Office Applications and Entertainment, Magic Squares

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10.2   Concentric, Eccentric and Inlaid Magic Squares

20.2.1 Concentric Magic Squares (14 x 14)

A 14th order Concentric Magic Square consists of a Concentric Magic Square of the 12th order with a border around it, as illustrated below.

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14
a15 a16 a17 a18 a19 a20 a21 a22 a23 a24 a25 a26 a27 a28
a29 a30 a31 a32 a33 a34 a35 a36 a37 a38 a39 a40 a41 a42
a43 a44 a45 a46 a47 a48 a49 a50 a51 a52 a53 a54 a55 a56
a57 a58 a59 a60 a61 a62 a63 a64 a65 a66 a67 a68 a69 a70
a71 a72 a73 a74 a75 a76 a77 a78 a79 a80 a81 a82 a83 a84
a85 a86 a87 a88 a89 a90 a91 a92 a93 a94 a95 a96 a97 a98
a99 a100 a101 a102 a103 a104 a105 a106 a107 a108 a109 a110 a111 a112
a113 a114 a115 a116 a117 a118 a119 a120 a121 a122 a123 a124 a125 a126
a127 a128 a129 a130 a131 a132 a133 a134 a135 a136 a137 a138 a139 a140
a141 a142 a143 a144 a145 a146 a147 a148 a149 a150 a151 a152 a153 a154
a155 a156 a157 a158 a159 a160 a161 a162 a163 a164 a165 a166 a167 a168
a169 a170 a171 a172 a173 a174 a175 a176 a177 a178 a179 a180 a181 a182
a183 a184 a185 a186 a187 a188 a189 a190 a191 a192 a193 a194 a195 a196

Based on the linear equations defining the border of a Concentric Magic Square of order 14: 

a(184)= 1379 - a(183)-a(185)-a(186)-a(187)-a(188)-a(189)-a(190)-a(191)-a(192)-a(193)-a(194)-a(195)-a(196)
a( 28)= 1379 - a( 14)-a( 42)-a( 56)-a( 70)-a( 84)-a( 98)-a(112)-a(126)-a(140)-a(154)-a(168)-a(182)-a(196)

a(1) = 197 - a(196)
a(2) = 197 - a(184)
a(3) = 197 - a(185)
a(4) = 197 - a(186)
a(5) = 197 - a(187)
a(6) = 197 - a(188)
a(7) = 197 - a(189)

a( 8) = 197 - a(190)
a( 9) = 197 - a(191)
a(10) = 197 - a(192)
a(11) = 197 - a(193)
a(12) = 197 - a(194)
a(13) = 197 - a(195)
a(14) = 197 - a(183)

a(15) = 197 - a(28)
a(29) = 197 - a(42)
a(43) = 197 - a(56)
a(57) = 197 - a(70)
a(71) = 197 - a(84)
a(85) = 197 - a(98)

a(99) = 197 - a(112)
a(113) = 197 - a(126)
a(127) = 197 - a(140)
a(141) = 197 - a(154)
a(155) = 197 - a(168)
a(169) = 197 - a(182)

a routine can be written to generate the borders for subject Concentric Magic Squares (ref. MgcSqrs14a).

Attachment 20.2.1 shows a few suitable borders for Concentric Magic Squares of order 14.

Each border shown corresponds with (12!)2 = 2,29 * 1017 borders with the same corner pairs, which can be obtained by permutation of the horizontal/vertical (non corner) pairs.

A full enumeration as executed for 8 x 8 Concentric Magic Squares in Section 8.8.2 is however beyond the scope of this section.

Note: The 12th order Concentric Magic Center Squares should be based on the consecutive integers 27, 28, ... 170.

20.2.2 Bordered Magic Squares (14 x 14)
       Miscellaneous Inlays

Also Non Concentric Magic Squares of the 12th order e.g. as described and constructed in Section 12.1. Section 12.2 and Section 22.5 can be used as Center Squares for 14th order Bordered Magic Squares.

The Embedded Magic Squares will have a Magic Sum s12 = 1182 and might be based on the consecutive integers 27, 28, ... 170.

Attachment 20.2.2 contains - based on some of the described Magic Squares of order 12 - a few examples of Bordered Magic Squares.

20.2.3 Bordered Magic Squares (14 x 14)
       Split Border

Alternatively a 14th order Bordered Magic Square with Magic Sum s14 = 1379 can be constructed based on:

  • a Concentric Magic Center Square of order 10 with Magic Sum s10 = 985;
  • 48 pairs, each summing to 197, surrounding the (Concentric) Magic Center Square;
  • a split of the supplementary rows and columns into three parts:
    two summing to s4 = 394 and one to s6 = 591.

as illustrated below:

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14
a15 a16 a17 a18 a19 a20 a21 a22 a23 a24 a25 a26 a27 a28
a29 a30 a31 a32 a33 a34 a35 a36 a37 a38 a39 a40 a41 a42
a43 a44 a45 a46 a47 a48 a49 a50 a51 a52 a53 a54 a55 a56
a57 a58 a59 a60 a61 a62 a63 a64 a65 a66 a67 a68 a69 a70
a71 a72 a73 a74 a75 a76 a77 a78 a79 a80 a81 a82 a83 a84
a85 a86 a87 a88 a89 a90 a91 a92 a93 a94 a95 a96 a97 a98
a99 a100 a101 a102 a103 a104 a105 a106 a107 a108 a109 a110 a111 a112
a113 a114 a115 a116 a117 a118 a119 a120 a121 a122 a123 a124 a125 a126
a127 a128 a129 a130 a131 a132 a133 a134 a135 a136 a137 a138 a139 a140
a141 a142 a143 a144 a145 a146 a147 a148 a149 a150 a151 a152 a153 a154
a155 a156 a157 a158 a159 a160 a161 a162 a163 a164 a165 a166 a167 a168
a169 a170 a171 a172 a173 a174 a175 a176 a177 a178 a179 a180 a181 a182
a183 a184 a185 a186 a187 a188 a189 a190 a191 a192 a193 a194 a195 a196

The supplementary rows and columns can be described by following linear equations:

Typical Corner Section (4 x 4):

a'(1) a'(2) a'(3) a'(4)
a'(5) a'(6) a'(7) a'(8)
a'(9) a'(10) - -
a'(11) a'(12) - -

a'( 4) = 394 - a'( 3) - a'(2) - a'(1)
a'( 5) = 197 - a'( 2)
a'( 6) = 197 - a'( 1)
a'( 7) = 197 - a'( 3)
a'( 8) = 197 - a'( 4)
a'(11) = 394 - a'( 9) - a'(5) - a'(1)
a'(10) = 197 - a'( 9)
a'(12) = 197 - a'(11)

Typical Border Rectangle (2 x 6):

a'(1) a'(2) a'(3) a'(4) a'(5) a'(6)
a'(7) a'(8) a'(9) a'(10) a'(11) a'(12)

a'( 6) = 591 - a'(5) - a'(4) - a'(3) - a'(2) - a'(1)
a'( 7) = 197 - a'(1)
a'( 8) = 197 - a'(2)
a'( 9) = 197 - a'(3)
a'(10) = 197 - a'(4)
a'(11) = 197 - a'(5)
a'(12) = 197 - a'(6)

Based on the equations above, procedures can be developed:

  • to generate, based on the distinct integers {1 ... 196}, four Corner Segments (4 x 4);
  • to complete the exterior border with four Magic Rectangles (2 x 6);
  • to construct, based on the remaining 100 distinct integers, the border of the Concentric Center Square (10 x 10);
  • to construct, based on the remaining 64 distinct integers, the embedded Concentric Magic Square of order 8.

The first occuring Bordered Magic Square is shown below:

1 4 194 195 24 25 34 168 169 171 190 188 10 6
193 196 3 2 173 172 163 29 28 26 7 9 191 187
8 189 48 145 144 143 142 141 66 57 49 50 182 15
192 5 138 123 80 120 79 119 76 122 69 59 11 186
42 155 136 68 108 84 85 110 116 88 129 61 30 167
43 154 135 130 83 94 95 101 104 114 67 62 31 166
51 146 134 73 86 99 106 92 97 111 124 63 39 158
150 47 65 125 90 96 93 103 102 107 72 132 162 35
152 45 64 71 115 105 100 98 91 82 126 133 164 33
153 44 60 70 109 113 112 87 81 89 127 137 165 32
175 22 58 128 117 77 118 78 121 75 74 139 14 183
27 170 147 52 53 54 55 56 131 140 148 149 178 19
174 179 21 20 36 37 46 156 157 159 13 16 185 180
18 23 176 177 161 160 151 41 40 38 184 181 17 12

The border shown above corresponds with (4! * 12582912) * (4! * 14404) = 3,12 * 1022 borders as:

  1. The Corner Squares can be arranged in 4! ways and belong each to a collection of 64, 32, 96 or 64 Corner Squares;
  2. The Rectangles can be arranged in 4! ways and belong each to a collection of 2 * 6! = 1440 Rectangles.

Based on the distinct integers applied in the border shown above, numerous suitable sets of unique Corner Squares can be found.

20.2.4 Bordered Magic Squares (14 x 14)
       Composed Border

The 14th order Composed Magic Square shown below, with Magic Sum s14 = 1379 , consists of:

  • a Border composed out of:
    - 4 Associated Magic Squares of order 4 with Magic Sum s4 = 394
    - 4 Associated Magic Rectangles order 4 x 6 with s4 = 394 and s6 = 591
  • a Concentric Magic Center Square composed of:
    - an order 6 Concentric Border with Magic Sum s6 = 591
    - an order 4 Pan Magic Center Square with Magic Sum s4 = 394
196 189 5 4 164 163 146 37 40 41 188 181 13 12
3 6 190 195 39 36 43 152 159 162 11 14 182 187
2 7 191 194 35 38 45 154 161 158 10 15 183 186
193 192 8 1 156 157 160 51 34 33 185 184 16 9
132 71 69 122 83 113 106 105 95 89 155 48 46 145
130 73 70 121 103 82 85 110 117 94 153 50 47 144
104 81 86 123 101 107 120 79 88 96 129 56 61 148
74 111 116 93 99 87 80 115 112 98 49 136 141 68
76 127 124 67 97 118 109 90 77 100 53 150 147 44
75 128 126 65 108 84 91 92 102 114 52 151 149 42
172 165 29 28 143 142 119 60 64 63 180 173 21 20
27 30 166 171 59 62 66 125 140 139 19 22 174 179
26 31 167 170 58 57 72 131 135 138 18 23 175 178
169 168 32 25 134 133 137 78 55 54 177 176 24 17

The Composed Square shown above corresponds with n14 = 1.769.472 * (4! * 3844) * (4! * 384 * 11523) = 1,30 * 1031 squares as:

  1. The Center Square belongs to a collection of 8 * (4!)2 * 384 = 1.769.472 Center Squares;
  2. The Corner Squares can be arranged in 4! ways and belong each to a collection of 384 Corner Squares;
  3. The Border Rectangles can be arranged in 4! ways and belong each to a collection of 384 or 1152 Rectangles.

Based on the principles described in previous sections, a fast procedure (MgcSqrs14d) can be developed:

  • to read the previously generated 6 x 6 Concentric Magic Center Square;
  • to generate, based on the remainder of the pairs, four 4 x 4 Associated Magic Squares;
  • to complete the Composed Border of order 14 with four 4 x 6 Associated Magic Rectangles.

Attachment 20.2.41 shows miscellaneous order 14 Composed Border Magic Squares. Each (unique) square shown corresponds with numerous squares (order of magnitude n14).

Note:

If the applied properties are changed to:

  • the opposite Semi Magic Corner Squares (4 x 4) are Anti Symmetric and Complementary;
  • the opposite Magic Rectangles (4 x 6) are Anti Symmetric and Complementary;
  • the Magic Center Square (6 x 6) is Almost Associated;

the 14th order Composed Magic Square will be Almost Associated.

Attachment 20.2.42 shows miscellanous order 14 Almost Associated Composed Magic Squares (ref. MgcSqrs14e).

20.2.5 Eccentric Magic Squares (14 x 14)

A 14th order Eccentric Magic Square consists of one Magic Corner Square of the 12th order, supplemented with two rows and two columns - further referred to as 'border' - as illustrated by following example.

1 2 4 3 189 188 187 186 185 26 22 5 190 191
195 196 193 194 8 9 10 11 12 171 175 192 7 6
182 15 30 39 162 163 37 42 156 159 29 151 48 166
14 183 158 167 35 34 160 155 41 38 168 46 149 31
179 18 45 152 59 139 61 132 137 56 52 62 140 147
178 19 161 36 58 138 136 65 60 141 145 135 57 50
177 20 40 157 54 143 72 124 121 77 123 75 129 67
174 23 47 150 142 55 73 125 76 120 74 122 68 130
25 172 153 44 144 53 128 69 85 88 87 108 116 107
24 173 154 43 134 63 119 78 109 112 110 89 81 90
21 176 33 164 64 133 127 70 83 114 98 101 104 91
13 184 165 32 131 66 71 126 86 111 103 92 97 102
180 17 169 28 51 146 80 117 113 84 93 106 99 96
16 181 27 170 148 49 118 79 115 82 100 95 94 105

The square shown above has been constructed based on the transformation of a Bordered Magic Square with an embedded 12th order Eccentric Magic Center Square as illustrated in Attachment 20.5.1.

The applied border corresponds with (10!)2 = 1,32 * 1013 borders with the same corner pairs, which can be obtained by permutation of the horizontal/vertical (non corner) pairs.

A full enumeration as executed for 8 x 8 Eccentric Magic Squares in Section 8.8.5 is however beyond the scope of this section.

Note: The 12th order Eccentric Magic Corner Squares should be based on the consecutive integers 27, 28, ... 170.

20.2.6 Inlaid Magic Squares (14 x 14)
       Order 5 Pan Magic Square Inlays (4 ea)

The 14th order Inlaid Magic Square shown below:

1 191 190 189 188 187 186 185 26 22 5 4 3 2
182 40 66 79 92 144 110 98 148 126 115 104 60 15
180 91 140 48 64 78 122 86 103 56 156 124 114 17
179 72 76 90 139 44 134 74 132 112 102 55 152 18
178 138 43 68 84 88 158 50 54 151 128 120 100 19
177 80 96 136 42 67 62 146 116 108 52 150 127 20
174 165 166 167 169 161 27 159 29 37 35 34 33 23
25 164 163 162 160 168 38 170 36 28 30 31 32 172
24 49 71 82 93 137 51 135 157 131 118 105 53 173
21 94 141 41 73 83 147 39 106 57 149 133 119 176
16 65 85 95 142 45 123 63 125 121 107 58 153 181
14 143 46 69 77 97 111 75 59 154 129 113 109 183
13 81 89 145 47 70 99 87 117 101 61 155 130 184
195 6 7 8 9 10 11 12 171 175 192 193 194 196
421 553
432 564

is an example of a Bordered Magic Square with Magic Sum s14 = 1379, containing an order 12 Inlaid Magic Center Square (s12 = 1182).

The Magic Center Square has beeen obtained by transformation of an order 12 Inlaid Magic Square with Associated Border as described in detail in Section 22.6.

Attachment 20.2.6 shows a few examples of subject 14th order Inlaid Bordered Magic Squares.

20.2.7 Inlaid Magic Squares (14 x 14)
       Order 4 Pan Magic Square Inlays (9 ea)

The 14th order Inlaid Magic Square shown below:

86 87 77 80 180 166 167 157 160 17 54 55 45 48
81 76 90 83 179 161 156 170 163 18 49 44 58 51
88 85 79 78 178 168 165 159 158 19 56 53 47 46
75 82 84 89 177 155 162 164 169 20 43 50 52 57
192 191 190 189 11 188 187 184 25 12 4 3 2 1
70 71 61 64 176 102 103 93 96 21 134 135 125 128
65 60 74 67 175 97 92 106 99 22 129 124 138 131
72 69 63 62 26 104 101 95 94 171 136 133 127 126
59 66 68 73 24 91 98 100 105 173 123 130 132 137
5 6 7 8 185 9 10 13 172 186 193 194 195 196
150 151 141 144 23 38 39 29 32 174 118 119 109 112
145 140 154 147 16 33 28 42 35 181 113 108 122 115
152 149 143 142 15 40 37 31 30 182 120 117 111 110
139 146 148 153 14 27 34 36 41 183 107 114 116 121
330 650 202
266 394 522
586 138 458

is an example of an Inlaid Magic Square with Magic Sum s14 = 1379, with:

  • Nine each 4th order Pan Magic Square Inlays with different Magic Sum
  • Twentysix supplementary pairs, each summing to s14 / 7 = 197

The Inlaid Magic Squares can be obtained by transformation of Bordered Magic Squares with Composed Magic Center Squares (ref. Attachment 20.2.2).

20.2.8 Composed Magic Squares (14 x 14)
       Overlapping Sub Squares

Following 14th order Composed Magic Square, with overlapping sub squares and Magic Sum s14 = 1379, contains following sub squares:

  • One 6th order (Almost Associated) Magic Center Square C;
  • Two each other overlapping 8th order Eccentric Magic Squares A1 and A2;
  • Two each other overlapping 10th order Eccentric Magic Squares B1 and B2;
  • Two 4th order (Simple) Magic Squares M1 and M2;

as illustrated below:

118 83 85 108 25 171 170 28 168 31 164 38 154 36
84 109 107 94 172 26 27 169 29 166 33 159 161 43
103 90 88 113 1 195 191 7 9 189 4 192 158 39
89 112 114 79 196 2 6 190 188 8 5 193 40 157
97 100 182 15 141 60 131 59 139 61 187 10 156 41
150 47 16 181 68 73 70 128 123 129 34 163 42 155
117 80 24 173 121 122 64 78 72 134 186 11 54 143
116 81 23 174 63 125 119 133 75 76 185 12 46 151
120 77 162 35 62 74 69 127 124 135 184 13 32 165
82 115 22 175 136 137 138 66 58 56 3 194 142 55
126 71 180 18 21 160 177 19 30 183 106 86 92 110
57 140 179 17 176 37 20 178 167 14 93 102 98 101
67 144 132 52 146 50 148 48 45 153 104 95 99 96
53 130 65 145 51 147 49 149 152 44 91 111 105 87

Attachment 20.2.81 shows, miscellaneous order 14 Composed Magic Squares (ref. MgcSqrs14b).

The corresponding Composed Magic Squares of order 18 contain, in addition to the sub squares mentioned above, following Corner Squares:

  • Two 6th order Eccentric Magic Squares F1 and F2 with embedded M1 and M2;
  • Two 12th order Eccentric Magic Squares D1 and D2 with embedded B1 and B2;

Attachment 20.2.82 shows, for the sake of completeness, an example of a Composed Magic Squares of order 18 (ref. MgcSqrs14c).

20.2.9 Composed Magic Squares (14 x 14)
       Center Cross

The 14th order Composed Magic Square shown below:

155 40 41 150 160 45 182 15 140 54 63 133 142 59
42 157 156 47 37 152 180 17 57 143 134 64 55 138
34 36 39 153 164 165 179 18 52 53 56 137 146 147
163 161 158 44 33 32 178 19 145 144 141 60 51 50
29 30 31 162 169 170 177 20 46 48 49 135 154 159
168 167 166 35 28 27 174 23 151 149 148 62 43 38
191 190 189 188 187 186 1 2 185 26 22 5 4 3
6 7 8 9 10 11 195 196 12 171 175 192 193 194
120 72 84 117 122 76 25 172 105 88 98 102 104 94
77 125 113 80 75 121 24 173 92 109 99 95 93 103
70 71 74 119 128 129 21 176 89 90 91 100 110 111
127 126 123 78 69 68 16 181 108 107 106 97 87 86
65 66 67 118 136 139 14 183 82 83 85 101 116 124
132 131 130 79 61 58 13 184 115 114 112 96 81 73

is an example of a Composed Magic Square with Magic Sum s14 = 1379, composed of:

  • Four each 6th order Magic Corner Squares (s6 = 6 * s14 / 14 = 591)
  • Twentysix supplementary pairs, each summing to 2 * s14 / 14 = 197

The Composed Magic Squares can be obtained by transformation of Bordered Magic Squares with Composed Magic Center Squares (ref. Section 22.6).

Attachment 27.2 shows a few examples of subject 14th order Composed and related Bordered Magic Squares.

An alternative method to construct Center Cross based Magic Squares has been illustrated in Section 29.2.

10.2.10 Order 5, 6 and 7 Single Magic Square Inlays

The construction of Order 14 Simple Magic Squares with Order 5, 6, 7 or 8 Single Magic Square Inlays will be described in Sections 20.2.2, 3, 4 and 5.

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