16x16 magic square

Basic pattern method (1) c

 

Use 4x the same Franklin panmagic 8x8 square and two reflecting grids to construct a most perfect Franklin panmagic 16x16 square.

 

 

Take 1x number

1 60 22 47 2 59 21 48 1 60 22 47 2 59 21 48
56 13 35 26 55 14 36 25 56 13 35 26 55 14 36 25
43 18 64 5 44 17 63 6 43 18 64 5 44 17 63 6
30 39 9 52 29 40 10 51 30 39 9 52 29 40 10 51
3 58 24 45 4 57 23 46 3 58 24 45 4 57 23 46
54 15 33 28 53 16 34 27 54 15 33 28 53 16 34 27
41 20 62 7 42 19 61 8 41 20 62 7 42 19 61 8
32 37 11 50 31 38 12 49 32 37 11 50 31 38 12 49
1 60 22 47 2 59 21 48 1 60 22 47 2 59 21 48
56 13 35 26 55 14 36 25 56 13 35 26 55 14 36 25
43 18 64 5 44 17 63 6 43 18 64 5 44 17 63 6
30 39 9 52 29 40 10 51 30 39 9 52 29 40 10 51
3 58 24 45 4 57 23 46 3 58 24 45 4 57 23 46
54 15 33 28 53 16 34 27 54 15 33 28 53 16 34 27
41 20 62 7 42 19 61 8 41 20 62 7 42 19 61 8
32 37 11 50 31 38 12 49 32 37 11 50 31 38 12 49

 

 

+64x number

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
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
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
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
1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0

 

 

+128x number

0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0

 

 

Most perfect Franklin panmagic 16x16 square

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

 

 

This 16x16 magic square is panmagic, 2x2 compact and each 1/4 row/column/diagonal gives 1/4 of the magic sum. Notify that the 16x16 magic square has the tight 'Willem Barink' structure.

 

 

Use basic pattern method (1) to construct magic squares of order is multiple of 4 from 8x8 to infinity. See 8x8, 12x12, 16x16a, 16x16b, 16x16c, 20x20, 24x24a, 24x24b, 28x28, 32x32a, 32x32b, 32x32c and 32x32d

 

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16x16, Basic pattern method (1c).xls
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16x16 magic square