## Trimagic Squares

A trimagic square is a bimagic square, where in addition, the rows, columns and diagonals have the same sum, when all the numbers are taken to the third power. The first trimagic square was given by Gaston Tarry in 1905 vor. It had the order of 128. Cazalas then improved this method and succeeded in creating trimagic squares of orders 64 and 81

A another great step was made in 1976, when Benson and Jacoby published a trimagic square of order 32.

The great sensation took place in 2002, when Walter Trump published a trimagic square of order 12. In the meantime it is well-known that this is the lowest order for which a trimagic square can exist.

 1 22 33 41 62 66 79 83 104 112 123 144 9 119 45 115 107 93 52 38 30 100 26 136 75 141 35 48 57 14 131 88 97 110 4 70 74 8 106 49 12 43 102 133 96 39 137 71 140 101 124 42 60 37 108 85 103 21 44 5 122 76 142 86 67 126 19 78 59 3 69 23 55 27 95 135 130 89 56 15 10 50 118 90 132 117 68 91 11 99 46 134 54 77 28 13 73 64 2 121 109 32 113 36 24 143 81 72 58 98 84 116 138 16 129 7 29 61 47 87 80 34 105 6 92 127 18 53 139 40 111 65 51 63 31 20 25 128 17 120 125 114 82 94

His trimagic square is self-complementary and the numbers are horizontally symmetrical. This means that numbers which are symmetrically in the same row always have the same sum n2 + 1, which is 145. This trimagic square has the constant sums S12=870, S122=83 810 and S123=9 082 800.

Read more on the multimagic pages of Christian Boyer about the discovery of this sensational trimagic square.

The first known 16th-order trimagic square was created in 2005 by Chen Qin-wu and Chen Mu-tian. This square is also self-complementary and the numbers are again horizontally symmetrical.

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