### Shift method (2)

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It is possible to use the shiftmethod to get a more tight structure of the panmagic 15x15 square.

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You have to puzzle to get the first row.Â Construct a 3x5 = 5x3 matrix:

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Â  Matrix 3x5Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â =Â Â Â Â Â Â Â  Â Matrix 5x3Â

 0 9 12 Â 21 Â Â 0 9 12 Â 21 1 14 6 Â 21 Â Â 1 14 6 Â 21 11 2 8 Â 21 Â Â 11 2 8 Â 21 13 3 5 Â 21 Â Â 13 3 5 Â 21 10 7 4 Â 21 Â Â 10 7 4 Â 21 Â Â Â Â Â Â Â Â Â Â Â Â 35 35 35 Â Â Â Â 35 35 35 Â Â

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The magic sumÂ of 0Â up to 14 is 105.Â In the matrixÂ theÂ sum of each column isÂ (5/15 x 105 =) 35Â and the sumÂ of each row isÂ (3/15 x 105 =) 21 is.Â Put the numbers in the first row:

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Â First row according toÂ 3x5 matrix

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

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Â First row according toÂ 5x3 matrix

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

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Construct row 2Â up toÂ 15Â by shifting the first rowÂ each timeÂ 4 places to the left.

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

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We have constructed the first grid. The second grid is a reflection (rotated by a quarter and mirrored) of the first grid. Take 15xÂ number +1 from first grid and add 1xÂ number from second grid to get a 15x15 panmagic square.

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Â Take 15xÂ number + 1

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

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Â + 1x number

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

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Â = 15x15 panmagicÂ squareÂ

 4 122 110 18 76 141 180 71 219 205 194 37 158 102 58 82 143 177 73 214 197 185 33 151 96 60 11 129 115 29 221 204 190 44 157 98 57 13 124 107 20 78 136 171 75 153 91 51 15 131 114 25 89 142 173 72 223 199 182 35 133 109 17 80 138 166 66 225 206 189 40 164 97 53 12 149 172 68 222 208 184 32 155 93 46 6 135 116 24 85 210 191 39 160 104 52 8 132 118 19 77 140 168 61 216 95 48 1 126 120 26 84 145 179 67 218 207 193 34 152 117 28 79 137 170 63 211 201 195 41 159 100 59 7 128 175 74 217 203 192 43 154 92