Symmetric transformation
Marios Mamzeris shows us that you can transform a odd square with sequencial numbers into a symmetric magic square in two steps (https://www.oddmagicsquares.com/):
Step 1, horizontal swap
| < | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | |
| < | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | |
| < | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 | 51 | |
| < | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | |
| < | 69 | 70 | 71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 | 81 | 82 | 83 | 84 | 85 | |
| < | 86 | 87 | 88 | 89 | 90 | 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 | 101 | 102 | |
| < | 103 | 104 | 105 | 106 | 107 | 108 | 109 | 110 | 111 | 112 | 113 | 114 | 115 | 116 | 117 | 118 | 119 | |
| < | 120 | 121 | 122 | 123 | 124 | 125 | 126 | 127 | 128 | 129 | 130 | 131 | 132 | 133 | 134 | 135 | 136 | |
| 137 | 138 | 139 | 140 | 141 | 142 | 143 | 144 | 145 | 146 | 147 | 148 | 149 | 150 | 151 | 152 | 153 | ||
| 154 | 155 | 156 | 157 | 158 | 159 | 160 | 161 | 162 | 163 | 164 | 165 | 166 | 167 | 168 | 169 | 170 | > | |
| 171 | 172 | 173 | 174 | 175 | 176 | 177 | 178 | 179 | 180 | 181 | 182 | 183 | 184 | 185 | 186 | 187 | > | |
| 188 | 189 | 190 | 191 | 192 | 193 | 194 | 195 | 196 | 197 | 198 | 199 | 200 | 201 | 202 | 203 | 204 | > | |
| 205 | 206 | 207 | 208 | 209 | 210 | 211 | 212 | 213 | 214 | 215 | 216 | 217 | 218 | 219 | 220 | 221 | > | |
| 222 | 223 | 224 | 225 | 226 | 227 | 228 | 229 | 230 | 231 | 232 | 233 | 234 | 235 | 236 | 237 | 238 | > | |
| 239 | 240 | 241 | 242 | 243 | 244 | 245 | 246 | 247 | 248 | 249 | 250 | 251 | 252 | 253 | 254 | 255 | > | |
| 256 | 257 | 258 | 259 | 260 | 261 | 262 | 263 | 264 | 265 | 266 | 267 | 268 | 269 | 270 | 271 | 272 | > | |
| 273 | 274 | 275 | 276 | 277 | 278 | 279 | 280 | 281 | 282 | 283 | 284 | 285 | 286 | 287 | 288 | 289 | > |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 1 | |
| 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 18 | 19 | |
| 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 | 51 | 35 | 36 | 37 | |
| 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 52 | 53 | 54 | 55 | |
| 74 | 75 | 76 | 77 | 78 | 79 | 80 | 81 | 82 | 83 | 84 | 85 | 69 | 70 | 71 | 72 | 73 | |
| 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 | 101 | 102 | 86 | 87 | 88 | 89 | 90 | 91 | |
| 110 | 111 | 112 | 113 | 114 | 115 | 116 | 117 | 118 | 119 | 103 | 104 | 105 | 106 | 107 | 108 | 109 | |
| 128 | 129 | 130 | 131 | 132 | 133 | 134 | 135 | 136 | 120 | 121 | 122 | 123 | 124 | 125 | 126 | 127 | |
| 137 | 138 | 139 | 140 | 141 | 142 | 143 | 144 | 145 | 146 | 147 | 148 | 149 | 150 | 151 | 152 | 153 | |
| 163 | 164 | 165 | 166 | 167 | 168 | 169 | 170 | 154 | 155 | 156 | 157 | 158 | 159 | 160 | 161 | 162 | |
| 181 | 182 | 183 | 184 | 185 | 186 | 187 | 171 | 172 | 173 | 174 | 175 | 176 | 177 | 178 | 179 | 180 | |
| 199 | 200 | 201 | 202 | 203 | 204 | 188 | 189 | 190 | 191 | 192 | 193 | 194 | 195 | 196 | 197 | 198 | |
| 217 | 218 | 219 | 220 | 221 | 205 | 206 | 207 | 208 | 209 | 210 | 211 | 212 | 213 | 214 | 215 | 216 | |
| 235 | 236 | 237 | 238 | 222 | 223 | 224 | 225 | 226 | 227 | 228 | 229 | 230 | 231 | 232 | 233 | 234 | |
| 253 | 254 | 255 | 239 | 240 | 241 | 242 | 243 | 244 | 245 | 246 | 247 | 248 | 249 | 250 | 251 | 252 | |
| 271 | 272 | 256 | 257 | 258 | 259 | 260 | 261 | 262 | 263 | 264 | 265 | 266 | 267 | 268 | 269 | 270 | |
| 289 | 273 | 274 | 275 | 276 | 277 | 278 | 279 | 280 | 281 | 282 | 283 | 284 | 285 | 286 | 287 | 288 |
Step 2, vertical swap
| ^ | ^ | ^ | ^ | ^ | ^ | ^ | ^ | |||||||||
| 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 1 |
| 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 18 | 19 |
| 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 | 51 | 35 | 36 | 37 |
| 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 52 | 53 | 54 | 55 |
| 74 | 75 | 76 | 77 | 78 | 79 | 80 | 81 | 82 | 83 | 84 | 85 | 69 | 70 | 71 | 72 | 73 |
| 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 | 101 | 102 | 86 | 87 | 88 | 89 | 90 | 91 |
| 110 | 111 | 112 | 113 | 114 | 115 | 116 | 117 | 118 | 119 | 103 | 104 | 105 | 106 | 107 | 108 | 109 |
| 128 | 129 | 130 | 131 | 132 | 133 | 134 | 135 | 136 | 120 | 121 | 122 | 123 | 124 | 125 | 126 | 127 |
| 137 | 138 | 139 | 140 | 141 | 142 | 143 | 144 | 145 | 146 | 147 | 148 | 149 | 150 | 151 | 152 | 153 |
| 163 | 164 | 165 | 166 | 167 | 168 | 169 | 170 | 154 | 155 | 156 | 157 | 158 | 159 | 160 | 161 | 162 |
| 181 | 182 | 183 | 184 | 185 | 186 | 187 | 171 | 172 | 173 | 174 | 175 | 176 | 177 | 178 | 179 | 180 |
| 199 | 200 | 201 | 202 | 203 | 204 | 188 | 189 | 190 | 191 | 192 | 193 | 194 | 195 | 196 | 197 | 198 |
| 217 | 218 | 219 | 220 | 221 | 205 | 206 | 207 | 208 | 209 | 210 | 211 | 212 | 213 | 214 | 215 | 216 |
| 235 | 236 | 237 | 238 | 222 | 223 | 224 | 225 | 226 | 227 | 228 | 229 | 230 | 231 | 232 | 233 | 234 |
| 253 | 254 | 255 | 239 | 240 | 241 | 242 | 243 | 244 | 245 | 246 | 247 | 248 | 249 | 250 | 251 | 252 |
| 271 | 272 | 256 | 257 | 258 | 259 | 260 | 261 | 262 | 263 | 264 | 265 | 266 | 267 | 268 | 269 | 270 |
| 289 | 273 | 274 | 275 | 276 | 277 | 278 | 279 | 280 | 281 | 282 | 283 | 284 | 285 | 286 | 287 | 288 |
| v | v | v | v | v | v | v | v |
| 20 | 39 | 58 | 77 | 96 | 115 | 134 | 144 | 10 | 155 | 174 | 193 | 212 | 231 | 250 | 269 | 288 |
| 38 | 57 | 76 | 95 | 114 | 133 | 143 | 170 | 28 | 173 | 192 | 211 | 230 | 249 | 268 | 287 | 1 |
| 56 | 75 | 94 | 113 | 132 | 142 | 169 | 171 | 46 | 191 | 210 | 229 | 248 | 267 | 286 | 17 | 19 |
| 74 | 93 | 112 | 131 | 141 | 168 | 187 | 189 | 64 | 209 | 228 | 247 | 266 | 285 | 16 | 18 | 37 |
| 92 | 111 | 130 | 140 | 167 | 186 | 188 | 207 | 82 | 227 | 246 | 265 | 284 | 15 | 34 | 36 | 55 |
| 110 | 129 | 139 | 166 | 185 | 204 | 206 | 225 | 100 | 245 | 264 | 283 | 14 | 33 | 35 | 54 | 73 |
| 128 | 138 | 165 | 184 | 203 | 205 | 224 | 243 | 118 | 263 | 282 | 13 | 32 | 51 | 53 | 72 | 91 |
| 137 | 164 | 183 | 202 | 221 | 223 | 242 | 261 | 136 | 281 | 12 | 31 | 50 | 52 | 71 | 90 | 109 |
| 163 | 182 | 201 | 220 | 222 | 241 | 260 | 279 | 145 | 11 | 30 | 49 | 68 | 70 | 89 | 108 | 127 |
| 181 | 200 | 219 | 238 | 240 | 259 | 278 | 9 | 154 | 29 | 48 | 67 | 69 | 88 | 107 | 126 | 153 |
| 199 | 218 | 237 | 239 | 258 | 277 | 8 | 27 | 172 | 47 | 66 | 85 | 87 | 106 | 125 | 152 | 162 |
| 217 | 236 | 255 | 257 | 276 | 7 | 26 | 45 | 190 | 65 | 84 | 86 | 105 | 124 | 151 | 161 | 180 |
| 235 | 254 | 256 | 275 | 6 | 25 | 44 | 63 | 208 | 83 | 102 | 104 | 123 | 150 | 160 | 179 | 198 |
| 253 | 272 | 274 | 5 | 24 | 43 | 62 | 81 | 226 | 101 | 103 | 122 | 149 | 159 | 178 | 197 | 216 |
| 271 | 273 | 4 | 23 | 42 | 61 | 80 | 99 | 244 | 119 | 121 | 148 | 158 | 177 | 196 | 215 | 234 |
| 289 | 3 | 22 | 41 | 60 | 79 | 98 | 117 | 262 | 120 | 147 | 157 | 176 | 195 | 214 | 233 | 252 |
| 2 | 21 | 40 | 59 | 78 | 97 | 116 | 135 | 280 | 146 | 156 | 175 | 194 | 213 | 232 | 251 | 270 |
See 3x3, 5x5, 7x7, 9x9, 11x11, 13x13, 15x15, 17x17, 19x19, 21x21, 23x23, 25x25, 27x27, 29x29 and 31x31
