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 15.0   Special Magic Squares, Bimagic Squares, Medjig Solutions 15.9   Bimagic Squares (16 x 16) Harry White illustrates on his website that, under certain limiting conditions, it is possible to construct order 2n Bimagic Squares based on order n Bimagic Squares. 15.9.1 General As described in Section 6.8, for any integer n, a Magic Square C of order 2n can be constructed from any n x n Medjig-Square A and any n x n Magic Square B, by application of the equations: cj = bi + n2 aj with i = 1, 2, ... n2, j = 1, 2, ... 4n2 and Sa = 3n. To make the results for Bimagic Squares (n >= 8) comparable with the results published by other authors, following alternative convention will be applied: cj = 4*(bi - 1) + aj with i = 1, 2, ... n2, j = 1, 2, ... 4n2 and Sa = 5n. as illustrated by the numerical example shown below:
Bimagic Square B (8 x 8)
 60 4 9 49 23 30 38 47 11 14 58 63 37 17 24 36 33 21 20 40 62 59 15 10 46 39 31 22 52 12 1 57 8 64 53 13 43 34 26 19 55 50 6 3 25 45 44 32 29 41 48 28 2 7 51 54 18 27 35 42 16 56 61 5
Medjig Square A (8 x 8)
 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 1 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 1 2 1 4 3 2 1 4 3 2 1 4 3 2 1 4 3 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 2 1 4 3 2 1 4 3 2 1 4 3 2 1 4 3 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 2 1 4 3 2 1 4 3 2 1 4 3 2 1 4 3 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 2 1 4 3 2 1 4 3 2 1 4 3 2 1 4 3 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 1 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 1
Bimagic Square C (16 x 16)
 237 238 15 16 33 34 195 196 89 90 119 120 149 150 187 188 240 239 14 13 36 35 194 193 92 91 118 117 152 151 186 185 41 42 55 56 229 230 251 252 145 146 67 68 93 94 143 144 44 43 54 53 232 231 250 249 148 147 66 65 96 95 142 141 130 129 84 83 78 77 160 159 246 245 236 235 58 57 40 39 131 132 81 82 79 80 157 158 247 248 233 234 59 60 37 38 182 181 156 155 122 121 88 87 206 205 48 47 2 1 228 227 183 184 153 154 123 124 85 86 207 208 45 46 3 4 225 226 30 29 256 255 210 209 52 51 170 169 136 135 102 101 76 75 31 32 253 254 211 212 49 50 171 172 133 134 103 104 73 74 218 217 200 199 22 21 12 11 98 97 180 179 174 173 128 127 219 220 197 198 23 24 9 10 99 100 177 178 175 176 125 126 113 114 163 164 189 190 111 112 5 6 27 28 201 202 215 216 116 115 162 161 192 191 110 109 8 7 26 25 204 203 214 213 69 70 107 108 137 138 167 168 61 62 223 224 241 242 19 20 72 71 106 105 140 139 166 165 64 63 222 221 244 243 18 17
 with the corresponding Magic Sums Sb = 260, Sa = 40, Sc = 2056 and Sums of Squares Sb2 = 11180, Sa2 = 120, Sc2 = 351576. Unlike the resulting Magic Sum Sc, the resulting Sum of Squares Sc2 depends from both bi and aj as: Sc2 = Σ(4*(bi - 1) + aj)2     = 2*16*(bi - 1)2 + 8*Σ(bi - 1)*aj + Σaj2     = 32*10668 + 8*Sab + Sa2     = 341376 + 8*Sab + Sa2 With Sc2 = 351576 and Sa2 = 120 the square C will be bimagic if: Sab = (351576 - 341376 - 120)/8 = 1260 which should be verified while generating the Medjig Squares A for a certain Bimagic Square B. The method of constructing Bimagic Squares of order 16, based on Medjig Squares as defined above, can be summarised as follows: Read (preselected) 8 x 8 Bimagic Square B (192 * 136244 possibilities); Construct 8 x 8 Medjig-Squares A, while ensuring that Sa2 = 120 and Sab = 1260; Construct the 16 x 16 Bimagic Squares C by applying the equations mentioned above. The example shown above is based on Bimagic Square 1 of Attachment 17.6.41 as described in Section 17.6.7. Attachment 15.8.1 shows 48 order 16 Bimagic Squares C, based on the applied order 8 Bimagic Square B, which could be generated with procedure MgcSqrs16a. The same procedure generated for each order 8 Bimagic Square B shown in Attachment 17.6.41 48 order 16 Bimagic Squares C, resulting in 96 * 48 = 4608 squares. Notes: Procedure MgcSqrs16a, generating Medjig Squares A based on the equations describing the 16 x 16 Franklin Squares as described in Section 12.3, has been applied for the sake of convenience (21 parameters only). With the check of Sa2 = 120 and Sab = 1260 disabled, 53120 Magic Squares C could be generated with the same procedure for the same square B. A general procedure for the generation of Medjig Squares A would require 172 parameters, which is beyond the scope of this section. Each Bimagic Square B corresponds with 24 * 4! / 2 = 192 transformations, which can be obtained by appropriate row and column permutations (ref. Section 6.3). Consequently each Bimagic Square C = {B, A} corresponds with 192 solutions, which can be obtained by applying the same row and column permutations on the Medjig Square A. Each (essential different) Bimagic Square C corresponds with 28 * 8! /2 = 5160960 transformations. 15.9.2 Pan Magic, Complete Bimagic Squares The method of constructing Pan Magic, Complete Bimagic Squares of order 16, based on Medjig Squares as defined above, can be summarised as follows: Read (preselected) 8 x 8 Pan Magic, Complete Bimagic Square B (29376 possibilities); Construct 8 x 8 Pan Magic, Complete Medjig-Squares A, while ensuring that Sa2 = 120 and Sab = 1260; Construct the 16 x 16 Pan Magic, Complete Bimagic Squares C by applying the equations mentioned above. (ref. Section 15.8.1) A numerical example is shown below:
Bimagic Square B (8 x 8)
 2 7 27 30 44 45 49 56 32 25 5 4 54 51 47 42 52 53 41 48 26 31 3 6 46 43 55 50 8 1 29 28 21 20 16 9 63 58 38 35 11 14 18 23 33 40 60 61 39 34 62 59 13 12 24 17 57 64 36 37 19 22 10 15
Medjig Square A (8 x 8)
 4 3 2 1 4 3 2 1 2 1 4 3 2 1 4 3 2 1 4 3 2 1 4 3 4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 1 2 1 4 3 2 1 4 3 2 1 4 3 2 1 4 3 4 3 2 1 4 3 2 1 3 4 1 2 3 4 1 2 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 3 4 1 2 3 4 1 2 4 3 2 1 4 3 2 1 2 1 4 3 2 1 4 3 2 1 4 3 2 1 4 3 4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 1 2 1 4 3 2 1 4 3 2 1 4 3 2 1 4 3 4 3 2 1 4 3 2 1
Bimagic Square C (16 x 16)
 8 7 26 25 108 107 118 117 174 173 180 179 194 193 224 223 6 5 28 27 106 105 120 119 176 175 178 177 196 195 222 221 128 127 98 97 20 19 14 13 214 213 204 203 186 185 168 167 126 125 100 99 18 17 16 15 216 215 202 201 188 187 166 165 207 208 209 210 163 164 189 190 101 102 123 124 9 10 23 24 205 206 211 212 161 162 191 192 103 104 121 122 11 12 21 22 183 184 169 170 219 220 197 198 29 30 3 4 113 114 111 112 181 182 171 172 217 218 199 200 31 32 1 2 115 116 109 110 83 84 77 78 63 64 33 34 249 250 231 232 149 150 139 140 81 82 79 80 61 62 35 36 251 252 229 230 151 152 137 138 43 44 53 54 71 72 89 90 129 130 159 160 237 238 243 244 41 42 55 56 69 70 91 92 131 132 157 158 239 240 241 242 156 155 134 133 248 247 234 233 50 49 48 47 94 93 68 67 154 153 136 135 246 245 236 235 52 51 46 45 96 95 66 65 228 227 254 253 144 143 146 145 74 73 88 87 38 37 60 59 226 225 256 255 142 141 148 147 76 75 86 85 40 39 58 57
 The example shown above is based on the first Bimagic Square B of a subset of unique Bimagic Squares, complete with bimagic semi-diagonals, as published by Walter Trump in 2014 (10496 ea). Attachment 15.8.2a shows 56 order 16 Pan Magic Bimagic Squares C (8 Complete) based on the applied order 8 Bimagic Square B, which could be generated with procedure MgcSqrs16a. The same procedure generated for 3712 of the 10496 order 8 Bimagic Squares of subject collection a total of 203776 (128 * 24 + 3584 * 56) order 16 Pan Magic Bimagic Squares C of which 28672 Complete (8 * 3584). Attachment 15.8.2b shows 8 order 16 Pan Magic Complete Bimagic Squares C based on the applied order 8 Bimagic Square B, which could be generated with a more strict procedure MgcSqrs16b. The same procedure generated for 3584 of the 10496 order 8 Bimagic Complete Squares of subject collection a total of 28672 (8 * 3584) order 16 Pan Magic Complete Bimagic Squares C. Notes: For all 28672 order 16 Pan Magic, Complete Bimagic Squares C found above, also the semi-diagonals are bimagic. It can be noticed that the Pan Magic, Complete Medjig Square A shown in the example above is Associated as well. The resulting Square C is however Pan Magic Complete as the 8 x 8 Square B is Pan Magic Complete. 15.9.3 Associated Bimagic Squares The method of constructing Associated Bimagic Squares of order 16, based on Medjig Squares as defined above, can be summarised as follows: Read (preselected) 8 x 8 Associated Bimagic Square B (192 * 841 possibilities); Construct 8 x 8 Associated Medjig-Squares A, while ensuring that Sa2 = 120 and Sab = 1260; Construct the 16 x 16 Associated Bimagic Squares C by applying the equations mentioned above. (ref. Section 15.8.1) A numerical example is shown below:
Bimagic Square B' (8 x 8)
 2 7 27 30 56 49 45 44 32 25 5 4 42 47 51 54 52 53 41 48 6 3 31 26 46 43 55 50 28 29 1 8 57 64 36 37 15 10 22 19 39 34 62 59 17 24 12 13 11 14 18 23 61 60 40 33 21 20 16 9 35 38 58 63
Medjig Square A' (8 x 8)
 4 3 2 1 4 3 2 1 3 4 1 2 3 4 1 2 2 1 4 3 2 1 4 3 1 2 3 4 1 2 3 4 4 3 2 1 4 3 2 1 3 4 1 2 3 4 1 2 2 1 4 3 2 1 4 3 1 2 3 4 1 2 3 4 3 4 1 2 3 4 1 2 4 3 2 1 4 3 2 1 1 2 3 4 1 2 3 4 2 1 4 3 2 1 4 3 3 4 1 2 3 4 1 2 4 3 2 1 4 3 2 1 1 2 3 4 1 2 3 4 2 1 4 3 2 1 4 3 2 1 4 3 2 1 4 3 1 2 3 4 1 2 3 4 4 3 2 1 4 3 2 1 3 4 1 2 3 4 1 2 2 1 4 3 2 1 4 3 1 2 3 4 1 2 3 4 4 3 2 1 4 3 2 1 3 4 1 2 3 4 1 2 1 2 3 4 1 2 3 4 2 1 4 3 2 1 4 3 3 4 1 2 3 4 1 2 4 3 2 1 4 3 2 1 1 2 3 4 1 2 3 4 2 1 4 3 2 1 4 3 3 4 1 2 3 4 1 2 4 3 2 1 4 3 2 1
Bimagic Square C' (16 x 16)
 8 7 26 25 108 107 118 117 223 224 193 194 179 180 173 174 6 5 28 27 106 105 120 119 221 222 195 196 177 178 175 176 128 127 98 97 20 19 14 13 167 168 185 186 203 204 213 214 126 125 100 99 18 17 16 15 165 166 187 188 201 202 215 216 207 208 209 210 163 164 189 190 24 23 10 9 124 123 102 101 205 206 211 212 161 162 191 192 22 21 12 11 122 121 104 103 183 184 169 170 219 220 197 198 112 111 114 113 4 3 30 29 181 182 171 172 217 218 199 200 110 109 116 115 2 1 32 31 226 225 256 255 142 141 148 147 57 58 39 40 85 86 75 76 228 227 254 253 144 143 146 145 59 60 37 38 87 88 73 74 154 153 136 135 246 245 236 235 65 66 95 96 45 46 51 52 156 155 134 133 248 247 234 233 67 68 93 94 47 48 49 50 41 42 55 56 69 70 91 92 242 241 240 239 158 157 132 131 43 44 53 54 71 72 89 90 244 243 238 237 160 159 130 129 81 82 79 80 61 62 35 36 138 137 152 151 230 229 252 251 83 84 77 78 63 64 33 34 140 139 150 149 232 231 250 249
 The example shown above is based on an order 8 Associated Bimagic Square B', being the Euler transformation of the order 8 Pan Magic Complete Bimagic Square B (with bimagic semi-diagonals) as used in Section 15.8.2 above. The Associated Medjig Square A' has been obtained by a comparable transformation on the corresponding Pan Magic, Complete Medjig Square A. Attachment 15.8.3 shows 8 order 16 Associated Bimagic Squares, based on transformation of the corresponding Pan Magic Complete Bimagic Squares as shown in Attachment 15.8.2b. Notes: The resulting order 16 Associated Bimagic Square C' could have been obtained by applying the Euler transformation directly on the order 16 Pan Magic, Complete Bimagic Square C with bimagic semi-diagonals. However the example shown above illustrates that it is possible to construct order 2n Associated Bimagic Squares based on order n Associated Bimagic Squares. As the semi-diagonals of all 28672 order 16 Pan Magic, Complete Bimagic Squares C found in Section 15.8.2 above are bimagic, 28672 order 16 Associated Bimagic Squares can be constructed based on the method described above. It can be noticed that the Associated Medjig Square A' shown in the example above is Pan Magic, Complete as well. The resulting Square C' is however Associated as the 8 x 8 Square B' is Associated. 15.9.4 Summary The obtained results regarding the miscellaneous types of order 16 Bimagic Squares as deducted and discussed in previous sections are summarized in following table:
 Type Characteristics Subroutine Results Bimagic Simple Pan Magic (Occasionally Complete) Pan Magic Complete Associated -
 Next section will provide both classical and modern construction methods for Magic Squares of Squares.