Use (as first grid) a splitted 4x4 panmagic square to construct a 16x16 most perfect (Franklin pan)magic square.
Choose one of the 4x4 panmagic squares and split it as follows:
16 |
3 |
10 |
5 |
16 |
5 |
3 |
10 |
||||||||
2 |
13 |
8 |
11 |
13 |
8 |
2 |
11 |
||||||||
7 |
12 |
1 |
14 |
12 |
1 |
7 |
14 |
||||||||
9 |
6 |
15 |
4 |
9 |
4 |
6 |
15 |
Fill in
the empty cells, crosswise, with the same numbers of the 4x4 square:
16 |
4 |
9 |
5 |
15 |
3 |
10 |
6 |
||
1 |
13 |
8 |
12 |
2 |
14 |
7 |
11 |
||
8 |
12 |
1 |
13 |
7 |
11 |
2 |
14 |
||
9 |
5 |
16 |
4 |
10 |
6 |
15 |
3 |
Fill the first grid of the 16x16 magic square with the two 4x4 squares.
The second grid is a fixed grid.
Take 1x number from first grid
16 | 4 | 9 | 5 | 15 | 3 | 10 | 6 | 16 | 4 | 9 | 5 | 15 | 3 | 10 | 6 |
1 | 13 | 8 | 12 | 2 | 14 | 7 | 11 | 1 | 13 | 8 | 12 | 2 | 14 | 7 | 11 |
8 | 12 | 1 | 13 | 7 | 11 | 2 | 14 | 8 | 12 | 1 | 13 | 7 | 11 | 2 | 14 |
9 | 5 | 16 | 4 | 10 | 6 | 15 | 3 | 9 | 5 | 16 | 4 | 10 | 6 | 15 | 3 |
16 | 4 | 9 | 5 | 15 | 3 | 10 | 6 | 16 | 4 | 9 | 5 | 15 | 3 | 10 | 6 |
1 | 13 | 8 | 12 | 2 | 14 | 7 | 11 | 1 | 13 | 8 | 12 | 2 | 14 | 7 | 11 |
8 | 12 | 1 | 13 | 7 | 11 | 2 | 14 | 8 | 12 | 1 | 13 | 7 | 11 | 2 | 14 |
9 | 5 | 16 | 4 | 10 | 6 | 15 | 3 | 9 | 5 | 16 | 4 | 10 | 6 | 15 | 3 |
16 | 4 | 9 | 5 | 15 | 3 | 10 | 6 | 16 | 4 | 9 | 5 | 15 | 3 | 10 | 6 |
1 | 13 | 8 | 12 | 2 | 14 | 7 | 11 | 1 | 13 | 8 | 12 | 2 | 14 | 7 | 11 |
8 | 12 | 1 | 13 | 7 | 11 | 2 | 14 | 8 | 12 | 1 | 13 | 7 | 11 | 2 | 14 |
9 | 5 | 16 | 4 | 10 | 6 | 15 | 3 | 9 | 5 | 16 | 4 | 10 | 6 | 15 | 3 |
16 | 4 | 9 | 5 | 15 | 3 | 10 | 6 | 16 | 4 | 9 | 5 | 15 | 3 | 10 | 6 |
1 | 13 | 8 | 12 | 2 | 14 | 7 | 11 | 1 | 13 | 8 | 12 | 2 | 14 | 7 | 11 |
8 | 12 | 1 | 13 | 7 | 11 | 2 | 14 | 8 | 12 | 1 | 13 | 7 | 11 | 2 | 14 |
9 | 5 | 16 | 4 | 10 | 6 | 15 | 3 | 9 | 5 | 16 | 4 | 10 | 6 | 15 | 3 |
+ 16x number from second grid
1 | 15 | 0 | 14 | 1 | 15 | 0 | 14 | 0 | 14 | 1 | 15 | 0 | 14 | 1 | 15 |
0 | 14 | 1 | 15 | 0 | 14 | 1 | 15 | 1 | 15 | 0 | 14 | 1 | 15 | 0 | 14 |
15 | 1 | 14 | 0 | 15 | 1 | 14 | 0 | 14 | 0 | 15 | 1 | 14 | 0 | 15 | 1 |
14 | 0 | 15 | 1 | 14 | 0 | 15 | 1 | 15 | 1 | 14 | 0 | 15 | 1 | 14 | 0 |
3 | 13 | 2 | 12 | 3 | 13 | 2 | 12 | 2 | 12 | 3 | 13 | 2 | 12 | 3 | 13 |
2 | 12 | 3 | 13 | 2 | 12 | 3 | 13 | 3 | 13 | 2 | 12 | 3 | 13 | 2 | 12 |
13 | 3 | 12 | 2 | 13 | 3 | 12 | 2 | 12 | 2 | 13 | 3 | 12 | 2 | 13 | 3 |
12 | 2 | 13 | 3 | 12 | 2 | 13 | 3 | 13 | 3 | 12 | 2 | 13 | 3 | 12 | 2 |
5 | 11 | 4 | 10 | 5 | 11 | 4 | 10 | 4 | 10 | 5 | 11 | 4 | 10 | 5 | 11 |
4 | 10 | 5 | 11 | 4 | 10 | 5 | 11 | 5 | 11 | 4 | 10 | 5 | 11 | 4 | 10 |
11 | 5 | 10 | 4 | 11 | 5 | 10 | 4 | 10 | 4 | 11 | 5 | 10 | 4 | 11 | 5 |
10 | 4 | 11 | 5 | 10 | 4 | 11 | 5 | 11 | 5 | 10 | 4 | 11 | 5 | 10 | 4 |
7 | 9 | 6 | 8 | 7 | 9 | 6 | 8 | 6 | 8 | 7 | 9 | 6 | 8 | 7 | 9 |
6 | 8 | 7 | 9 | 6 | 8 | 7 | 9 | 7 | 9 | 6 | 8 | 7 | 9 | 6 | 8 |
9 | 7 | 8 | 6 | 9 | 7 | 8 | 6 | 8 | 6 | 9 | 7 | 8 | 6 | 9 | 7 |
8 | 6 | 9 | 7 | 8 | 6 | 9 | 7 | 9 | 7 | 8 | 6 | 9 | 7 | 8 | 6 |
= most perfect (Franklin pan)magic 16x16 square
32 | 244 | 9 | 229 | 31 | 243 | 10 | 230 | 16 | 228 | 25 | 245 | 15 | 227 | 26 | 246 |
1 | 237 | 24 | 252 | 2 | 238 | 23 | 251 | 17 | 253 | 8 | 236 | 18 | 254 | 7 | 235 |
248 | 28 | 225 | 13 | 247 | 27 | 226 | 14 | 232 | 12 | 241 | 29 | 231 | 11 | 242 | 30 |
233 | 5 | 256 | 20 | 234 | 6 | 255 | 19 | 249 | 21 | 240 | 4 | 250 | 22 | 239 | 3 |
64 | 212 | 41 | 197 | 63 | 211 | 42 | 198 | 48 | 196 | 57 | 213 | 47 | 195 | 58 | 214 |
33 | 205 | 56 | 220 | 34 | 206 | 55 | 219 | 49 | 221 | 40 | 204 | 50 | 222 | 39 | 203 |
216 | 60 | 193 | 45 | 215 | 59 | 194 | 46 | 200 | 44 | 209 | 61 | 199 | 43 | 210 | 62 |
201 | 37 | 224 | 52 | 202 | 38 | 223 | 51 | 217 | 53 | 208 | 36 | 218 | 54 | 207 | 35 |
96 | 180 | 73 | 165 | 95 | 179 | 74 | 166 | 80 | 164 | 89 | 181 | 79 | 163 | 90 | 182 |
65 | 173 | 88 | 188 | 66 | 174 | 87 | 187 | 81 | 189 | 72 | 172 | 82 | 190 | 71 | 171 |
184 | 92 | 161 | 77 | 183 | 91 | 162 | 78 | 168 | 76 | 177 | 93 | 167 | 75 | 178 | 94 |
169 | 69 | 192 | 84 | 170 | 70 | 191 | 83 | 185 | 85 | 176 | 68 | 186 | 86 | 175 | 67 |
128 | 148 | 105 | 133 | 127 | 147 | 106 | 134 | 112 | 132 | 121 | 149 | 111 | 131 | 122 | 150 |
97 | 141 | 120 | 156 | 98 | 142 | 119 | 155 | 113 | 157 | 104 | 140 | 114 | 158 | 103 | 139 |
152 | 124 | 129 | 109 | 151 | 123 | 130 | 110 | 136 | 108 | 145 | 125 | 135 | 107 | 146 | 126 |
137 | 101 | 160 | 116 | 138 | 102 | 159 | 115 | 153 | 117 | 144 | 100 | 154 | 118 | 143 | 99 |
Use the basic pattern method (2) also to construct a most perfect 8x8 magic square.