Office Applications and Entertainment, Magic Squares of Subtraction

1 Construction of Magic Squares of Subtraction (16 x 16)

Associated, Res16 = 32, Pair Sum = 257

With square B Associated Magic and square A Associated, the resulting square C will be an Associated Magic Square of Subtraction, as illustrated below:

A, Res4 = 8 16 7 14 13 11 12 9 2 15 8 5 6 4 3 10 1

C, Res16 = 32, Pair Sum = 257 256 247 254 253 251 252 249 242 255 248 245 246 244 243 250 241 144 135 142 141 139 140 137 130 143 136 133 134 132 131 138 129 80 71 78 77 75 76 73 66 79 72 69 70 68 67 74 65 64 55 62 61 59 60 57 50 63 56 53 54 52 51 58 49 48 39 46 45 43 44 41 34 47 40 37 38 36 35 42 33 96 87 94 93 91 92 89 82 95 88 85 86 84 83 90 81 160 151 158 157 155 156 153 146 159 152 149 150 148 147 154 145 240 231 238 237 235 236 233 226 239 232 229 230 228 227 234 225 32 23 30 29 27 28 25 18 31 24 21 22 20 19 26 17 112 103 110 109 107 108 105 98 111 104 101 102 100 99 106 97 176 167 174 173 171 172 169 162 175 168 165 166 164 163 170 161 224 215 222 221 219 220 217 210 223 216 213 214 212 211 218 209 208 199 206 205 203 204 201 194 207 200 197 198 196 195 202 193 192 183 190 189 187 188 185 178 191 184 181 182 180 179 186 177 128 119 126 125 123 124 121 114 127 120 117 118 116 115 122 113 16 7 14 13 11 12 9 2 15 8 5 6 4 3 10 1

B, s4 = 34 16 9 5 4 3 6 10 15 2 7 11 14 13 12 8 1

With 384 possible squares for B and n4 for C_j (j = 1 ... 16), the resulting number of 16^th order Associated Squares of Subtraction with Res16 = 32 will be 384 * n4¹⁶ with n4 = 8 * 8 (ref. Attachment 4.1).

2 Construction of Magic Squares of Subtraction (16 x 16)

Res16 = 32, Main - and Broken Diagonals sum to s16 = 2056

With square B Pan Magic and square A Pan Magic, the resulting square C will be a Magic Square of Subtraction with Pan Magic Diagonals, as illustrated below:

A, Res4 = 8 12 8 2 14 3 15 9 5 16 4 6 10 7 11 13 1

C, Res16 = 32, s16 = 2056 188 184 178 190 179 191 185 181 192 180 182 186 183 187 189 177 204 200 194 206 195 207 201 197 208 196 198 202 199 203 205 193 44 40 34 46 35 47 41 37 48 36 38 42 39 43 45 33 92 88 82 94 83 95 89 85 96 84 86 90 87 91 93 81 108 104 98 110 99 111 105 101 112 100 102 106 103 107 109 97 28 24 18 30 19 31 25 21 32 20 22 26 23 27 29 17 252 248 242 254 243 255 249 245 256 244 246 250 247 251 253 241 140 136 130 142 131 143 137 133 144 132 134 138 135 139 141 129 220 216 210 222 211 223 217 213 224 212 214 218 215 219 221 209 172 168 162 174 163 175 169 165 176 164 166 170 167 171 173 161 76 72 66 78 67 79 73 69 80 68 70 74 71 75 77 65 60 56 50 62 51 63 57 53 64 52 54 58 55 59 61 49 12 8 2 14 3 15 9 5 16 4 6 10 7 11 13 1 124 120 114 126 115 127 121 117 128 116 118 122 119 123 125 113 156 152 146 158 147 159 153 149 160 148 150 154 151 155 157 145 236 232 226 238 227 239 233 229 240 228 230 234 231 235 237 225

B, s4 = 34 12 13 3 6 7 2 16 9 14 11 5 4 1 8 10 15

With 384 possible squares for B and n4 for C_j (j = 1 ... 16), the resulting number of 16^th order Squares of Subtraction with Res16 = 32 will be 384 * n4¹⁶ with n4 = 8 * 24 (ref. Attachment 4.2).