# Concentric magic squares

It should be mentioned that the definition of concentric magic squares is not unique. Sometimes the second condition of enclosing the inner numbers is neglected.

In this case only the condition remains that there is a new magic square after removing the border. Nowadays we call these squares concentric. But for hundreds of years no difference between these two types was made and concentric squares were called irregular.

So do Benson and Jacoby, who gave an example in their book New Recreations with Magic Squares on page 33..

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

The following magic squares was created with a method by Arnauld. A closer examination shows that it is not only concentric, but nested concentric or multi-concentric. So continuous removing of the borders will always create new concentric magic squares, until the inner magic square of order 4 is reached.

 11 99 50 4 96 95 7 10 92 41 1 12 88 14 86 85 17 83 19 100 98 49 33 77 48 28 74 43 52 3 21 22 23 64 36 35 67 78 79 80 70 69 76 57 45 46 54 25 32 31 30 39 75 47 55 56 44 26 62 71 81 72 38 34 66 65 37 63 29 20 93 59 58 24 53 73 27 68 42 8 40 82 13 87 15 16 84 18 89 61 60 2 51 97 5 6 94 91 9 90