# 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.

• The first k odd numbers are enter in the bottom row. Start to the right of the lower left corner and proceed to the right.
• The next odd number is entered in the center cell of the left column.
• The remaining k − 1 odd numbers are finally entered in the empty cells to the right of the center cell of the top row.
• Now the even numbers follow. Start in the upper right corner and enter k numbers down.
• Further k − 1 even numbers follow from the cell below the center cell of the left column.
• Now there is one remaining even number that is placed in the upper left corner.

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.

•  16 11 13 15 2 4 6 8 9 10 12 14 1 3 5 7
•  16 81 79 77 75 11 13 15 2 78 4 76 6 74 8 9 73 10 72 12 70 14 68 80 1 3 5 7 71 69 67 66

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:

•  16 81 79 77 75 11 13 15 2 78 28 25 27 18 4 76 20 6 74 22 8 9 23 73 10 24 72 12 26 70 14 17 19 21 68 80 1 3 5 7 71 69 67 66
•  16 81 79 77 75 11 13 15 2 78 28 65 63 61 25 27 18 4 76 62 20 6 74 60 22 8 9 23 59 73 10 24 58 72 12 26 56 70 14 64 17 19 21 57 55 54 68 80 1 3 5 7 71 69 67 66

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

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

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

•  8 25 23 7 2 22 12 17 10 4 5 11 13 15 21 6 16 9 14 20 24 1 3 19 18
•  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