### Shift method (1)

The 15x15 magic square is odd but also a multiple of 3. You can use the shift method to construct a 15x15 magic square but with boundary condition. Take as first row of the first and/or second grid 0-1-2-3-4-5-6-7- 8-9-10-11-12-13-14, than you get only a semi-magic 15x15 square. Take as first row of the first and/or second grid 0-2-1-3-4-5-8-7-6-11-10-9-13-12-14, and you get a panmagic 15x15 square.

The row 0-2-1-3-4-5-8-7-6-11-10-9-13-12-14 leads to a valid 15x15 panmagic square because 0+3+8+11+13 = 2+4+7+10+12 = 1+5+6+9+14 = 35, that is 1/3 of (0+1+2+3+4+5+6+7+8 +9+10+11+12+13+14=) 105.

Take 1x number from first grid (shift 2 to the left) +1

 0 2 1 3 4 5 8 7 6 11 10 9 13 12 14 1 3 4 5 8 7 6 11 10 9 13 12 14 0 2 4 5 8 7 6 11 10 9 13 12 14 0 2 1 3 8 7 6 11 10 9 13 12 14 0 2 1 3 4 5 6 11 10 9 13 12 14 0 2 1 3 4 5 8 7 10 9 13 12 14 0 2 1 3 4 5 8 7 6 11 13 12 14 0 2 1 3 4 5 8 7 6 11 10 9 14 0 2 1 3 4 5 8 7 6 11 10 9 13 12 2 1 3 4 5 8 7 6 11 10 9 13 12 14 0 3 4 5 8 7 6 11 10 9 13 12 14 0 2 1 5 8 7 6 11 10 9 13 12 14 0 2 1 3 4 7 6 11 10 9 13 12 14 0 2 1 3 4 5 8 11 10 9 13 12 14 0 2 1 3 4 5 8 7 6 9 13 12 14 0 2 1 3 4 5 8 7 6 11 10 12 14 0 2 1 3 4 5 8 7 6 11 10 9 13

+15x number from second grid (shift 2 to the right)

 0 2 1 3 4 5 8 7 6 11 10 9 13 12 14 12 14 0 2 1 3 4 5 8 7 6 11 10 9 13 9 13 12 14 0 2 1 3 4 5 8 7 6 11 10 11 10 9 13 12 14 0 2 1 3 4 5 8 7 6 7 6 11 10 9 13 12 14 0 2 1 3 4 5 8 5 8 7 6 11 10 9 13 12 14 0 2 1 3 4 3 4 5 8 7 6 11 10 9 13 12 14 0 2 1 2 1 3 4 5 8 7 6 11 10 9 13 12 14 0 14 0 2 1 3 4 5 8 7 6 11 10 9 13 12 13 12 14 0 2 1 3 4 5 8 7 6 11 10 9 10 9 13 12 14 0 2 1 3 4 5 8 7 6 11 6 11 10 9 13 12 14 0 2 1 3 4 5 8 7 8 7 6 11 10 9 13 12 14 0 2 1 3 4 5 4 5 8 7 6 11 10 9 13 12 14 0 2 1 3 1 3 4 5 8 7 6 11 10 9 13 12 14 0 2

= panmagic 15x15 square

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

Use the shift method to construct magic squares of odd order from 5x5 to infinity.

See

15x15, shift method (1).xls