Stifel

Michael Stifel (1487 - 1567) was a German theologian, mathematician and reformer who also dealt with magic squares in his book on arithmetic. Stifel also fills the square separately according to even and odd numbers. For order n=n=2k+1=9 follows k=4.

Finally, empty cells are filled with the complements of the numbers already entered. These complements are always entered in the horizontally or vertically opposite row or column. The only exception is the two upper corners. As always with bordered magic squares, their complement must be entered in the diagonally opposite corner.

  • 161113152
    4
    6
    8
    9
    10
    12
    14
    1357
  • 16817977751113152
    784
    766
    748
    973
    1072
    1270
    1468
    80135771696766

This procedure is now continued for each border from the outside towards the center. For the border of the embedded inner square of order n=7, the following intermediate square follows:

  • 16817977751113152
    78282527184
    76206
    74228
    92373
    102472
    122670
    1417192168
    80135771696766
  • 16817977751113152
    78286563612527184
    7662206
    7460228
    9235973
    10245872
    12265670
    146417192157555468
    80135771696766

The same procedure is followed with the remaining borders until the complete multi-bordered magic square has finally been created.

16817977751113152
78286563612527184
76623653513530206
74605040453832228
92333394143495973
102434443742485872
122652293147465670
146417192157555468
80135771696766

This method works for all odd orders. Two other examples for n=5 and n=7 are shown in the following figure.

  • 8252372
    221217104
    511131521
    61691420
    24131918
  • 124947459112
    4620373519144
    4434242922166
    7172325273343
    8182821263242
    10361315313040
    48135413938