|
Before we explore the possibilities to produce a 12x12 magic square, it is necessary to analyze the
principle of the basic key method.
Why does the basic key method of construction lead to a most perfect
16x16 (Franklin pan)magic square?
Why does the basic key method lead to most perfect (Franklin pan)magic squares that are multiples
of 8, for example the most perfect 16x16 Franklin pan magic square.
1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16 = 136.
(1st) sum = sum = 34 = 1/4 x 136
That is the condition which ensures that the sum of the digits of the quarter rows and the quarter
columns is a quarter of the magic sum.
(2nd) sum = sum = 34 = 1/4 x 136
That is the condition which ensures that the sum of the digits of the quarter diagonals (and the
[parallel] [mirrored] bent diagonals) is a quarter of the magic sum.
(3rd) An odd number of digits must always be between reverse vertical combinations of digits
(reverse vertical combinations may not lie next to each other).
This is the condition that allows it to be used as a second (column) square, thus rotate the first
(row) square by a quarter turn to the left and all the digits from 1 to 256 can be found in the
square.
How to use the basic key method of construction for the 12x12
square?
For squares that are odd multiples of 4, for example the 12x12 square, it is not possible to
comply with all three above-mentioned conditions. You must choose between the following:
[option a]
(1st) ) sum = sum = 26 = 1/3 x 78
|
1
|
10
|
11
|
4
|
8
|
6
|
5
|
7
|
12
|
3
|
2
|
9
|
|
12
|
3
|
2
|
9
|
5
|
7
|
8
|
6
|
1
|
10
|
11
|
4
|
This is the condition which ensures that the sum of the digits of 1/3 rows and 1/3 columns
are 1/3 of the magic sum, but the sum of the digits of 1/2 rows and 1/2 columns are not ½
of the magic sum.
(2nd) sum = sum = 39 = 1/2 x 78
|
1
|
10
|
11
|
4
|
8
|
6
|
|
12
|
3
|
2
|
9
|
5
|
7
|
This is the condition which ensures that the sum of the digits of ½ diagonals (and the [parallel]
[mirrored] bent diagonals) is ½ of the magic sum.
(3rd) This is the condition that allows it to be used as second (column) square, thus rotate the
first (row) square by a quarter turn to the left, and you will find all the digits from 1 to 144 can
be found in the square.
1x digit from row pattern
|
1
|
10
|
11
|
4
|
8
|
6
|
5
|
7
|
12
|
3
|
2
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9
|
|
12
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3
|
2
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9
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5
|
7
|
8
|
6
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1
|
10
|
11
|
4
|
|
1
|
10
|
11
|
4
|
8
|
6
|
5
|
7
|
12
|
3
|
2
|
9
|
|
12
|
3
|
2
|
9
|
5
|
7
|
8
|
6
|
1
|
10
|
11
|
4
|
|
1
|
10
|
11
|
4
|
8
|
6
|
5
|
7
|
12
|
3
|
2
|
9
|
|
12
|
3
|
2
|
9
|
5
|
7
|
8
|
6
|
1
|
10
|
11
|
4
|
|
1
|
10
|
11
|
4
|
8
|
6
|
5
|
7
|
12
|
3
|
2
|
9
|
|
12
|
3
|
2
|
9
|
5
|
7
|
8
|
6
|
1
|
10
|
11
|
4
|
|
1
|
10
|
11
|
4
|
8
|
6
|
5
|
7
|
12
|
3
|
2
|
9
|
|
12
|
3
|
2
|
9
|
5
|
7
|
8
|
6
|
1
|
10
|
11
|
4
|
|
1
|
10
|
11
|
4
|
8
|
6
|
5
|
7
|
12
|
3
|
2
|
9
|
|
12
|
3
|
2
|
9
|
5
|
7
|
8
|
6
|
1
|
10
|
11
|
4
|
+ 12x digit from column pattern
|
9
|
4
|
9
|
4
|
9
|
4
|
9
|
4
|
9
|
4
|
9
|
4
|
|
2
|
11
|
2
|
11
|
2
|
11
|
2
|
11
|
2
|
11
|
2
|
11
|
|
3
|
10
|
3
|
10
|
3
|
10
|
3
|
10
|
3
|
10
|
3
|
10
|
|
12
|
1
|
12
|
1
|
12
|
1
|
12
|
1
|
12
|
1
|
12
|
1
|
|
7
|
6
|
7
|
6
|
7
|
6
|
7
|
6
|
7
|
6
|
7
|
6
|
|
5
|
8
|
5
|
8
|
5
|
8
|
5
|
8
|
5
|
8
|
5
|
8
|
|
6
|
7
|
6
|
7
|
6
|
7
|
6
|
7
|
6
|
7
|
6
|
7
|
|
8
|
5
|
8
|
5
|
8
|
5
|
8
|
5
|
8
|
5
|
8
|
5
|
|
4
|
9
|
4
|
9
|
4
|
9
|
4
|
9
|
4
|
9
|
4
|
9
|
|
11
|
2
|
11
|
2
|
11
|
2
|
11
|
2
|
11
|
2
|
11
|
2
|
|
10
|
3
|
10
|
3
|
10
|
3
|
10
|
3
|
10
|
3
|
10
|
3
|
|
1
|
12
|
1
|
12
|
1
|
12
|
1
|
12
|
1
|
12
|
1
|
12
|
= 12x12 panmagic square (option a)
|
97
|
46
|
107
|
40
|
104
|
42
|
101
|
43
|
108
|
39
|
98
|
45
|
|
24
|
123
|
14
|
129
|
17
|
127
|
20
|
126
|
13
|
130
|
23
|
124
|
|
25
|
118
|
35
|
112
|
32
|
114
|
29
|
115
|
36
|
111
|
26
|
117
|
|
144
|
3
|
134
|
9
|
137
|
7
|
140
|
6
|
133
|
10
|
143
|
4
|
|
73
|
70
|
83
|
64
|
80
|
66
|
77
|
67
|
84
|
63
|
74
|
69
|
|
60
|
87
|
50
|
93
|
53
|
91
|
56
|
90
|
49
|
94
|
59
|
88
|
|
61
|
82
|
71
|
76
|
68
|
78
|
65
|
79
|
72
|
75
|
62
|
81
|
|
96
|
51
|
86
|
57
|
89
|
55
|
92
|
54
|
85
|
58
|
95
|
52
|
|
37
|
106
|
47
|
100
|
44
|
102
|
41
|
103
|
48
|
99
|
38
|
105
|
|
132
|
15
|
122
|
21
|
125
|
19
|
128
|
18
|
121
|
22
|
131
|
16
|
|
109
|
34
|
119
|
28
|
116
|
30
|
113
|
31
|
120
|
27
|
110
|
33
|
|
12
|
135
|
2
|
141
|
5
|
139
|
8
|
138
|
1
|
142
|
11
|
136
|
[option b]
(1st) sum = sum = 39 = 1/2 x 78
|
1
|
11
|
10
|
2
|
7
|
8
|
12
|
4
|
3
|
9
|
6
|
5
|
|
12
|
2
|
3
|
11
|
6
|
5
|
1
|
9
|
10
|
4
|
7
|
8
|
This is the condition which ensures that the sum of the digits of ½ rows and ½ columns are ½ of the
magic sum.
(2nd) The sum of the ½ twisting rows is not 39, but the sum of the whole twisting rows is 78. That is
the boundary condition which ensures that the sum of the digits of the whole diagonal is the magic sum,
but the sum of the digits of ½ diagonals (and the [parallel] [mirrored] bent diagonals) is not ½ of the
magic sum.
(3rd) Odd number of digits are between reverse vertical combinations, so rotating the first (row) square
by a quarter turn to the left enables you to use it as a second (column) square and all the digits from 1 to
144 can be found in the square.
1x digit from row pattern
|
1
|
11
|
10
|
2
|
7
|
8
|
12
|
4
|
3
|
9
|
6
|
5
|
|
12
|
2
|
3
|
11
|
6
|
5
|
1
|
9
|
10
|
4
|
7
|
8
|
|
1
|
11
|
10
|
2
|
7
|
8
|
12
|
4
|
3
|
9
|
6
|
5
|
|
12
|
2
|
3
|
11
|
6
|
5
|
1
|
9
|
10
|
4
|
7
|
8
|
|
1
|
11
|
10
|
2
|
7
|
8
|
12
|
4
|
3
|
9
|
6
|
5
|
|
12
|
2
|
3
|
11
|
6
|
5
|
1
|
9
|
10
|
4
|
7
|
8
|
|
1
|
11
|
10
|
2
|
7
|
8
|
12
|
4
|
3
|
9
|
6
|
5
|
|
12
|
2
|
3
|
11
|
6
|
5
|
1
|
9
|
10
|
4
|
7
|
8
|
|
1
|
11
|
10
|
2
|
7
|
8
|
12
|
4
|
3
|
9
|
6
|
5
|
|
12
|
2
|
3
|
11
|
6
|
5
|
1
|
9
|
10
|
4
|
7
|
8
|
|
1
|
11
|
10
|
2
|
7
|
8
|
12
|
4
|
3
|
9
|
6
|
5
|
|
12
|
2
|
3
|
11
|
6
|
5
|
1
|
9
|
10
|
4
|
7
|
8
|
+ 12x digit from column pattern
|
5
|
8
|
5
|
8
|
5
|
8
|
5
|
8
|
5
|
8
|
5
|
8
|
|
6
|
7
|
6
|
7
|
6
|
7
|
6
|
7
|
6
|
7
|
6
|
7
|
|
9
|
4
|
9
|
4
|
9
|
4
|
9
|
4
|
9
|
4
|
9
|
4
|
|
3
|
10
|
3
|
10
|
3
|
10
|
3
|
10
|
3
|
10
|
3
|
10
|
|
4
|
9
|
4
|
9
|
4
|
9
|
4
|
9
|
4
|
9
|
4
|
9
|
|
12
|
1
|
12
|
1
|
12
|
1
|
12
|
1
|
12
|
1
|
12
|
1
|
|
8
|
5
|
8
|
5
|
8
|
5
|
8
|
5
|
8
|
5
|
8
|
5
|
|
7
|
6
|
7
|
6
|
7
|
6
|
7
|
6
|
7
|
6
|
7
|
6
|
|
2
|
11
|
2
|
11
|
2
|
11
|
2
|
11
|
2
|
11
|
2
|
11
|
|
10
|
3
|
10
|
3
|
10
|
3
|
10
|
3
|
10
|
3
|
10
|
3
|
|
11
|
2
|
11
|
2
|
11
|
2
|
11
|
2
|
11
|
2
|
11
|
2
|
|
1
|
12
|
1
|
12
|
1
|
12
|
1
|
12
|
1
|
12
|
1
|
12
|
= panmagic 12x12 square (option b)
|
49
|
95
|
58
|
86
|
55
|
92
|
60
|
88
|
51
|
93
|
54
|
89
|
|
72
|
74
|
63
|
83
|
66
|
77
|
61
|
81
|
70
|
76
|
67
|
80
|
|
97
|
47
|
106
|
38
|
103
|
44
|
108
|
40
|
99
|
45
|
102
|
41
|
|
36
|
110
|
27
|
119
|
30
|
113
|
25
|
117
|
34
|
112
|
31
|
116
|
|
37
|
107
|
46
|
98
|
43
|
104
|
48
|
100
|
39
|
105
|
42
|
101
|
|
144
|
2
|
135
|
11
|
138
|
5
|
133
|
9
|
142
|
4
|
139
|
8
|
|
85
|
59
|
94
|
50
|
91
|
56
|
96
|
52
|
87
|
57
|
90
|
53
|
|
84
|
62
|
75
|
71
|
78
|
65
|
73
|
69
|
82
|
64
|
79
|
68
|
|
13
|
131
|
22
|
122
|
19
|
128
|
24
|
124
|
15
|
129
|
18
|
125
|
|
120
|
26
|
111
|
35
|
114
|
29
|
109
|
33
|
118
|
28
|
115
|
32
|
|
121
|
23
|
130
|
14
|
127
|
20
|
132
|
16
|
123
|
21
|
126
|
17
|
|
12
|
134
|
3
|
143
|
6
|
137
|
1
|
141
|
10
|
136
|
7
|
140
|
Ultra magic 12x12 square
It is also possible to produce a 12x12 square consisting of 4x4 sub-squares with the following
structure (per sub-square):
Two digits with the same colour give the same sum, that is the lowest digit plus the highest digit
of the magic 12x12 square: 1 + 144 = 145.
You need the following basic key (see yellow marked), which leads to an ultra panmagic 12x12
square:
| 1x digit |
|
|
|
|
|
|
|
|
|
| 12 |
2 |
2 |
12 |
5 |
6 |
6 |
5 |
10 |
4 |
4 |
10 |
| 1 |
11 |
11 |
1 |
8 |
7 |
7 |
8 |
3 |
9 |
9 |
3 |
| 12 |
2 |
2 |
12 |
5 |
6 |
6 |
5 |
10 |
4 |
4 |
10 |
| 1 |
11 |
11 |
1 |
8 |
7 |
7 |
8 |
3 |
9 |
9 |
3 |
| 12 |
2 |
2 |
12 |
5 |
6 |
6 |
5 |
10 |
4 |
4 |
10 |
| 1 |
11 |
11 |
1 |
8 |
7 |
7 |
8 |
3 |
9 |
9 |
3 |
| 12 |
2 |
2 |
12 |
5 |
6 |
6 |
5 |
10 |
4 |
4 |
10 |
| 1 |
11 |
11 |
1 |
8 |
7 |
7 |
8 |
3 |
9 |
9 |
3 |
| 12 |
2 |
2 |
12 |
5 |
6 |
6 |
5 |
10 |
4 |
4 |
10 |
| 1 |
11 |
11 |
1 |
8 |
7 |
7 |
8 |
3 |
9 |
9 |
3 |
| 12 |
2 |
2 |
12 |
5 |
6 |
6 |
5 |
10 |
4 |
4 |
10 |
| 1 |
11 |
11 |
1 |
8 |
7 |
7 |
8 |
3 |
9 |
9 |
3 |
+ 12 x (digit -/- 1) |
|
|
|
|
|
|
|
| 10 |
3 |
10 |
3 |
10 |
3 |
10 |
3 |
10 |
3 |
10 |
3 |
| 4 |
9 |
4 |
9 |
4 |
9 |
4 |
9 |
4 |
9 |
4 |
9 |
| 4 |
9 |
4 |
9 |
4 |
9 |
4 |
9 |
4 |
9 |
4 |
9 |
| 10 |
3 |
10 |
3 |
10 |
3 |
10 |
3 |
10 |
3 |
10 |
3 |
| 5 |
8 |
5 |
8 |
5 |
8 |
5 |
8 |
5 |
8 |
5 |
8 |
| 6 |
7 |
6 |
7 |
6 |
7 |
6 |
7 |
6 |
7 |
6 |
7 |
| 6 |
7 |
6 |
7 |
6 |
7 |
6 |
7 |
6 |
7 |
6 |
7 |
| 5 |
8 |
5 |
8 |
5 |
8 |
5 |
8 |
5 |
8 |
5 |
8 |
| 12 |
1 |
12 |
1 |
12 |
1 |
12 |
1 |
12 |
1 |
12 |
1 |
| 2 |
11 |
2 |
11 |
2 |
11 |
2 |
11 |
2 |
11 |
2 |
11 |
| 2 |
11 |
2 |
11 |
2 |
11 |
2 |
11 |
2 |
11 |
2 |
11 |
| 12 |
1 |
12 |
1 |
12 |
1 |
12 |
1 |
12 |
1 |
12 |
1 |
= ultra panmagic 12x12 square |
|
|
|
|
| 120 |
26 |
110 |
36 |
113 |
30 |
114 |
29 |
118 |
28 |
112 |
34 |
| 37 |
107 |
47 |
97 |
44 |
103 |
43 |
104 |
39 |
105 |
45 |
99 |
| 48 |
98 |
38 |
108 |
41 |
102 |
42 |
101 |
46 |
100 |
40 |
106 |
| 109 |
35 |
119 |
25 |
116 |
31 |
115 |
32 |
111 |
33 |
117 |
27 |
| 60 |
86 |
50 |
96 |
53 |
90 |
54 |
89 |
58 |
88 |
52 |
94 |
| 61 |
83 |
71 |
73 |
68 |
79 |
67 |
80 |
63 |
81 |
69 |
75 |
| 72 |
74 |
62 |
84 |
65 |
78 |
66 |
77 |
70 |
76 |
64 |
82 |
| 49 |
95 |
59 |
85 |
56 |
91 |
55 |
92 |
51 |
93 |
57 |
87 |
| 144 |
2 |
134 |
12 |
137 |
6 |
138 |
5 |
142 |
4 |
136 |
10 |
| 13 |
| 
