Office Applications and Entertainment, Magic Squares

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12.8   Quadrant Magic Squares (13 x 13)

The concept of Quadrant Magic Squares, as discussed in following sections for order 13 Magic Squares, was introduced by Harvey Heinz (2001/2002).

12.8.1 Definition and Terminology

An order 13 magic square can be divided into four overlapping quadrants of 7 x 7 cells.

Each quadrant might contain symmetric patterns of 13 cells (12 cells + the centre cell) which sum to the Magic Sum s1, as illustrated in following example:

P01 (Plus Magic)
o o o o o o o
o o o o o o o
o o o o o o o
o o o o o o o
o o o o o o o
o o o o o o o
o o o o o o o

For order 13 magic squares 38 patterns can be recognised, further referred to as Magic Pattern P01 thru P38, which are shown in Attachment 12.7.1.

An order 13 Magic Square is called Quadrant Magic if the same pattern(s) occur in all four quadrants.

The number of patterns which occur in all four quadrants will be referred to as nQ4.

12.8.2 Historical Results

The properties of the historical order 13 (Multi) Quadrant Magic Squares, as published by Harvey Heinz, are summarised below:

nQ4

Quadrant Magic Patterns

1

P34

3

P05, P13, P38

4

P02, P09, P21, P25

6

P05, P12, P14, P18, P27, P32

10

P02, P03, P09, P10, P15, P16, P21, P25, P26, P30
P01, P03, P05, P08, P11, P14, P19, P22, P24, P33
P01, P08, P09, P17, P18, P21, P22, P30, P32, P37
P01, P04, P05, P09, P12, P16, P17, P20, P28, P32

11

P02, P04, P05, P07, P15, P18, P21, P23, P26, P36, P37

14

P01, P02, P04, P05, P07, P09, P21, P23, P24, P25, P33, P35, P36, P37

Note: The squares listed above contain also other patterns, however not in all four quadrants.

Subject squares were filtered from the order 13 Pan Magic Squares, which can be constructed based on Latin Diagonal Squares, as discussed in Section 13.2.1.

Following sections will describe and illustrate how comparable squares, not necessarily listed in the table shown above, can be constructed or generated.

12.8.3 Equations Quadrant Magic Patterns

The quadrant properties defined in Section 12.7.1 above can be described by the linear equations as listed in Attachment 12.7.3.

Subject equations can be combined with the equations describing miscellaneous types of order 13 Magic Squares.

Subject equations might be incorporated in procedures to determine all occurring patterns of (previously) generated Quadrant Magic Squares.

12.8.4 Pan Magic, P34

Exhibit P34a describes how a controllable collection of Quadrant P34 Pan Magic Squares can be obtained based on a generalisation of the decomposition into Latin Squares of Example 1.

Attachment 12.7.41 shows the resulting (72) Quadrant P34 Pan Magic Squares as well as a summary of all patterns per square.

Square Nr 39 is Quadrant (P34, P38) Magic as illustrated below:

P34
60 88 116 131 159 18 46 74 102 130 145 4 32
77 92 120 148 7 35 63 91 106 134 162 21 49
81 109 137 165 24 52 67 95 123 151 10 38 53
98 126 154 13 28 56 84 112 140 168 14 42 70
115 143 158 17 45 73 101 129 144 3 31 59 87
119 147 6 34 62 90 105 133 161 20 48 76 104
136 164 23 51 66 94 122 150 9 37 65 80 108
153 12 27 55 83 111 139 167 26 41 69 97 125
157 16 44 72 100 128 156 2 30 58 86 114 142
5 33 61 89 117 132 160 19 47 75 103 118 146
22 50 78 93 121 149 8 36 64 79 107 135 163
39 54 82 110 138 166 25 40 68 96 124 152 11
43 71 99 127 155 1 29 57 85 113 141 169 15
P38
60 88 116 131 159 18 46 74 102 130 145 4 32
77 92 120 148 7 35 63 91 106 134 162 21 49
81 109 137 165 24 52 67 95 123 151 10 38 53
98 126 154 13 28 56 84 112 140 168 14 42 70
115 143 158 17 45 73 101 129 144 3 31 59 87
119 147 6 34 62 90 105 133 161 20 48 76 104
136 164 23 51 66 94 122 150 9 37 65 80 108
153 12 27 55 83 111 139 167 26 41 69 97 125
157 16 44 72 100 128 156 2 30 58 86 114 142
5 33 61 89 117 132 160 19 47 75 103 118 146
22 50 78 93 121 149 8 36 64 79 107 135 163
39 54 82 110 138 166 25 40 68 96 124 152 11
43 71 99 127 155 1 29 57 85 113 141 169 15

Square Nr 39 contains 36 patterns (all except P22 and P26) which are (separately) shown in Attachment 12.7.42.

Square Nr 52 is Quadrant (P21, P34) Magic as illustrated below:

P34
91 134 21 77 120 7 63 106 162 49 92 148 35
112 168 42 98 154 28 84 140 14 70 126 13 56
133 20 76 119 6 62 105 161 48 104 147 34 90
167 41 97 153 27 83 139 26 69 125 12 55 111
19 75 118 5 61 117 160 47 103 146 33 89 132
40 96 152 39 82 138 25 68 124 11 54 110 166
74 130 4 60 116 159 46 102 145 32 88 131 18
95 151 38 81 137 24 67 123 10 53 109 165 52
129 3 59 115 158 45 101 144 31 87 143 17 73
150 37 80 136 23 66 122 9 65 108 164 51 94
2 58 114 157 44 100 156 30 86 142 16 72 128
36 79 135 22 78 121 8 64 107 163 50 93 149
57 113 169 43 99 155 29 85 141 15 71 127 1
P21
91 134 21 77 120 7 63 106 162 49 92 148 35
112 168 42 98 154 28 84 140 14 70 126 13 56
133 20 76 119 6 62 105 161 48 104 147 34 90
167 41 97 153 27 83 139 26 69 125 12 55 111
19 75 118 5 61 117 160 47 103 146 33 89 132
40 96 152 39 82 138 25 68 124 11 54 110 166
74 130 4 60 116 159 46 102 145 32 88 131 18
95 151 38 81 137 24 67 123 10 53 109 165 52
129 3 59 115 158 45 101 144 31 87 143 17 73
150 37 80 136 23 66 122 9 65 108 164 51 94
2 58 114 157 44 100 156 30 86 142 16 72 128
36 79 135 22 78 121 8 64 107 163 50 93 149
57 113 169 43 99 155 29 85 141 15 71 127 1

Exhibit P34b describes a more generalised method based on which 158400 L2 - and R6 - Quadrant P34 Latin Diagonal Squares could be constructed, resulting in numerous Quadrant P34 Pan Magic Squares.

Note: For the classification of order 13 Latin Diagonal Squares reference is made to Section 13.2.1.

12.8.5 Ultra Magic, General

In Section 13.2.2 is described how order 13 Ultra Magic Squares can be constructed based on pairs of Orthogonal Latin Diagonal Squares A and T(A).

Note: Square A is of type R2, L2, L3, R3, L4, R4, L5, R5, L6 or R6 and T(A) is the transposed of A.

With the type R2 based routine UltraLat13 2678 Quadrant Ultra Magic Squares could be generated as summarised below:

n13 = f(nQ4)
nQ4 1 2 3 4 5 6 7 8 10 12 16 18 20
n13 1788 616 70 132 14 8 24 2 14 2 4 2 2
Patterns
nQ4 Quadrant Magic Patterns
1
P02 P03 P05 P06 P07 P08 P10 P14 P16 P19
P21 P22 P23 P24 P25 P26 P33 P34 P35 P36
P37 P38
2
P02, P35 P03, P35 P03, P36 P03, P38 P06, P14 P07, P22 P07, P35 P11, P29 P13, P27 P15, P31
P16, P33 P16, P35 P16, P37 P18, P30 P21, P26 P21, P34 P21, P36 P21, P37 P22, P25 P22, P35
P23, P25 P23, P26 P23, P38 P24, P36 P25, P34 P33, P34 P34, P36 P34, P38 P36, P38
3
P07, P25, P36 P13, P26, P27 P13, P27, P33 P13, P27, P34 P13, P27, P37
P15, P31, P33 P18, P19, P30 P18, P25, P30 P18, P30, P33 P18, P30, P34
P18, P30, P36
4
P01, P09, P17, P32 P04, P12, P20, P28 P11, P18, P29, P30 P21, P34, P36, P37
5
P01, P09, P17, P32, P35 P01, P09, P17, P32, P36 P04, P08, P12, P20, P28 P04, P12, P20, P28, P38
6
P07, P18, P22, P25, P30, P36 P13, P15, P27, P31, P33, P37 P13, P19, P23, P26, P27, P33
7
P01, P08, P09, P17, P18, P30, P32 P04, P08, P12, P20, P28, P33, P38
8 P01, P08, P09, P17, P18, P30, P32, P37
10 P01, P04, P05, P09, P12, P16, P17, P20, P28, P32
12 P01, P04, P05, P09, P12, P16, P17, P20, P28, P32, P34, P38
16 P01, P03, P04, P05, P08, P09, P12, P13, P16, P17, P18, P20, P27, P28, P30, P32
18 P01, P03, P04, P05, P08, P09, P12, P13, P16, P17, P18, P20, P27, P28, P30, P32, P35, P37
20 P01, P03, P04, P05, P08, P09, P12, P13, P16, P17, P18, P20, P27, P28, P30, P32, P33, P34, P36, P38

Both squares with nQ4 = 20 contain 36 patterns (all except P23 and P25) which are (separately) shown in Attachment 12.7.44 for the first square.

12.8.6 Ultra Magic, P06, P29

The historical squares as listed in Section 12.7.2 above don't contain any Quadrant P06 or Quadrant P29 (Pan) Magic Squares.

Attachment 12.7.45 shows the 40 Quadrant P06 Ultra Magic Squares, as found with UltraLat13, as well as a summary of all patterns per square.

Square Nr 18 and 23 are Quadrant (P06, P14) Ultra Magic, as illustrated below for Square Nr 18:

P06
113 70 14 86 146 103 62 163 134 41 128 32 13
58 169 139 44 118 34 3 116 75 20 82 145 102
2 115 71 26 87 148 92 60 159 142 49 124 30
98 56 158 141 45 130 35 5 105 73 16 90 153
38 10 111 69 15 89 149 104 61 161 131 47 120
151 94 64 166 137 43 119 37 6 117 74 18 79
122 27 8 107 77 23 85 147 93 63 162 143 48
91 152 96 53 164 133 51 127 33 4 106 76 19
50 123 39 9 109 66 21 81 155 101 59 160 132
17 80 154 97 65 165 135 40 125 29 12 114 72
140 46 121 28 11 110 78 22 83 144 99 55 168
68 25 88 150 95 54 167 136 52 126 31 1 112
157 138 42 129 36 7 108 67 24 84 156 100 57
P14
113 70 14 86 146 103 62 163 134 41 128 32 13
58 169 139 44 118 34 3 116 75 20 82 145 102
2 115 71 26 87 148 92 60 159 142 49 124 30
98 56 158 141 45 130 35 5 105 73 16 90 153
38 10 111 69 15 89 149 104 61 161 131 47 120
151 94 64 166 137 43 119 37 6 117 74 18 79
122 27 8 107 77 23 85 147 93 63 162 143 48
91 152 96 53 164 133 51 127 33 4 106 76 19
50 123 39 9 109 66 21 81 155 101 59 160 132
17 80 154 97 65 165 135 40 125 29 12 114 72
140 46 121 28 11 110 78 22 83 144 99 55 168
68 25 88 150 95 54 167 136 52 126 31 1 112
157 138 42 129 36 7 108 67 24 84 156 100 57

Attachment 12.7.46 shows the 44 Quadrant P29 Ultra Magic Squares, as found with UltraLat13, as well as a summary of all patterns per square.

It can be noticed that all 44 Quadrant P29 Ultra Magic Squares are Quadrant (P11, P29) Ultra Magic Squares.

Square Nr 17 and 28 are Quadrant (P11, P18, P29, P30) Ultra Magic, as illustrated below for Square Nr 28:

P11
141 107 17 83 149 64 27 124 104 158 8 74 49
35 127 102 159 4 70 45 142 105 20 91 145 60
41 138 113 23 89 146 56 31 123 103 157 7 78
59 39 119 99 165 10 76 42 134 109 19 90 144
77 40 137 117 15 86 152 62 37 120 95 161 6
148 58 38 118 98 169 2 73 48 140 115 16 82
3 69 44 136 116 14 85 156 54 34 126 101 167
88 154 55 30 122 97 168 1 72 52 132 112 22
164 9 75 50 133 108 18 84 155 53 33 130 93
26 80 151 61 36 128 94 160 5 71 51 131 111
92 163 13 67 47 139 114 24 81 147 57 32 129
110 25 79 150 65 28 125 100 166 11 68 43 135
121 96 162 12 66 46 143 106 21 87 153 63 29
P18
141 107 17 83 149 64 27 124 104 158 8 74 49
35 127 102 159 4 70 45 142 105 20 91 145 60
41 138 113 23 89 146 56 31 123 103 157 7 78
59 39 119 99 165 10 76 42 134 109 19 90 144
77 40 137 117 15 86 152 62 37 120 95 161 6
148 58 38 118 98 169 2 73 48 140 115 16 82
3 69 44 136 116 14 85 156 54 34 126 101 167
88 154 55 30 122 97 168 1 72 52 132 112 22
164 9 75 50 133 108 18 84 155 53 33 130 93
26 80 151 61 36 128 94 160 5 71 51 131 111
92 163 13 67 47 139 114 24 81 147 57 32 129
110 25 79 150 65 28 125 100 166 11 68 43 135
121 96 162 12 66 46 143 106 21 87 153 63 29
P29
141 107 17 83 149 64 27 124 104 158 8 74 49
35 127 102 159 4 70 45 142 105 20 91 145 60
41 138 113 23 89 146 56 31 123 103 157 7 78
59 39 119 99 165 10 76 42 134 109 19 90 144
77 40 137 117 15 86 152 62 37 120 95 161 6
148 58 38 118 98 169 2 73 48 140 115 16 82
3 69 44 136 116 14 85 156 54 34 126 101 167
88 154 55 30 122 97 168 1 72 52 132 112 22
164 9 75 50 133 108 18 84 155 53 33 130 93
26 80 151 61 36 128 94 160 5 71 51 131 111
92 163 13 67 47 139 114 24 81 147 57 32 129
110 25 79 150 65 28 125 100 166 11 68 43 135
121 96 162 12 66 46 143 106 21 87 153 63 29
P30
141 107 17 83 149 64 27 124 104 158 8 74 49
35 127 102 159 4 70 45 142 105 20 91 145 60
41 138 113 23 89 146 56 31 123 103 157 7 78
59 39 119 99 165 10 76 42 134 109 19 90 144
77 40 137 117 15 86 152 62 37 120 95 161 6
148 58 38 118 98 169 2 73 48 140 115 16 82
3 69 44 136 116 14 85 156 54 34 126 101 167
88 154 55 30 122 97 168 1 72 52 132 112 22
164 9 75 50 133 108 18 84 155 53 33 130 93
26 80 151 61 36 128 94 160 5 71 51 131 111
92 163 13 67 47 139 114 24 81 147 57 32 129
110 25 79 150 65 28 125 100 166 11 68 43 135
121 96 162 12 66 46 143 106 21 87 153 63 29

Note: Quadrant (P06, P29) Magic Squares don't occur in the collection of R2 based Ultra Magic Squares.

12.8.7 Borderred, P02, P03, P08, P14

Exhibit P14 describes how Quadrant P14 Bordered Magic Squares, with Associated Compact Pan Magic Centre Squares, can be constructed based on Semi Latin Squares as discussed in Section 13.2.4.

The resulting key squares (standard position) are shown below:

P14 (Option 1)
169 130 143 156 161 162 7 6 5 4 3 2 157
105 20 16 17 148 149 151 152 23 24 155 80 65
117 145 102 86 70 139 123 107 62 46 30 25 53
118 142 58 35 42 98 75 82 138 115 122 28 52
131 129 111 134 127 34 57 50 71 94 87 41 39
144 106 73 96 89 110 133 126 33 56 49 64 26
79 93 29 45 61 69 85 101 109 125 141 77 91
104 67 121 114 137 44 37 60 81 74 97 103 66
92 54 83 76 99 120 113 136 43 36 59 116 78
12 51 48 55 32 88 95 72 128 135 112 119 158
11 38 140 124 108 63 47 31 100 84 68 132 159
10 90 154 153 22 21 19 18 147 146 15 150 160
13 40 27 14 9 8 163 164 165 166 167 168 1
P14 (Option 2)
168 130 143 157 161 163 8 6 5 4 3 1 156
105 20 16 17 148 149 151 152 23 24 155 80 65
117 145 102 86 70 139 123 107 62 46 30 25 53
118 142 58 35 42 98 75 82 138 115 122 28 52
131 129 111 134 127 34 57 50 71 94 87 41 39
144 106 73 96 89 110 133 126 33 56 49 64 26
79 93 29 45 61 69 85 101 109 125 141 77 91
104 67 121 114 137 44 37 60 81 74 97 103 66
92 54 83 76 99 120 113 136 43 36 59 116 78
12 51 48 55 32 88 95 72 128 135 112 119 158
11 38 140 124 108 63 47 31 100 84 68 132 159
10 90 154 153 22 21 19 18 147 146 15 150 160
14 40 27 13 9 7 162 164 165 166 167 169 2

Each of the two (unique) squares shown above correspond with 8 * (10!)2 * n11 * n9 Quadrant P14 Bordered Magic Squares (n11 = all possible order 11 borders, n9 = all suitable order 9 centre squares).

Note:
Any order 9 centre square for which the elements of the 3 x 3 corner squares sum to s9 can be used for the construction of Quadrant P14 Bordered Magic Squares.

Exhibit P08 describes how Quadrant P08 Bordered Magic Squares, with Associated Compact Pan Magic Centre Squares, can be constructed based on Semi Latin Squares as discussed in Section 13.2.4.

By applying exterior borders as constructed for Quadrant P14 Bordered Magic Squares, the resulting square will be Quadrant (P08, P14) Magic as illustrated below:

P08
169 130 143 156 161 162 7 6 5 4 3 2 157
105 153 25 22 21 93 20 129 147 152 154 19 65
117 54 141 72 42 62 110 83 100 34 121 116 53
118 38 84 57 114 125 95 35 46 133 76 132 52
131 28 30 126 99 68 50 137 109 88 58 142 39
144 51 122 101 32 43 139 73 81 63 111 119 26
79 90 74 47 134 115 85 55 36 123 96 80 91
104 15 59 107 89 97 31 127 138 69 48 155 66
92 103 112 82 61 33 120 102 71 44 140 67 78
12 106 94 37 124 135 75 45 56 113 86 64 158
11 146 49 136 70 87 60 108 128 98 29 24 159
10 151 145 148 149 77 150 41 23 18 16 17 160
13 40 27 14 9 8 163 164 165 166 167 168 1
P14
169 130 143 156 161 162 7 6 5 4 3 2 157
105 153 25 22 21 93 20 129 147 152 154 19 65
117 54 141 72 42 62 110 83 100 34 121 116 53
118 38 84 57 114 125 95 35 46 133 76 132 52
131 28 30 126 99 68 50 137 109 88 58 142 39
144 51 122 101 32 43 139 73 81 63 111 119 26
79 90 74 47 134 115 85 55 36 123 96 80 91
104 15 59 107 89 97 31 127 138 69 48 155 66
92 103 112 82 61 33 120 102 71 44 140 67 78
12 106 94 37 124 135 75 45 56 113 86 64 158
11 146 49 136 70 87 60 108 128 98 29 24 159
10 151 145 148 149 77 150 41 23 18 16 17 160
13 40 27 14 9 8 163 164 165 166 167 168 1

The square shown above corresponds with 8 * (10!)2 * (7!)2 * n9 Quadrant (P08, P14) Bordered Magic Squares (n9 = all suitable order 9 centre squares).

Attachment 12.7.71 shows a few Quadrant (P08, P14) Bordered Magic Squares, as found with routine Priem11a, based on a sub collection of 0rder 9 Associated Compact Pan Magic Squares.

Exhibit P03 describes how Quadrant P03 Bordered Magic Squares, with Associated Magic Centre Squares (third-rows and third-columns summing to s9/3), can be constructed based on Semi Latin Squares as discussed in Section 13.2.4.

By applying:

  • exterior borders as constructed for Quadrant P14 Bordered Magic Squares and
  • order 11 Bordered Center Squares resulting in Quadrant P08 Bordered Magic Squares

the resulting square will be Quadrant (P03, P08, P14) Magic.

Attachment 12.7.76 shows a few Quadrant (P03, P08, P14) Bordered Magic Squares, as found with routine Priem13b, based on a sub collection of 0rder 11 Quadrant P08 Bordered Magic Centre Squares.

Attachment 12.7.77 shows a few Quadrant (P02, P03, P14) Bordered Magic Squares, based on a sub collection of 0rder 11 Quadrant P02 Bordered Magic Centre Squares, as found with routine Priem11b.

Each square shown corresponds with 8 * (8!)2 * n11 Quadrant (P03, P08, P14) Bordered Magic Squares (n11 = all suitable order 11 centre squares).

12.8.8 Summary

The obtained results regarding the miscellaneous types of order 13 Quadrant Magic Squares as deducted and discussed in previous sections are summarised in following table:

Type

Characteristics

Subroutine

Results

P34

Pan Magic

CnstrP34a

Attachment 12.7.41

P34, P38

Pan   Magic, Occurring Patterns

Attachment 12.7.42

nQ4 = 20

Ultra Magic, Occurring Patterns

UltraLat13

Attachment 12.7.44

P06

Ultra Magic

Attachment 12.7.45

P29

Ultra Magic

Attachment 12.7.46

P08, P14

Bordered

Priem11a

Attachment 12.7.71

P03, P08, P14

Bordered

Priem13b

Attachment 12.7.76

P02, P03, P14

Bordered

Priem11b

Attachment 12.7.77

Comparable routines as listed above, can be used to generate alternative types of order 13 Magic Squares.

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