# Multi-bordered magic squares

The following bordered magic square was created by Michael Stifel in 1544. A closer examination shows that it is not only bordered, but nested bordered or multi-bordered. You can also remove the border of the inner magic square of order 5 and will get another magic square of order 3. So continuous removing of the borders will always create new magic squares.

•  12 49 47 45 9 11 2 46 20 37 35 19 14 4 44 34 24 29 22 16 6 7 17 23 25 27 33 43 8 18 28 21 26 32 42 10 36 13 15 31 30 40 48 1 3 5 41 39 38
•  20 37 35 19 14 34 24 29 22 16 17 23 25 27 33 18 28 21 26 32 36 13 15 31 30
•  24 29 22 23 25 27 28 21 26

Another wonderful example of multi-concentric magic squares of order 12 is given by Kraitchik, which he shows in his book Mathematical Recreations on page 167. Four borders can be removed and each new inner square is magic again.

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