4
8
5
1
6
2
7
3
8
4
7
7
1
7
2
8
3
5
4
6
5
3
6
4
7
1
8
2
4
4
1
4
2
3
3
2
4
1
5
8
6
7
7
6
8
5
2
2
1
2
2
1
3
4
4
3
5
6
6
5
7
8
8
7
8
8
1
8
2
7
3
6
4
5
5
4
6
3
7
2
8
1
6
6
1
6
2
5
3
8
4
7
5
2
6
1
7
4
8
3
1
2
3
4
5
6
7
8
3
4
1
2
7
8
5
6
5
6
7
8
1
2
3
4
7
8
5
6
3
4
1
2
4
3
2
1
8
7
6
5
2
1
4
3
6
5
8
7
8
7
6
5
4
3
2
1
6
5
8
7
2
1
4
3
27
27
1
27
10
20
19
9
28
2
37
63
46
56
55
45
64
38
53
53
1
53
10
62
19
39
28
48
37
17
46
26
55
3
64
12
47
47
1
47
10
40
19
61
28
54
37
11
46
4
55
25
64
18
60
60
1
60
10
51
19
42
28
33
37
32
46
23
55
14
64
5
34
34
1
34
10
41
19
52
28
59
37
6
46
13
55
24
64
31
16
16
1
16
10
7
19
30
28
21
37
44
46
35
55
58
64
49
22
22
1
22
10
29
19
8
28
15
37
50
46
57
55
36
64
43
1
10
19
28
37
46
55
64
27
20
9
2
63
56
45
38
53
62
39
48
17
26
3
12
47
40
61
54
11
4
25
18
60
51
42
33
32
23
14
5
34
41
52
59
6
13
24
31
16
7
30
21
44
35
58
49
22
29
8
15
50
57
36
43
· Method of superposition › Euler's method
94
94
10
94
2
18
93
12
19
0
17
17
10
17
2
90
93
4
19
13
3
3
10
3
2
14
93
15
19
92
10
2
93
19
94
18
12
0
17
90
4
13
3
14
15
92
α
α
α
α
δ
δ
α
δ
β
γ
β
γ
α
δ
δ
δ
α
δ
γ
β
β
γ
δ
α
δ
δ
α
δ
γ
β
β
γ
δ
α
α
α
α
α
γ
γ
β
β
δ
δ
α
γ
β
δ
δ
β
γ
α
δ
β
γ
α
α
γ
β
δ
c
c
a
c
d
b
d
b
a
c
b
b
a
b
d
c
d
c
a
b
d
d
a
d
d
a
d
a
a
d
a
d
d
a
c
b
b
c
b
c
c
b
d
a
a
d
δc
δc
αa
δc
γd
βb
βd
γb
δa
αc
δb
δb
αa
δb
γd
βc
βd
γc
δa
αb
αd
αd
αa
αd
γd
γa
βd
βa
δa
δd
αa
γd
βd
δa
δc
βb
γb
αc
δb
βc
γc
αb
αd
γa
βa
δd
15
15
1
15
12
6
8
10
13
3
14
14
1
14
12
7
8
11
13
2
4
4
1
4
12
9
8
5
13
16
1
12
8
13
15
6
10
3
14
7
11
2
4
9
5
16
δ
δ
α
δ
γ
β
α
γ
γ
α
β
δ
δ
β
δ
δ
α
δ
β
γ
δ
α
γ
β
α
α
α
α
β
β
δ
δ
γ
γ
δ
δ
α
δ
β
γ
δ
α
γ
β
α
β
δ
γ
δ
γ
α
β
α
β
δ
γ
δ
γ
α
β
b
b
a
b
d
c
a
b
d
c
d
d
a
d
d
a
a
d
d
a
c
c
a
c
d
b
a
c
d
b
a
d
a
d
b
c
b
c
d
a
d
a
c
b
c
b
δb
δb
αa
δb
βd
γc
δa
αb
γd
βc
αd
αd
αa
αd
βd
βa
δa
δd
γd
γa
δc
δc
αa
δc
βd
γb
δa
αc
γd
βb
αa
βd
δa
γd
δb
γc
αb
βc
αd
βa
δd
γa
δc
γb
αc
βb
14
14
1
14
8
11
13
2
12
7
4
4
1
4
8
5
13
16
12
9
15
15
1
15
8
10
13
3
12
6
1
8
13
12
14
11
2
7
4
5
16
9
15
10
3
6
30
30
0
30
24
6
18
12
12
18
6
24
30
0
0
0
0
0
24
24
18
12
12
18
6
6
30
30
30
30
0
30
24
24
18
12
12
18
6
6
30
0
30
30
0
30
24
6
18
18
12
12
6
24
30
0
0
0
0
0
24
6
18
18
12
12
6
24
30
30
0
24
18
12
6
30
30
6
12
18
24
0
0
24
12
18
6
30
30
24
12
18
6
0
30
6
18
12
24
0
0
6
18
12
24
30
5
5
1
5
6
2
1
5
6
5
6
2
1
2
4
4
1
4
6
3
1
3
6
3
6
4
1
4
3
3
1
3
6
4
1
4
6
4
6
3
1
3
2
2
1
2
6
5
1
2
6
2
6
5
1
5
6
6
1
6
6
1
1
6
6
1
6
1
1
6
1
6
1
6
6
1
5
2
5
5
2
2
4
3
3
3
4
4
3
4
4
4
3
3
2
5
2
2
5
5
6
1
6
1
1
6
35
35
1
35
30
8
19
17
18
23
12
26
31
2
4
4
1
4
30
27
19
15
18
21
12
10
31
34
33
33
1
33
30
28
19
16
18
22
12
9
31
3
32
32
1
32
30
11
19
20
18
14
12
29
31
5
6
6
1
6
30
7
19
24
18
13
12
25
31
36
1
30
19
18
12
31
35
8
17
23
26
2
4
27
15
21
10
34
33
28
16
22
9
3
32
11
20
14
29
5
6
7
24
13
25
36
24
24
6
24
30
0
12
12
18
18
0
30
24
6
24
24
6
24
30
0
12
18
18
12
0
30
24
6
6
6
6
6
30
30
12
18
18
12
0
0
24
24
24
24
6
24
30
0
12
18
18
12
0
30
24
6
6
6
6
6
30
30
12
12
18
18
0
0
24
24
6
30
12
18
0
24
24
0
12
18
30
6
24
0
18
12
30
6
6
30
18
12
0
24
24
0
18
12
30
6
6
30
12
18
0
24
6
6
2
6
5
1
5
1
2
6
5
1
2
6
3
3
2
3
5
3
5
4
2
4
5
4
2
3
4
4
2
4
5
4
5
3
2
3
5
3
2
4
1
1
2
1
5
6
5
6
2
1
5
6
2
1
5
5
2
5
5
2
5
2
2
5
5
2
2
5
2
5
5
2
5
2
6
1
1
6
1
6
3
3
4
4
4
3
4
4
3
3
3
4
1
6
6
1
6
1
5
2
2
5
2
5
30
30
8
30
35
1
17
13
20
24
5
31
26
12
27
27
8
27
35
3
17
22
20
16
5
34
26
9
10
10
8
10
35
34
17
21
20
15
5
3
26
28
25
25
8
25
35
6
17
24
20
13
5
36
26
7
11
11
8
11
35
32
17
14
20
23
5
2
26
29
8
35
17
20
5
26
30
1
13
24
31
12
27
3
22
16
34
9
10
34
21
15
3
28
25
6
24
13
36
7
11
32
14
23
2
29
6
6
6
6
30
0
12
18
24
24
18
12
0
30
24
24
6
24
30
0
12
12
24
6
18
18
0
30
6
6
6
6
30
30
12
18
24
24
18
12
0
0
24
24
6
24
30
30
12
12
24
6
18
18
0
0
24
24
6
24
30
0
12
18
24
6
18
12
0
30
6
30
12
24
18
0
6
0
18
24
12
30
24
0
12
6
18
30
6
30
18
24
12
0
24
30
12
6
18
0
24
0
18
6
12
30
6
6
2
6
2
1
5
1
2
6
5
6
5
1
3
3
2
3
2
4
5
3
2
4
5
3
5
4
5
5
2
5
2
5
5
2
2
5
5
2
5
2
4
4
2
4
2
3
5
4
2
3
5
4
5
3
1
1
2
1
2
6
5
6
2
1
5
1
5
6
2
2
5
2
5
5
6
1
1
6
6
1
3
4
3
4
3
4
5
5
2
5
2
2
4
3
4
3
4
3
1
6
6
1
1
6
12
12
8
12
32
1
17
19
26
30
23
18
5
31
27
27
8
27
32
4
17
15
26
10
23
21
5
34
11
11
8
11
32
35
17
20
26
29
23
14
5
2
28
28
8
28
32
33
17
16
26
9
23
22
5
3
25
25
8
25
32
6
17
24
26
7
23
13
5
36
8
32
17
26
23
5
12
1
19
30
18
31
27
4
15
10
21
34
11
35
20
29
14
2
28
33
16
9
22
3
25
6
24
7
13
36
56
56
0
56
48
8
16
40
32
24
24
32
40
16
8
48
56
0
0
0
0
0
48
48
16
16
32
32
24
24
40
40
8
8
56
56
56
56
0
56
48
8
16
40
32
24
24
32
40
16
8
48
56
0
56
56
0
56
48
8
16
40
32
24
24
32
40
16
8
48
56
0
0
0
0
0
48
48
16
16
32
32
24
24
40
40
8
8
56
56
56
56
0
56
48
8
16
40
32
24
24
32
40
16
8
48
56
0
0
0
0
0
48
48
16
16
32
32
24
24
40
40
8
8
56
56
0
48
16
32
24
40
8
56
56
8
40
24
32
16
48
0
0
48
16
32
24
40
8
56
56
8
40
24
32
16
48
0
56
8
40
24
32
16
48
0
0
48
16
32
24
40
8
56
56
8
40
24
32
16
48
0
0
48
16
32
24
40
8
56
7
7
1
7
8
2
1
7
8
2
8
2
1
7
8
2
1
7
3
3
1
3
8
6
1
3
8
6
8
6
1
3
8
6
1
3
5
5
1
5
8
4
1
5
8
4
8
4
1
5
8
4
1
5
4
4
1
4
8
5
1
4
8
5
8
5
1
4
8
5
1
4
6
6
1
6
8
3
1
6
8
3
8
3
1
6
8
3
1
6
2
2
1
2
8
7
1
2
8
7
8
7
1
2
8
7
1
2
8
8
1
8
8
1
1
8
8
1
8
1
1
8
8
1
1
8
1
8
1
8
8
1
8
1
7
2
7
2
2
7
2
7
3
6
3
6
6
3
6
3
5
4
5
4
4
5
4
5
4
5
4
5
5
4
5
4
6
3
6
3
3
6
3
6
2
7
2
7
7
2
7
2
8
1
8
1
1
8
1
8
63
63
1
63
56
10
17
47
40
26
32
34
41
23
16
50
57
7
3
3
1
3
56
54
17
19
40
38
32
30
41
43
16
14
57
59
61
61
1
61
56
12
17
45
40
28
32
36
41
21
16
42
57
5
60
60
1
60
56
13
17
44
40
29
32
37
41
20
16
53
57
4
6
6
1
6
56
51
17
22
40
35
32
27
41
46
16
11
57
62
58
58
1
58
56
15
17
42
40
31
32
39
41
18
16
55
57
2
8
8
1
8
56
47
17
24
40
33
32
25
41
48
16
9
57
64
1
56
17
40
32
41
16
57
63
10
47
26
34
23
50
7
3
54
19
38
30
43
14
59
61
12
45
28
36
21
42
5
60
13
44
29
37
20
53
4
6
51
22
35
27
46
11
62
58
15
42
31
39
18
55
2
8
47
24
33
25
48
9
64
56
56
0
56
48
8
56
0
8
48
16
40
32
24
40
16
24
32
0
0
0
0
48
48
56
56
8
8
16
16
32
32
40
40
24
24
56
56
0
56
48
8
56
0
8
48
16
40
32
24
40
16
24
32
48
48
0
48
48
0
56
8
8
56
16
32
32
16
40
24
24
40
8
8
0
8
48
56
56
48
8
0
16
24
32
40
40
32
24
16
48
48
0
48
48
0
56
8
8
56
16
32
32
16
40
24
24
40
8
8
0
8
48
56
56
48
8
0
16
24
32
40
40
32
24
16
0
48
56
8
16
32
40
24
56
8
0
48
40
24
16
32
0
48
56
8
16
32
40
24
56
8
0
48
40
24
16
32
48
0
8
56
32
16
24
40
8
56
48
0
24
40
32
16
48
0
8
56
32
16
24
40
8
56
48
0
24
40
32
16
7
7
1
7
8
2
1
7
8
2
7
1
2
8
7
1
2
8
8
8
1
8
8
1
1
8
8
1
7
2
2
7
7
2
2
7
2
2
1
2
8
7
1
2
8
7
7
8
2
1
7
8
2
1
3
3
1
3
8
6
1
3
8
6
7
5
2
4
7
5
2
4
5
5
1
5
8
4
1
5
8
4
7
3
2
6
7
3
2
6
6
6
1
6
8
3
1
6
8
3
7
4
2
5
7
4
2
5
4
4
1
4
8
5
1
4
8
5
7
6
2
3
7
6
2
3
1
8
1
8
7
2
7
2
7
2
7
2
1
8
1
8
8
1
8
1
2
7
2
7
2
7
2
7
8
1
8
1
3
6
3
6
5
4
5
4
5
4
5
4
3
6
3
6
6
3
6
3
4
5
4
5
4
5
4
5
6
3
6
3
63
63
1
63
56
10
57
7
16
50
23
41
34
32
47
17
26
40
8
8
1
8
56
49
57
64
16
9
23
18
34
39
47
42
26
31
58
58
1
58
56
15
57
2
16
55
23
48
34
25
47
24
26
33
51
51
1
51
56
6
57
11
16
62
23
37
34
20
47
29
26
44
13
13
1
13
56
60
57
53
16
4
23
27
34
46
47
35
26
22
54
54
1
54
56
3
57
14
16
59
23
36
34
21
47
28
26
45
12
12
1
12
56
61
57
52
16
5
23
30
34
43
47
38
26
19
1
56
57
16
23
34
47
26
63
10
7
50
41
32
17
40
8
49
64
9
18
39
42
31
58
15
2
55
48
25
24
33
51
6
11
62
37
20
29
44
13
60
53
4
27
46
35
22
54
3
14
59
36
21
28
45
12
61
52
5
30
43
38
19
· Method of borders › Bordering method for order 3
b∗
b∗
u
b∗
a
0
v
b
v∗
v∗
u
v∗
a
a∗
v
u∗
u
a
v
b∗
0
b
v∗
a∗
u∗
2
2
1
2
−4
0
3
−2
−3
−3
1
−3
−4
4
3
−1
1
−4
3
2
0
−2
−3
4
−1
7
7
6
7
1
5
8
3
2
2
6
2
1
9
8
4
6
1
8
7
5
3
2
9
4
· Method of borders › Bordering method for order 5
d∗
d∗
u
d∗
v
d
e∗
e∗
u
e∗
v
e
f∗
f∗
u
f∗
v
f
v∗
v∗
u
v∗
a
a∗
b
b∗
c
c∗
v
u∗
u
a
b
c
v
d∗
d
e∗
e
f∗
f
v∗
a∗
b∗
c∗
u∗
5
5
10
5
12
-5
-11
-11
10
-11
12
11
8
8
10
8
12
-8
-12
-12
10
-12
-7
7
-9
9
-6
6
12
-10
10
-7
-9
-6
12
5
-5
-11
11
8
-8
-12
7
9
6
-10
18
18
23
18
25
8
2
2
23
2
25
24
21
21
23
21
25
5
1
1
23
1
6
20
4
22
7
19
25
3
23
6
4
7
25
18
8
2
24
21
5
1
20
22
19
3
7
7
10
7
12
-7
-11
-11
10
-11
12
11
6
6
10
6
12
-6
-12
-12
10
-12
-5
5
-9
9
-8
8
12
-10
10
-5
-9
-8
12
7
-7
-11
11
6
-6
-12
5
9
8
-10
20
20
23
20
25
6
2
2
23
2
25
24
19
19
23
19
25
7
1
1
23
1
8
18
4
22
5
21
25
3
23
8
4
5
25
20
6
2
24
19
7
1
18
22
21
3
· Method of borders › Bordering method for order 5
12, 10
12, 10
u, v
12, 10
a, b, c
-6, -7, -9
d, e, f
-11, 5, 8
12, 10
12, 10
u, v
12, 10
a, b, c
-5, -8, -9
d, e, f
-11, 6, 7
11, 5
11, 5
u, v
11, 5
a, b, c
6, -10, -12
d, e, f
-9, 7, 8
10, 6
10, 6
u, v
10, 6
a, b, c
5, -9, -12
d, e, f
-11, 7, 8
10, 6
10, 6
u, v
10, 6
a, b, c
7, -11, -12
d, e, f
-9, 5, 8
9, 7
9, 7
u, v
9, 7
a, b, c
5, -10, -11
d, e, f
-12, 6, 8
9, 7
9, 7
u, v
9, 7
a, b, c
6, -10, -12
d, e, f
-11, 5, 8
8, 6
8, 6
u, v
8, 6
a, b, c
7, -10, -11
d, e, f
-12, 5, 9
8, 6
8, 6
u, v
8, 6
a, b, c
9, -11, -12
d, e, f
-10, 5, 7
7, 5
7, 5
u, v
7, 5
a, b, c
9, -10, -11
d, e, f
-12, 6, 8
u, v
a, b, c
d, e, f
12, 10
-6, -7, -9
-11, 5, 8
12, 10
-5, -8, -9
-11, 6, 7
11, 5
6, -10, -12
-9, 7, 8
10, 6
5, -9, -12
-11, 7, 8
10, 6
7, -11, -12
-9, 5, 8
9, 7
5, -10, -11
-12, 6, 8
9, 7
6, -10, -12
-11, 5, 8
8, 6
7, -10, -11
-12, 5, 9
8, 6
9, -11, -12
-10, 5, 7
7, 5
9, -10, -11
-12, 6, 8
· Method of borders › Continuous enumeration methods
67
67
8
67
80
22
78
64
76
62
75
61
12
26
14
28
16
24
10
15
69
69
8
69
80
55
78
32
76
52
75
51
12
36
14
34
16
27
10
13
71
71
8
71
80
57
78
47
76
38
75
45
12
40
14
35
16
25
10
11
73
73
8
73
80
59
78
49
76
43
75
41
12
39
14
33
16
23
10
9
5
5
8
5
80
19
78
29
76
42
75
37
12
44
14
53
16
63
10
77
3
3
8
3
80
17
78
48
76
30
75
31
12
46
14
50
16
65
10
79
1
1
8
1
80
58
78
18
76
20
75
21
12
56
14
54
16
60
10
81
72
72
8
72
80
2
78
4
76
6
75
7
12
70
14
68
16
66
10
74
8
80
78
76
75
12
14
16
10
67
22
64
62
61
26
28
24
15
69
55
32
52
51
36
34
27
13
71
57
47
38
45
40
35
25
11
73
59
49
43
41
39
33
23
9
5
19
29
42
37
44
53
63
77
3
17
48
30
31
46
50
65
79
1
58
18
20
21
56
54
60
81
72
2
4
6
7
70
68
66
74
· Method of borders › Continuous enumeration methods
240
240
15
240
16
17
18
18
15
18
16
239
19
19
15
19
16
238
237
237
15
237
16
20
236
236
15
236
16
21
22
22
15
22
16
235
23
23
15
23
16
234
233
233
15
233
16
24
232
232
15
232
16
25
26
26
15
26
16
231
27
27
15
27
16
230
229
229
15
229
16
28
228
228
15
228
16
29
30
30
15
30
16
227
241
241
15
241
1
256
255
2
254
3
4
253
5
252
251
6
250
7
8
249
9
248
10
247
246
11
245
12
244
13
243
14
16
242
15
1
255
254
4
5
251
250
8
9
10
246
245
244
243
16
240
17
18
239
19
238
237
20
236
21
22
235
23
234
233
24
232
25
26
231
27
230
229
28
228
29
30
227
241
256
2
3
253
252
6
7
249
248
247
11
12
13
14
242
· Method of borders › Continuous enumeration methods
56
56
7
56
8
9
10
10
7
10
8
55
11
11
7
11
8
54
53
53
7
53
8
12
52
52
7
52
8
13
14
14
7
14
8
51
57
57
7
57
1
64
2
63
62
3
61
4
60
5
59
6
8
58
7
1
2
62
61
60
59
8
56
9
10
55
11
54
53
12
52
13
14
51
57
64
63
3
4
5
6
58
· Method of borders › Continuous enumeration methods
83
83
9
83
10
18
16
16
9
16
10
85
87
87
9
87
10
14
12
12
9
12
10
89
11
11
9
11
10
90
93
93
9
93
10
8
6
6
9
6
10
95
97
97
9
97
10
4
91
91
9
91
100
1
2
99
98
3
5
96
94
7
88
13
15
86
84
17
10
92
9
100
2
98
5
94
88
15
84
10
83
18
16
85
87
14
12
89
11
90
93
8
6
95
97
4
91
1
99
3
96
7
13
86
17
92
Order 3
3
3
8
3
1
5
6
7
4
4
8
4
1
9
6
2
8
1
6
3
5
7
4
9
2
Order 3×3
63
63
63
63
63
63
63
63
0
0
0
0
0
0
45
45
45
45
45
45
63
63
63
63
63
63
63
63
0
0
0
0
0
0
45
45
45
45
45
45
18
18
63
18
63
18
63
18
0
36
0
36
0
36
45
54
45
54
45
54
18
18
63
18
63
18
63
18
0
36
0
36
0
36
45
54
45
54
45
54
18
18
63
18
63
18
63
18
0
36
0
36
0
36
45
54
45
54
45
54
27
27
63
27
63
27
63
27
0
72
0
72
0
72
45
9
45
9
45
9
27
27
63
27
63
27
63
27
0
72
0
72
0
72
45
9
45
9
45
9
27
27
63
27
63
27
63
27
0
72
0
72
0
72
45
9
45
9
45
9
63
63
63
0
0
0
45
45
45
63
63
63
0
0
0
45
45
45
63
63
63
0
0
0
45
45
45
18
18
18
36
36
36
54
54
54
18
18
18
36
36
36
54
54
54
18
18
18
36
36
36
54
54
54
27
27
27
72
72
72
9
9
9
27
27
27
72
72
72
9
9
9
27
27
27
72
72
72
9
9
9
Order 3×3
66
66
71
66
64
68
69
70
8
3
1
5
6
7
53
48
46
50
51
52
67
67
71
67
64
72
69
65
8
4
1
9
6
2
53
49
46
54
51
47
26
26
71
26
64
19
69
24
8
44
1
37
6
42
53
62
46
55
51
60
21
21
71
21
64
23
69
25
8
39
1
41
6
43
53
57
46
59
51
61
22
22
71
22
64
27
69
20
8
40
1
45
6
38
53
58
46
63
51
56
35
35
71
35
64
28
69
33
8
80
1
73
6
78
53
17
46
10
51
15
30
30
71
30
64
32
69
34
8
75
1
77
6
79
53
12
46
14
51
16
31
31
71
31
64
36
69
29
8
76
1
81
6
74
53
13
46
18
51
11
71
64
69
8
1
6
53
46
51
66
68
70
3
5
7
48
50
52
67
72
65
4
9
2
49
54
47
26
19
24
44
37
42
62
55
60
21
23
25
39
41
43
57
59
61
22
27
20
40
45
38
58
63
56
35
28
33
80
73
78
17
10
15
30
32
34
75
77
79
12
14
16
31
36
29
76
81
74
13
18
11
Order 3
7
7
2
7
9
5
4
3
6
6
2
6
9
1
4
8
2
9
4
7
5
3
6
1
8
Order 4
12
12
1
12
14
7
11
2
8
13
6
6
1
6
14
9
11
16
8
3
15
15
1
15
14
4
11
5
8
10
1
14
11
8
12
7
2
13
6
9
16
3
15
4
5
10
Order 3 × 4
7
7
2
7
9
5
4
3
119
124
126
122
121
120
92
97
99
95
94
93
65
70
72
68
67
66
6
6
2
6
9
1
4
8
119
123
126
118
121
125
92
96
99
91
94
98
65
69
72
64
67
71
101
101
2
101
9
108
4
103
119
56
126
63
121
58
92
11
99
18
94
13
65
110
72
117
67
112
106
106
2
106
9
104
4
102
119
61
126
59
121
57
92
16
99
14
94
12
65
115
72
113
67
111
105
105
2
105
9
100
4
107
119
60
126
55
121
62
92
15
99
10
94
17
65
114
72
109
67
116
47
47
2
47
9
54
4
49
119
74
126
81
121
76
92
137
99
144
94
139
65
20
72
27
67
22
52
52
2
52
9
50
4
48
119
79
126
77
121
75
92
142
99
140
94
138
65
25
72
23
67
21
51
51
2
51
9
46
4
53
119
78
126
73
121
80
92
141
99
136
94
143
65
24
72
19
67
26
128
128
2
128
9
135
4
130
119
29
126
36
121
31
92
38
99
45
94
40
65
83
72
90
67
85
133
133
2
133
9
131
4
129
119
34
126
32
121
30
92
43
99
41
94
39
65
88
72
86
67
84
132
132
2
132
9
127
4
134
119
33
126
28
121
35
92
42
99
37
94
44
65
87
72
82
67
89
2
9
4
119
126
121
92
99
94
65
72
67
7
5
3
124
122
120
97
95
93
70
68
66
6
1
8
123
118
125
96
91
98
69
64
71
101
108
103
56
63
58
11
18
13
110
117
112
106
104
102
61
59
57
16
14
12
115
113
111
105
100
107
60
55
62
15
10
17
114
109
116
47
54
49
74
81
76
137
144
139
20
27
22
52
50
48
79
77
75
142
140
138
25
23
21
51
46
53
78
73
80
141
136
143
24
19
26
128
135
130
29
36
31
38
45
40
83
90
85
133
131
129
34
32
30
43
41
39
88
86
84
132
127
134
33
28
35
42
37
44
87
82
89
Order 4 × 3
28
28
17
28
30
23
27
18
24
29
129
140
142
135
139
130
136
141
49
60
62
55
59
50
56
61
22
22
17
22
30
25
27
32
24
19
129
134
142
137
139
144
136
131
49
54
62
57
59
64
56
51
31
31
17
31
30
20
27
21
24
26
129
143
142
132
139
133
136
138
49
63
62
52
59
53
56
58
97
97
17
97
30
110
27
107
24
104
129
65
142
78
139
75
136
72
49
33
62
46
59
43
56
40
108
108
17
108
30
103
27
98
24
109
129
76
142
71
139
66
136
77
49
44
62
39
59
34
56
45
102
102
17
102
30
105
27
112
24
99
129
70
142
73
139
80
136
67
49
38
62
41
59
48
56
35
111
111
17
111
30
100
27
101
24
106
129
79
142
68
139
69
136
74
49
47
62
36
59
37
56
42
81
81
17
81
30
94
27
91
24
88
129
1
142
14
139
11
136
8
49
113
62
126
59
123
56
120
92
92
17
92
30
87
27
82
24
93
129
12
142
7
139
2
136
13
49
124
62
119
59
114
56
125
86
86
17
86
30
89
27
96
24
83
129
6
142
9
139
16
136
3
49
118
62
121
59
128
56
115
95
95
17
95
30
84
27
85
24
90
129
15
142
4
139
5
136
10
49
127
62
116
59
117
56
122
17
30
27
24
129
142
139
136
49
62
59
56
28
23
18
29
140
135
130
141
60
55
50
61
22
25
32
19
134
137
144
131
54
57
64
51
31
20
21
26
143
132
133
138
63
52
53
58
97
110
107
104
65
78
75
72
33
46
43
40
108
103
98
109
76
71
66
77
44
39
34
45
102
105
112
99
70
73
80
67
38
41
48
35
111
100
101
106
79
68
69
74
47
36
37
42
81
94
91
88
1
14
11
8
113
126
123
120
92
87
82
93
12
7
2
13
124
119
114
125
86
89
96
83
6
9
16
3
118
121
128
115
95
84
85
90
15
4
5
10
127
116
117
122
140
140
1
140
142
7
139
2
8
141
9
132
134
15
131
10
16
133
17
124
126
23
123
18
24
125
6
6
1
6
142
137
139
144
8
3
9
14
134
129
131
136
16
11
17
22
126
121
123
128
24
19
143
143
1
143
142
4
139
5
8
138
9
135
134
12
131
13
16
130
17
127
126
20
123
21
24
122
25
25
1
25
142
118
139
115
8
32
9
33
134
110
131
107
16
40
17
41
126
102
123
99
24
48
116
116
1
116
142
31
139
26
8
117
9
108
134
39
131
34
16
109
17
100
126
47
123
42
24
101
30
30
1
30
142
113
139
120
8
27
9
38
134
105
131
112
16
35
17
46
126
97
123
104
24
43
119
119
1
119
142
28
139
29
8
114
9
111
134
36
131
37
16
106
17
103
126
44
123
45
24
98
49
49
1
49
142
94
139
91
8
56
9
57
134
86
131
83
16
64
17
65
126
78
123
75
24
72
92
92
1
92
142
55
139
50
8
93
9
84
134
63
131
58
16
85
17
76
126
71
123
66
24
77
54
54
1
54
142
89
139
96
8
51
9
62
134
81
131
88
16
59
17
70
126
73
123
80
24
67
95
95
1
95
142
52
139
53
8
90
9
87
134
60
131
61
16
82
17
79
126
68
123
69
24
74
1
142
139
8
9
134
131
16
17
126
123
24
140
7
2
141
132
15
10
133
124
23
18
125
6
137
144
3
14
129
136
11
22
121
128
19
143
4
5
138
135
12
13
130
127
20
21
122
25
118
115
32
33
110
107
40
41
102
99
48
116
31
26
117
108
39
34
109
100
47
42
101
30
113
120
27
38
105
112
35
46
97
104
43
119
28
29
114
111
36
37
106
103
44
45
98
49
94
91
56
57
86
83
64
65
78
75
72
92
55
50
93
84
63
58
85
76
71
66
77
54
89
96
51
62
81
88
59
70
73
80
67
95
52
53
90
87
60
61
82
79
68
69
74
75
75
69
75
74
72
79
65
68
78
29
115
114
32
119
25
28
118
61
83
82
64
87
57
60
86
66
66
69
66
74
77
79
76
68
71
29
26
114
117
119
116
28
31
61
58
82
85
87
84
60
63
80
80
69
80
74
67
79
70
68
73
29
120
114
27
119
30
28
113
61
88
82
59
87
62
60
81
21
21
69
21
74
122
79
127
68
20
29
53
114
90
119
95
28
52
61
13
82
130
87
135
60
12
123
123
69
123
74
24
79
17
68
126
29
91
114
56
119
49
28
94
61
131
82
16
87
9
60
134
18
18
69
18
74
125
79
124
68
23
29
50
114
93
119
92
28
55
61
10
82
133
87
132
60
15
128
128
69
128
74
19
79
22
68
121
29
96
114
51
119
54
28
89
61
136
82
11
87
14
60
129
45
45
69
45
74
98
79
103
68
44
29
5
114
138
119
143
28
4
61
37
82
106
87
111
60
36
99
99
69
99
74
48
79
41
68
102
29
139
114
8
119
1
28
142
61
107
82
40
87
33
60
110
42
42
69
42
74
101
79
100
68
47
29
2
114
141
119
140
28
7
61
34
82
109
87
108
60
39
104
104
69
104
74
43
79
46
68
97
29
144
114
3
119
6
28
137
61
112
82
35
87
38
60
105
69
74
79
68
29
114
119
28
61
82
87
60
75
72
65
78
115
32
25
118
83
64
57
86
66
77
76
71
26
117
116
31
58
85
84
63
80
67
70
73
120
27
30
113
88
59
62
81
21
122
127
20
53
90
95
52
13
130
135
12
123
24
17
126
91
56
49
94
131
16
9
134
18
125
124
23
50
93
92
55
10
133
132
15
128
19
22
121
96
51
54
89
136
11
14
129
45
98
103
44
5
138
143
4
37
106
111
36
99
48
41
102
139
8
1
142
107
40
33
110
42
101
100
47
2
141
140
7
34
109
108
39
104
43
46
97
144
3
6
137
112
35
38
105
· Method of composition › For squares of doubly even order
64
64
60
64
82
69
88
74
56
79
90
68
59
81
24
28
118
33
124
110
20
115
126
32
23
117
83
83
60
83
82
75
88
72
56
65
90
78
59
62
24
119
118
111
124
36
20
29
126
114
23
26
84
84
60
84
82
66
88
77
56
76
90
71
59
61
24
120
118
30
124
113
20
112
126
35
23
25
58
58
60
58
82
80
88
67
56
70
90
73
59
87
24
22
118
116
124
31
20
34
126
109
23
123
86
86
60
86
82
63
88
57
56
89
90
55
59
85
24
122
118
27
124
21
20
125
126
19
23
121
6
6
60
6
82
136
88
142
56
2
90
144
59
5
24
42
118
100
124
106
20
38
126
108
23
41
10
10
60
10
82
15
88
128
56
133
90
14
59
135
24
46
118
51
124
92
20
97
126
50
23
99
137
137
60
137
82
129
88
18
56
11
90
132
59
8
24
101
118
93
124
54
20
47
126
96
23
44
138
138
60
138
82
12
88
131
56
130
90
17
59
7
24
102
118
48
124
95
20
94
126
53
23
43
4
4
60
4
82
134
88
13
56
16
90
127
59
141
24
40
118
98
124
49
20
52
126
91
23
105
140
140
60
140
82
9
88
3
56
143
90
1
59
139
24
104
118
45
124
39
20
107
126
37
23
103
60
82
88
56
90
59
24
118
124
20
126
23
64
69
74
79
68
81
28
33
110
115
32
117
83
75
72
65
78
62
119
111
36
29
114
26
84
66
77
76
71
61
120
30
113
112
35
25
58
80
67
70
73
87
22
116
31
34
109
123
86
63
57
89
55
85
122
27
21
125
19
121
6
136
142
2
144
5
42
100
106
38
108
41
10
15
128
133
14
135
46
51
92
97
50
99
137
129
18
11
132
8
101
93
54
47
96
44
138
12
131
130
17
7
102
48
95
94
53
43
4
134
13
16
127
141
40
98
49
52
91
105
140
9
3
143
1
139
104
45
39
107
37
103
2
2
0
2
1
3
0
1
2
3
3
3
0
3
1
2
0
1
3
2
3
3
0
3
2
1
0
2
3
1
5
5
1
5
0
2
1
3
3
3
1
3
0
2
1
4
Col 4
3
1
0
1
5
2
3
3
2
4
3
5
5
1
5
0
3
1
2
4
4
1
4
0
3
1
3
Col 4
2
1
0
1
5
3
2
4
3
3
2
4
4
2
4
0
3
2
1
5
5
2
5
0
3
2
3
Col 4
1
2
0
2
4
3
1
5
3
3
1
Medjig 2×2
2
2
0
2
1
3
3
1
2
0
3
3
0
3
1
2
3
0
2
1
1
1
0
1
1
0
3
2
2
3
0
1
3
2
2
3
1
0
3
2
0
1
1
0
2
3
Medjig 2×2
3
3
0
3
1
2
2
1
3
0
1
1
0
1
1
0
2
3
3
2
2
2
0
2
1
3
2
0
3
1
0
1
2
3
3
2
1
0
1
0
3
2
2
3
0
1
Medjig 2×2
3
3
0
3
2
1
3
0
1
2
0
0
0
0
2
2
3
3
1
1
3
3
0
3
2
1
3
0
1
2
0
2
3
1
3
1
0
2
0
2
3
1
3
1
0
2
Order 4
8
8
1
8
14
11
4
5
15
10
13
13
1
13
14
2
4
16
15
3
12
12
1
12
14
7
4
9
15
6
1
14
4
15
8
11
5
10
13
2
16
3
12
7
9
6
Medjig 4 × 4
2
2
0
2
1
3
3
1
2
0
0
2
1
3
3
1
2
0
3
3
0
3
1
2
3
0
2
1
0
3
1
2
3
0
2
1
1
1
0
1
1
0
3
2
2
3
0
1
1
0
3
2
2
3
0
0
0
0
1
1
3
3
2
2
0
0
1
1
3
3
2
2
2
2
0
2
1
3
3
1
2
0
0
2
1
3
3
1
2
0
3
3
0
3
1
2
3
0
2
1
0
3
1
2
3
0
2
1
1
1
0
1
1
0
3
2
2
3
0
1
1
0
3
2
2
3
0
1
3
2
0
1
3
2
2
3
1
0
2
3
1
0
3
2
0
1
3
2
0
1
1
0
2
3
1
0
2
3
0
1
3
2
0
1
3
2
2
3
1
0
2
3
1
0
3
2
0
1
3
2
0
1
1
0
2
3
1
0
2
3
Order 8
33
33
1
33
17
49
62
30
46
14
4
36
20
52
63
31
47
15
56
56
1
56
17
40
62
11
46
27
4
53
20
37
63
10
47
26
24
24
1
24
17
8
62
43
46
59
4
21
20
5
63
42
47
58
13
13
1
13
17
29
62
50
46
34
4
16
20
32
63
51
47
35
45
45
1
45
17
61
62
18
46
2
4
48
20
64
63
19
47
3
60
60
1
60
17
44
62
7
46
23
4
57
20
41
63
6
47
22
28
28
1
28
17
12
62
39
46
55
4
25
20
9
63
38
47
54
1
17
62
46
4
20
63
47
33
49
30
14
36
52
31
15
56
40
11
27
53
37
10
26
24
8
43
59
21
5
42
58
13
29
50
34
16
32
51
35
45
61
18
2
48
64
19
3
60
44
7
23
57
41
6
22
28
12
39
55
25
9
38
54
Order 4
8
8
1
8
14
11
4
5
15
10
13
13
1
13
14
2
4
16
15
3
12
12
1
12
14
7
4
9
15
6
1
14
4
15
8
11
5
10
13
2
16
3
12
7
9
6
Medjig 4 × 4
2
2
0
2
1
3
0
3
1
2
2
1
3
0
3
1
2
0
0
0
0
0
1
3
0
0
1
2
2
3
3
1
3
0
2
3
1
1
0
1
1
2
0
3
1
1
2
0
3
2
3
1
2
2
2
2
0
2
1
1
0
0
1
2
2
3
3
1
3
2
2
1
3
3
0
3
1
0
0
3
1
1
2
0
3
2
3
3
2
0
3
3
0
3
1
2
0
0
1
1
2
2
3
3
3
0
2
1
1
1
0
1
1
0
0
3
1
2
2
1
3
0
3
2
2
3
0
1
0
1
2
3
3
2
2
3
3
2
1
0
1
0
0
3
0
2
3
1
0
3
1
2
3
1
0
2
1
2
2
1
0
2
3
1
2
1
3
0
3
1
0
2
3
0
3
2
0
1
2
3
0
1
1
0
3
2
1
0
2
3
Order 8
33
33
1
33
17
49
14
62
30
46
36
20
52
4
63
31
47
15
8
8
1
8
17
56
14
11
30
43
36
53
52
21
63
10
47
58
24
24
1
24
17
40
14
59
30
27
36
5
52
37
63
26
47
42
45
45
1
45
17
29
14
2
30
34
36
64
52
32
63
35
47
19
61
61
1
61
17
13
14
50
30
18
36
16
52
48
63
51
47
3
60
60
1
60
17
44
14
7
30
23
36
41
52
57
63
6
47
22
28
28
1
28
17
12
14
55
30
39
36
25
52
9
63
38
47
54
1
17
14
30
36
52
63
47
33
49
62
46
20
4
31
15
8
56
11
43
53
21
10
58
24
40
59
27
5
37
26
42
45
29
2
34
64
32
35
19
61
13
50
18
16
48
51
3
60
44
7
23
41
57
6
22
28
12
55
39
25
9
38
54
Medjig 3 × 3
1
1
2
1
3
0
0
3
2
1
0
3
2
1
3
3
2
3
3
1
0
1
2
2
0
2
2
0
0
0
2
0
3
2
0
0
2
3
0
3
2
1
3
3
2
3
3
2
0
2
2
0
0
0
2
2
0
0
2
0
3
1
0
3
2
1
0
1
2
3
2
3
0
2
0
2
1
0
3
1
3
1
3
1
1
2
2
0
0
2
0
3
3
1
3
2
2
0
0
2
0
1
3
1
1
3
Medjig 3 × 3
2
2
0
2
3
1
0
2
1
3
3
1
2
0
3
3
0
3
3
0
0
3
1
2
3
0
2
1
2
2
0
2
3
1
0
1
1
0
3
2
2
3
0
0
0
0
3
1
0
3
1
1
3
3
2
1
2
2
0
2
3
3
0
0
1
2
3
0
2
2
0
3
0
1
3
2
2
1
2
3
1
0
3
0
3
2
0
1
2
1
1
0
2
3
0
1
3
1
3
1
2
3
0
2
0
2
Medjig 3 × 3
2
2
0
2
1
3
0
2
3
1
3
1
2
0
3
3
0
3
1
1
0
0
3
1
3
3
2
1
0
0
0
0
1
2
0
2
3
3
3
0
2
2
3
3
0
3
1
2
0
3
3
0
3
0
2
1
1
1
0
1
1
0
0
2
3
1
3
2
2
3
0
1
0
3
3
2
2
3
2
1
1
0
3
1
0
1
3
1
0
2
2
3
0
2
3
2
3
0
0
1
1
0
2
1
2
3
Order 3
3
3
8
3
1
5
6
7
4
4
8
4
1
9
6
2
8
1
6
3
5
7
4
9
2
Medjig 3 × 3
1
1
2
1
3
0
0
3
2
1
0
3
2
1
3
3
2
3
3
1
0
1
2
2
0
2
2
0
0
0
2
0
3
2
0
0
2
3
0
3
2
1
3
3
2
3
3
2
0
2
2
0
0
0
2
2
0
0
2
0
3
1
0
3
2
1
0
1
2
3
2
3
0
2
0
2
1
0
3
1
3
1
3
1
1
2
2
0
0
2
0
3
3
1
3
2
2
0
0
2
0
1
3
1
1
3
Order 6
17
17
26
17
35
8
1
28
19
10
6
33
24
15
30
30
26
30
35
12
1
14
19
23
6
25
24
7
3
3
26
3
35
21
1
5
19
32
6
34
24
16
31
31
26
31
35
22
1
27
19
9
6
2
24
20
4
4
26
4
35
13
1
36
19
18
6
11
24
29
26
35
1
19
6
24
17
8
28
10
33
15
30
12
14
23
25
7
3
21
5
32
34
16
31
22
27
9
2
20
4
13
36
18
11
29
Order 5
23
23
17
23
24
5
1
7
8
14
15
16
4
4
17
4
24
6
1
13
8
20
15
22
10
10
17
10
24
12
1
19
8
21
15
3
11
11
17
11
24
18
1
25
8
2
15
9
17
24
1
8
15
23
5
7
14
16
4
6
13
20
22
10
12
19
21
3
11
18
25
2
9
Medjig 5 × 5
2
2
0
2
1
3
3
0
1
2
0
2
1
3
3
0
1
2
3
1
2
0
3
3
0
3
1
0
3
2
1
3
0
0
1
2
3
0
1
2
3
1
2
2
1
1
0
1
1
2
3
1
1
0
0
3
1
1
3
3
1
1
3
3
2
0
0
0
0
0
1
2
3
3
1
1
0
1
1
2
3
2
1
0
3
1
2
3
1
1
0
1
1
3
3
0
1
2
0
0
1
3
3
3
1
1
3
0
2
2
3
3
0
3
1
0
3
3
1
2
0
2
1
0
3
0
1
2
3
1
2
2
1
1
0
1
1
2
3
0
1
1
0
3
1
1
3
1
1
3
3
3
2
0
3
3
0
3
1
2
3
1
1
3
0
1
1
0
3
1
1
3
3
0
2
1
1
1
0
1
1
0
3
2
1
0
0
3
1
2
3
2
1
0
3
2
2
3
0
1
3
1
0
1
3
1
3
2
2
3
0
2
2
3
0
2
1
0
3
0
2
3
0
2
0
2
1
2
1
2
1
0
3
1
3
1
3
0
0
2
3
1
1
2
2
0
1
3
1
3
0
2
0
3
3
1
0
2
3
0
3
2
2
0
0
2
1
2
1
2
0
1
3
1
1
3
3
0
3
2
1
3
1
0
1
3
0
1
1
0
2
0
3
2
2
0
2
3
Order 10
67
67
17
67
42
92
99
24
49
74
1
51
26
76
83
8
33
58
90
40
65
15
98
98
17
98
42
23
99
55
49
80
1
7
26
57
83
14
33
64
90
41
65
66
48
48
17
48
42
73
99
30
49
5
1
82
26
32
83
89
33
39
90
91
65
16
4
4
17
4
42
54
99
81
49
31
1
38
26
63
83
70
33
20
90
47
65
97
29
29
17
29
42
79
99
6
49
56
1
13
26
88
83
95
33
45
90
22
65
72
85
85
17
85
42
10
99
87
49
62
1
69
26
19
83
21
33
71
90
28
65
53
35
35
17
35
42
60
99
12
49
37
1
94
26
44
83
46
33
96
90
78
65
3
86
86
17
86
42
61
99
43
49
93
1
100
26
25
83
27
33
77
90
9
65
34
36
36
17
36
42
11
99
68
49
18
1
95
26
75
83
52
33
2
90
59
65
84
17
42
99
49
1
26
83
33
90
65
67
92
24
74
51
76
8
58
40
15
98
23
55
80
7
57
14
64
41
66
48
73
30
5
82
32
89
39
91
16
4
54
81
31
38
63
70
20
47
97
29
79
6
56
13
88
95
45
22
72
85
10
87
62
69
19
21
71
28
53
35
60
12
37
94
44
46
96
78
3
86
61
43
93
100
25
27
77
9
34
36
11
68
18
95
75
52
2
59
84
· Variations of the magic square › Extra constraints
113
113
17
113
89
59
71
5
47
47
17
47
89
29
71
101
17
89
71
113
59
5
47
29
101
· Variations of the magic square › Extra constraints
88
88
96
88
11
69
89
91
68
16
61
61
96
61
11
86
89
18
68
99
19
19
96
19
11
98
89
66
68
81
96
11
89
68
88
69
91
16
61
86
18
99
19
98
66
81
M = 32768 · Variations of the magic square › Multiplicative magic squares
8
8
16
8
512
32
4
128
256
256
16
256
512
2
4
64
16
512
4
8
32
128
256
2
64
M = 216
36
36
2
36
9
6
12
1
3
3
2
3
9
4
12
18
2
9
12
36
6
1
3
4
18
M = 6720
40
40
1
40
6
28
20
2
56
3
14
14
1
14
6
5
20
24
56
4
12
12
1
12
6
8
20
7
56
10
1
6
20
56
40
28
2
3
14
5
24
4
12
8
7
10
M = 6,227,020,800
24
24
27
24
50
52
66
3
84
40
13
54
2
70
32
11
56
56
27
56
50
9
66
20
84
44
13
36
2
65
32
6
55
55
27
55
50
72
66
91
84
1
13
16
2
36
32
30
4
4
27
4
50
24
66
45
84
60
13
77
2
12
32
26
10
10
27
10
50
22
66
48
84
39
13
5
2
48
32
63
78
78
27
78
50
7
66
8
84
18
13
40
2
33
32
60
27
50
66
84
13
2
32
24
52
3
40
54
70
11
56
9
20
44
36
65
6
55
72
91
1
16
36
30
4
24
45
60
77
12
26
10
22
48
39
5
48
63
78
7
8
18
40
33
60
Skalli multiplicative 7×7 of complex numbers · Variations of the magic square › Multiplicative magic squares of complex numbers
63
63
21
63
+14i
−35i
−70
28
+30i
+114i
−9i
−14i
−105
2
−217i
+6i
16
3
+50i
−11i
4
211
−14i
+357i
14
−123
−8i
−87i
31
31
21
31
+14i
−15i
−70
13
+30i
−13i
−93
−103
−9i
+69i
−105
−261
−217i
−213i
16
49
+50i
−49i
4
−46
−14i
+2i
14
−6
−8i
+2i
102
102
21
102
+14i
−84i
−70
−28
+30i
−14i
−93
43
−9i
+247i
−105
−10
−217i
−2i
16
5
+50i
+9i
4
31
−14i
−27i
14
−77
−8i
+91i
−22
−22
21
−22
+14i
−6i
−70
7
+30i
+7i
−93
8
−9i
+14i
−105
50
−217i
+20i
16
−525
+50i
−492i
4
−28
−14i
−42i
14
−73
−8i
+17i
54
54
21
54
+14i
+68i
−70
138
+30i
−165i
−93
−56
−9i
−98i
−105
−63
−217i
+35i
16
4
+50i
−8i
4
2
−14i
−4i
14
70
−8i
−53i
24
24
21
24
+14i
+22i
−70
−46
+30i
−16i
−93
6
−9i
−4i
−105
17
−217i
+20i
16
110
+50i
+160i
4
84
−14i
−189i
14
42
−8i
−14i
21
+14i
−70
+30i
−93
−9i
−105
−217i
16
+50i
4
−14i
14
−8i
63
−35i
28
+114i
−14i
2
+6i
3
−11i
211
+357i
−123
−87i
31
−15i
13
−13i
−103
+69i
−261
−213i
49
−49i
−46
+2i
−6
+2i
102
−84i
−28
−14i
43
+247i
−10
−2i
5
+9i
31
−27i
−77
+91i
−22
−6i
7
+7i
8
+14i
50
+20i
−525
−492i
−28
−42i
−73
+17i
54
+68i
138
−165i
−56
−98i
−63
+35i
4
−8i
2
−4i
70
−53i
24
+22i
−46
−16i
6
−4i
17
+20i
110
+160i
84
−189i
42
−14i
First knownadditive-multiplicative magic square 8×8 found by W. W. Horner in 1955 Sum = 840 Product = 2058068231856000
105
105
162
105
207
152
51
100
26
29
133
138
120
243
116
39
25
34
92
92
162
92
207
27
51
91
26
136
133
45
120
38
116
150
25
261
57
57
162
57
207
30
51
174
26
225
133
108
120
23
116
119
25
104
58
58
162
58
207
75
51
171
26
90
133
17
120
52
116
216
25
161
13
13
162
13
207
68
51
184
26
189
133
50
120
87
116
135
25
114
200
200
162
200
207
203
51
15
26
76
133
117
120
102
116
46
25
81
153
153
162
153
207
78
51
54
26
69
133
232
120
175
116
19
25
60
162
207
51
26
133
120
116
25
105
152
100
29
138
243
39
34
92
27
91
136
45
38
150
261
57
30
174
225
108
23
119
104
58
75
171
90
17
52
216
161
13
68
184
189
50
87
135
114
200
203
15
76
117
102
46
81
153
78
54
69
232
175
19
60
Smallest known additive-multiplicative semimagic square 4×4 found by L. Morgenstern in 2007 Sum = 247 Product = 3369600
30
30
156
30
18
144
48
60
25
13
16
16
156
16
18
20
48
130
25
81
45
45
156
45
18
65
48
9
25
128
156
18
48
25
30
144
60
13
16
20
130
81
45
65
9
128
Smallest known additive-multiplicative magic square 7×7 found by Sébastien Miquel(Sébastien Miquel) in August 2016 Sum = 465 Product = 150885504000 · Variations of the magic square › Additive-multiplicative magic and semimagic squares
20
20
126
20
66
70
50
16
90
54
48
189
1
110
84
6
100
100
126
100
66
2
50
22
90
98
48
36
1
72
84
135
96
96
126
96
66
60
50
81
90
4
48
10
1
49
84
165
3
3
126
3
66
63
50
30
90
176
48
120
1
45
84
28
99
99
126
99
66
180
50
14
90
25
48
7
1
108
84
32
21
21
126
21
66
24
50
252
90
18
48
55
1
80
84
15
126
66
50
90
48
1
84
20
70
16
54
189
110
6
100
2
22
98
36
72
135
96
60
81
4
10
49
165
3
63
30
176
120
45
28
99
180
14
25
7
108
32
21
24
252
18
55
80
15
A
A
S
A
A
R
T
E
O
P
R
O
T
T
S
T
A
E
T
N
O
E
R
T
O
O
S
O
A
P
T
E
O
R
R
A
R
R
S
R
A
O
T
T
O
A
R
S
S
A
T
O
R
A
R
E
P
O
T
E
N
E
T
O
P
E
R
A
R
O
T
A
S
8
8
6
8
66
11
848
544
938
839
1
1
6
1
66
11
848
383
938
839
2
2
6
2
66
73
848
774
938
447
6
66
848
938
8
11
544
839
1
11
383
839
2
73
774
447
E
E
H
E
E
Q
S
A
E
L
S
S
H
S
E
E
H
E
S
G
B
B
H
B
H
E
S
E
B
E
Q
A
L
S
E
G
B
D
D
A
D
D
A
A
R
M
A
A
A
A
A
D
R
A
A
M
D
M
M
A
M
D
A
A
D
M
A
A
D
A
M
D
A
R
A
A
R
A
D
M
A
D
A

References

  1. "Earlier Known Uses of Some of the Words of Mathematics (M)"
    http://jeff560.tripod.com/m.html
  2. The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English
    https://books.google.com/books?id=iuoZSkSOBQsC&pg=PA130
  3. The most famous Arabic book on magic, named "Shams Al-ma'arif (Arabic: كتاب شمس المعارف), for Ahmed bin Ali Al-boni, who
  4. Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures
    https://doi.org/10.1007%2F978-1-4020-4425-0_9350
  5. Magic Squares and Cubes
    https://archive.org/details/MagicSquaresAndCubes_754
  6. Journal of the American Oriental Society
    http://www.chinesehsc.org/downloads/cammann/camman_the_evolution_of_magic_squares_in_china.pdf
  7. The Legacy of the Luoshu
  8. MacTutor History of Mathematics Archive
    http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Yang_Hui.html
  9. The Influence of Chinese Mathematical Arts on Seki Kowa by Shigeru Jochi, MA, School of Oriental and African Studies, Un
    https://core.ac.uk/download/pdf/161528551.pdf
  10. A history of Japanese mathematics
    https://archive.org/details/in.ernet.dli.2015.161063
  11. A history of Japanese mathematics
    https://archive.org/details/in.ernet.dli.2015.161063
  12. A history of Japanese mathematics
    https://archive.org/details/in.ernet.dli.2015.161063
  13. A history of Japanese mathematics
    https://archive.org/details/in.ernet.dli.2015.161063
  14. Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures
    https://doi.org/10.1007%2F978-1-4020-4425-0_9154
  15. Magic squares in Japanese mathematics
    https://books.google.com/books?id=sU1tAAAAMAAJ&q=Isomura++Kittoku
  16. Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures
    https://doi.org/10.1007%2F978-1-4020-4425-0_9778
  17. Indian Journal of History of Science
    https://web.archive.org/web/20180117131236/http://124.108.19.235:12000/jspui/bitstream/123456789/11663/1/Vol27_1_5_BDatta.pdf
  18. Historia Mathematica
    https://core.ac.uk/download/pdf/82500954.pdf
  19. J. P. Hogendijk, A. I. Sabra, The Enterprise of Science in Islam: New Perspectives, Published by MIT Press, 2003, ISBN 0
  20. Helaine Selin, Ubiratan D'Ambrosio, Mathematics Across Cultures: The History of Non-Western Mathematics, Published by Sp
  21. Archive for History of Exact Sciences
    http://doc.rero.ch/record/316928/files/407_2003_Article_71.pdf
  22. Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures
  23. Magic squares in the tenth century: Two Arabic treatises by Antaki and Buzjani
  24. Sesiano, J., Abūal-Wafā\rasp's treatise on magic squares (French), Z. Gesch. Arab.-Islam. Wiss. 12 (1998), 121–244.
  25. History of Religions
    https://doi.org/10.1086%2F462584
  26. Bulletin de la Société Vaudoise des Sciences Naturelles
  27. Peter, J. Barta, The Seal-Ring of Proportion and the magic rings (2016), pp. 6–9.
  28. Theoretical Influences of China on Arabic Alchemy
    https://books.google.com/books?id=4b2zC7fL558C
  29. Jābir ibn Hayyān, Book of the Scales. French translation in: Marcelin Berthelot (1827–1907), Histoire de sciences. La ch
  30. al-Ghazālī, Deliverance From Error (al-munqidh min al-ḍalāl ) ch. 145. Arabic: al-Munkidh min al-dalal. ed. J. Saliba –
  31. Medieval Textual Cultures: Agents of Transmission, Translation and Transformation
  32. The Latin version is Liber de septem figuris septem planetarum figurarum Geberi regis Indorum. This treatise is the iden
  33. Les carrés magiques dans les pays islamiques
  34. The mystery of numbers
  35. www.maa.org
    https://web.archive.org/web/20191205132327/https://www.maa.org/press/periodicals/convergence/the-magic-squares-of-manuel-moschopoulos-introduction
  36. History of Religions
    https://doi.org/10.1086%2F462589
  37. presently in the Biblioteca Vaticana (cod. Reg. Lat. 1283a)
  38. See Alfonso X el Sabio, Astromagia (Ms. Reg. lat. 1283a), a cura di A.D'Agostino, Napoli, Liguori, 1992
  39. Mars magic square appears in figure 1 of "Saturn and Melancholy: Studies in the History of Natural Philosophy, Religion,
  40. The squares can be seen on folios 20 and 21 of MS. 2433, at the Biblioteca Universitaria of Bologna. They also appear on
  41. In a 1981 article ("Zur Frühgeschichte der magischen Quadrate in Westeuropa" i.e. "Prehistory of Magic Squares in Wester
  42. This manuscript text (circa 1496–1508) is also at the Biblioteca Universitaria in Bologna. It can be seen in full at the
    http://www.uriland.it/matematica/DeViribus/Presentazione.html
  43. Pacioli states: A lastronomia summamente hanno mostrato li supremi di quella commo Ptolomeo, al bumasar ali, al fragano,
  44. Fermat, magic squares and the idea of self-supporting blocks
    https://helda.helsinki.fi/bitstream/handle/10138/322537/Muurinen_Ismo_gradu_2020.pdf?sequence=2
  45. A History of Algorithms: From the Pebble to the Microchip
    https://doi.org/10.1007%2F978-3-642-18192-4
  46. MacTutor History of Mathematics Archive
    http://www-history.mcs.st-andrews.ac.uk/Biographies/Franklin_Benjamin.html
  47. Mathematical Recreations and Essays
  48. Magic Squares and Cubes
    https://archive.org/details/MagicSquaresAndCubes_754
  49. "Virtual Home of Paul Muljadi"
    https://web.archive.org/web/20051109234521/http://www.muljadi.org/MagicSquares.htm
  50. "Magic cube with Dürer's square" Ali Skalli's magic squares and magic cubes
    https://sites.google.com/site/aliskalligvaen/home-page/-magic-cube-with-duerer-s-square
  51. "The magic square on the Passion façade: keys to understanding it"
    https://blog.sagradafamilia.org/en/divulgation/the-magic-square-the-passion-facade-keys-to-understanding-it/
  52. Letters: The Mathematical Intelligencer; 2003; 25; 4: pp. 6–7.
  53. "Magic cube with Gaudi's square"
    https://web.archive.org/web/20211215212634/https://sites.google.com/site/aliskalligvaen/home-page/-magic-cube-with-gaudi-s-square
  54. mathforum.org
    https://web.archive.org/web/20180302092216/http://mathforum.org/alejandre/magic.square/adler/adler5.html
  55. Mathematical Gazette
    https://web.archive.org/web/20171114111409/http://home.cc.umanitoba.ca/~loly/MathGaz.pdf
  56. The Quarterly Journal of Mathematics
    https://doi.org/10.1093%2Fqmath%2F10.1.296
  57. How many magic squares are there? by Walter Trump, Nürnberg, January 11, 2001
    http://www.trump.de/magic-squares/howmany.html
  58. Anything but square: from magic squares to Sudoku by Hardeep Aiden, Plus Magazine, March 1, 2006
    https://plus.maths.org/content/anything-square-magic-squares-sudoku
  59. PLOS ONE
    https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4431883
  60. Mathematical Recreations
    https://archive.org/details/mathematicalrecr0000krai
  61. The Mathematical Intelligencer
    https://doi.org/10.1007%2FBF03024415
  62. budshaw.ca
    https://budshaw.ca/Associative.html
  63. nrich.maths.org
    https://nrich.maths.org/1338
  64. Mathematical Circles Squared By Phillip E. Johnson, Howard Whitley Eves, p. 22
  65. http://oz.nthu.edu.tw/~u9621110/IT2010/txt/0929/canterburypuzzle00dudeuoft.pdf The Canterbury Puzzles and Other Curious
    http://oz.nthu.edu.tw/~u9621110/IT2010/txt/0929/canterburypuzzle00dudeuoft.pdf
  66. http://budshaw.ca/howMany.html, Bordered Square Numbers, S. Harry White, 2009
    http://budshaw.ca/howMany.html
  67. http://www.law05.si/iwms/presentations/Styan.pdf Some illustrated comments on 5×5 golden magic matrices and on 5×5 Stife
    http://www.law05.si/iwms/presentations/Styan.pdf
  68. Hartley, M. "Making Big Magic Squares".
    http://www.dr-mikes-math-games-for-kids.com/making-big-magic-squares.html
  69. http://budshaw.ca/2xNComposite.html, 2N Composite Squares, S. Harry White, 2009
    http://budshaw.ca/2xNComposite.html
  70. Karl Fulves, Self-working Number Magic (Dover Magic Books)
    http://www.markfarrar.co.uk/othmsq01.htm#reversible
  71. Arithmetica integra
    https://books.google.com/books?id=fndPsRv08R0C
  72. "8x8 multiplicative magic square of complex numbers" Ali Skalli's magic squares and magic cubes
    https://sites.google.com/site/aliskalligvaen/home-page/-multiplicative-of-complex-numbers-8x8
  73. "Multimagie.com – Additive-Multiplicative magic squares, 8th and 9th-order"
    http://multimagie.com/English/Multiplicative8_9.htm
  74. "Multimagie.com – Smallest additive-multiplicative magic square"
    http://multimagie.com/English/SmallestAddMult.htm
  75. Magic squares are given a whole new dimension, The Observer, April 3, 2011
    https://www.theguardian.com/science/2011/apr/03/magic-squares-geomagic-lee-sallows
  76. Les carrés magiques géométriques by Jean-Paul Delahaye, Pour La Science No. 428, June 2013
    https://www.pourlascience.fr/sd/mathematiques/les-carres-magiques-geometriques-7372.php
  77. Futility Closet
    https://www.futilitycloset.com/2017/01/19/area-magic-squares/
  78. Magic Designs, Robert B. Ely III, Journal of Recreational Mathematics volume 1 number 1, January 1968
  79. "MathWorld News: There Are No Magic Knight's Tours on the Chessboard"
    http://mathworld.wolfram.com/news/2003-08-06/magictours/
  80. Mayhematics, "12×12 Magic Knight's Tours"
    http://mayhematics.com/t/md.htm
  81. n
  82. See Juris Lidaka, The Book of Angels, Rings, Characters and Images of the Planets in Conjuring Spirits, C. Fangier ed. (
  83. Benedek Láng, Demons in Krakow, and Image Magic in a Magical Handbook, in Christian Demonology and Popular Mythology, Gá
  84. According to the correspondence principle, each of the seven planets is associated to a given metal: lead to Saturn, iro
  85. Dictionary of Mysticism and the Esoteric Traditions
  86. The Secret Lore of Magic
Image
Source: —
Tip: Wheel or +/− to zoom, drag to pan, Esc to close.