Office Applications and Entertainment, Magic Squares

26,0 Magic Squares, Higher Order, Associated Border

26.1 Introduction

In previous sections Composed Simple Inlaid Magic Squares have been discussed for miscellaneous orders.

Following sections will show some examples of Composed Inlaid Magic Squares with Associated Borders.

26.2 Magic Squares (8 x 8)

Inlaid Magic Squares of order 8, with four 3^th order Embedded Magic Squares with different Magic Sums and an Associated Border, have been described in Section 8.8.5.

Examples of such Inlaid Magic Squares of order 8 are shown in Attachment 8.6.14.

26.3 Magic Squares (9 x 9)

Inlaid Magic Squares of order 9, with four Embedded Magic Squares with different Magic Sums and an Associated Border, have been described in Section 9.7.6.

Examples of such Inlaid Magic Squares of order 9 are shown in Attachment 9.7.7.

Alternatively, the Magic Center Squares can be constructed by means of suitable selected Latin Squares as illustrated in Attachment 18.3.3, based on resp. order 3 and 4 Magic Lines for the integers 0 ... 8 as shown in Attachment 18.6.1.

Attachment 18.6.2 shows a few 9^th order Inlaid Magic Squares for miscellaneous possible Magic Sums s(3) and s(4), however with two each 3^th order Semi Magic Center Squares (6 magic lines).

Each square shown corresponds with numerous solutions, which can be obtained by variation of the four inlays and the border.

26.4 Magic Squares (11 x 11)

Inlaid Magic Squares of order 11, with Embedded Magic Squares of order 4 and 5 (overlapping) with different Magic Sums and an Associated Border, have been described in Section 11.3.2.

Examples of such Inlaid Magic Squares of order 11 are shown in Attachment 14.9.9b.

26.5 Magic Squares (12 x 12)

The 12^th order Inlaid Magic Square shown below, is composed out of an Associated Border and four each 5^th order Pan Magic Center Squares with Magic Sums s(1) = 291, s(2) = 423, s(3) = 302, and s(4) = 434.

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

291 423 302 434

The relation between the Magic Sums s(1), s(2), s(3) and s(4) is:

s(1) = 5 * s12 / 6 - s(4)
s(2) = 5 * s12 / 6 - s(3)

With s12 = 870 the Magic Sum of the 12^th order Inlaid Magic Square.

The 5^th order Pan Magic Center Squares can be constructed by means of suitable selected Latin Squares as illustrated in Attachment 18.3.3, based on 5^th order Magic Lines for the integers 0 ... 11 as shown in Attachment 18.3.1.

Attachment 18.3.2 shows a few 12^th order Inlaid Magic Squares for miscellaneous possible Magic Sums s(3) and s(4).

Each square shown corresponds with numerous solutions, which can be obtained by variation of the four inlays and the border

26.6 Magic Squares (13 x 13)

Inlaid Magic Squares of order 13, with Embedded Magic Squares of order 5 and 6 (overlapping) with different Magic Sums and an Associated Border, have been described in Section 12.7.4.

Examples of such Inlaid Magic Squares of order 11 are shown in Attachment 14.9.9.

26.7 Magic Squares (16 x 16)

The 16^th order Inlaid Magic Square shown below, is composed out of an Associated Border and four each 7^th order Ultra Magic Center Squares with Magic Sums s(1) = 952, s(2) = 840, s(3) = 959 and s(4) = 847.

69 80 68 71 73 75 78 66 252 17 175 99 246 216 138 61 60 250 221 129 63 19 166 104 108 54 24 170 109 241 223 131 140 111 243 214 136 58 29 161 172 141 49 31 163 102 248 218 220 168 106 253 209 143 51 22 28 211 134 56 26 173 97 255 178 190 187 185 183 180 192 181 225 95 147 6 40 122 205 12 10 45 113 207 227 86 152 204 198 232 90 157 1 47 115 156 159 3 38 120 202 237 81 124 125 193 239 83 150 8 42 92 88 154 13 33 127 195 230 44 35 118 200 234 93 145 15 236 21 32 162 110 251 217 135 52 213 247 212 144 50 30 171 105 165 59 25 167 100 256 210 142 133 98 254 219 137 55 20 176 101 132 64 18 174 107 249 215 53 169 103 244 224 130 62 27 245 222 139 57 23 164 112 242 76 65 77 74 72 70 67 79 240 82 158 11 41 119 196 229 7 36 128 194 238 91 153 37 203 233 87 148 16 34 126 85 146 14 43 121 199 228 96 117 116 208 226 94 155 9 39 149 89 151 4 48 114 206 235 197 46 123 201 231 84 160 2 5 191 179 182 184 186 189 177 188

952 840 959 847

The relation between the Magic Sums s(1), s(2), s(3) and s(4) is:

s(1) = 7 * s16 / 8 - s(4)
s(2) = 7 * s16 / 8 - s(3)

With s16 = 2056 the Magic Sum of the 16^th order Inlaid Magic Square.

Attachment 18.4.2 shows a few 16^th order Inlaid Magic Squares for miscellaneous possible Magic Sums s(3) and s(4), however with Pan Magic Center Squares (Non Symmetric).

The 7^th order Pan Magic Center Squares can be constructed by means of suitable selected Latin Squares as illustrated in Attachment 18.3.3, based on 7^th order Magic Lines for the integers 0 ... 15 as shown in Attachment 18.4.1.

Each square shown corresponds with numerous solutions, which can be obtained by variation of the four inlays and the border.

26.8 Magic Squares (20 x 20)

The 20^th order Inlaid Magic Square shown below, originally published by John Hendricks, is composed out of an Associated Border and four each 9^th order Pan Magic Center Squares with Magic Sums s(1), s(2), s(3) and s(4).

The relation between the Magic Sums s(1), s(2), s(3) and s(4) is:

s(1) = 9 * s20 / 10 - s(4)
s(2) = 9 * s20 / 10 - s(3)

With s20 = 4010 the Magic Sum of the 20^th order Inlaid Magic Square.

Attachment 18.5.2 shows a few 20^th order Inlaid Magic Squares for miscellaneous possible Magic Sums s(3) and s(4).

Attachment 18.5.3 shows a few more 20^th order Inlaid Magic Squares for miscellaneous possible Magic Sums s(3) and s(4), however with Simple Magic Center Squares for which one third of the diagonals sum to the Magic Sum.

The 9^th order (Pan) Magic Center Squares can be constructed by means of suitable selected Latin Squares as illustrated in Attachment 18.3.3, based on 9^th order Magic Lines for the integers 0 ... 19 as shown in Attachment 18.5.1.

Each square shown corresponds with numerous solutions, which can be obtained by variation of the four inlays and the border.