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 | > |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 1 | |
| 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 16 | 17 | |
| 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 31 | 32 | 33 | |
| 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 | 46 | 47 | 48 | 49 | |
| 66 | 67 | 68 | 69 | 70 | 71 | 72 | 73 | 74 | 75 | 61 | 62 | 63 | 64 | 65 | |
| 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 | 76 | 77 | 78 | 79 | 80 | 81 | |
| 98 | 99 | 100 | 101 | 102 | 103 | 104 | 105 | 91 | 92 | 93 | 94 | 95 | 96 | 97 | |
| 106 | 107 | 108 | 109 | 110 | 111 | 112 | 113 | 114 | 115 | 116 | 117 | 118 | 119 | 120 | |
| 129 | 130 | 131 | 132 | 133 | 134 | 135 | 121 | 122 | 123 | 124 | 125 | 126 | 127 | 128 | |
| 145 | 146 | 147 | 148 | 149 | 150 | 136 | 137 | 138 | 139 | 140 | 141 | 142 | 143 | 144 | |
| 161 | 162 | 163 | 164 | 165 | 151 | 152 | 153 | 154 | 155 | 156 | 157 | 158 | 159 | 160 | |
| 177 | 178 | 179 | 180 | 166 | 167 | 168 | 169 | 170 | 171 | 172 | 173 | 174 | 175 | 176 | |
| 193 | 194 | 195 | 181 | 182 | 183 | 184 | 185 | 186 | 187 | 188 | 189 | 190 | 191 | 192 | |
| 209 | 210 | 196 | 197 | 198 | 199 | 200 | 201 | 202 | 203 | 204 | 205 | 206 | 207 | 208 | |
| 225 | 211 | 212 | 213 | 214 | 215 | 216 | 217 | 218 | 219 | 220 | 221 | 222 | 223 | 224 |
Step 2, vertical swap
| ^ | ^ | ^ | ^ | ^ | ^ | ^ | ||||||||
| 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 1 |
| 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 16 | 17 |
| 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 31 | 32 | 33 |
| 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 | 46 | 47 | 48 | 49 |
| 66 | 67 | 68 | 69 | 70 | 71 | 72 | 73 | 74 | 75 | 61 | 62 | 63 | 64 | 65 |
| 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 | 76 | 77 | 78 | 79 | 80 | 81 |
| 98 | 99 | 100 | 101 | 102 | 103 | 104 | 105 | 91 | 92 | 93 | 94 | 95 | 96 | 97 |
| 106 | 107 | 108 | 109 | 110 | 111 | 112 | 113 | 114 | 115 | 116 | 117 | 118 | 119 | 120 |
| 129 | 130 | 131 | 132 | 133 | 134 | 135 | 121 | 122 | 123 | 124 | 125 | 126 | 127 | 128 |
| 145 | 146 | 147 | 148 | 149 | 150 | 136 | 137 | 138 | 139 | 140 | 141 | 142 | 143 | 144 |
| 161 | 162 | 163 | 164 | 165 | 151 | 152 | 153 | 154 | 155 | 156 | 157 | 158 | 159 | 160 |
| 177 | 178 | 179 | 180 | 166 | 167 | 168 | 169 | 170 | 171 | 172 | 173 | 174 | 175 | 176 |
| 193 | 194 | 195 | 181 | 182 | 183 | 184 | 185 | 186 | 187 | 188 | 189 | 190 | 191 | 192 |
| 209 | 210 | 196 | 197 | 198 | 199 | 200 | 201 | 202 | 203 | 204 | 205 | 206 | 207 | 208 |
| 225 | 211 | 212 | 213 | 214 | 215 | 216 | 217 | 218 | 219 | 220 | 221 | 222 | 223 | 224 |
| v | v | v | v | v | v | v |
| 18 | 35 | 52 | 69 | 86 | 103 | 112 | 9 | 122 | 139 | 156 | 173 | 190 | 207 | 224 |
| 34 | 51 | 68 | 85 | 102 | 111 | 135 | 25 | 138 | 155 | 172 | 189 | 206 | 223 | 1 |
| 50 | 67 | 84 | 101 | 110 | 134 | 136 | 41 | 154 | 171 | 188 | 205 | 222 | 15 | 17 |
| 66 | 83 | 100 | 109 | 133 | 150 | 152 | 57 | 170 | 187 | 204 | 221 | 14 | 16 | 33 |
| 82 | 99 | 108 | 132 | 149 | 151 | 168 | 73 | 186 | 203 | 220 | 13 | 30 | 32 | 49 |
| 98 | 107 | 131 | 148 | 165 | 167 | 184 | 89 | 202 | 219 | 12 | 29 | 31 | 48 | 65 |
| 106 | 130 | 147 | 164 | 166 | 183 | 200 | 105 | 218 | 11 | 28 | 45 | 47 | 64 | 81 |
| 129 | 146 | 163 | 180 | 182 | 199 | 216 | 113 | 10 | 27 | 44 | 46 | 63 | 80 | 97 |
| 145 | 162 | 179 | 181 | 198 | 215 | 8 | 121 | 26 | 43 | 60 | 62 | 79 | 96 | 120 |
| 161 | 178 | 195 | 197 | 214 | 7 | 24 | 137 | 42 | 59 | 61 | 78 | 95 | 119 | 128 |
| 177 | 194 | 196 | 213 | 6 | 23 | 40 | 153 | 58 | 75 | 77 | 94 | 118 | 127 | 144 |
| 193 | 210 | 212 | 5 | 22 | 39 | 56 | 169 | 74 | 76 | 93 | 117 | 126 | 143 | 160 |
| 209 | 211 | 4 | 21 | 38 | 55 | 72 | 185 | 90 | 92 | 116 | 125 | 142 | 159 | 176 |
| 225 | 3 | 20 | 37 | 54 | 71 | 88 | 201 | 91 | 115 | 124 | 141 | 158 | 175 | 192 |
| 2 | 19 | 36 | 53 | 70 | 87 | 104 | 217 | 114 | 123 | 140 | 157 | 174 | 191 | 208 |
See 3x3, 5x5, 7x7, 9x9, 11x11, 13x13, 15x15, 17x17, 19x19, 21x21, 23x23, 25x25, 27x27, 29x29 and 31x31
