### Shift method (2)

Put in the first row of the first grid the numbers 0 up to 8. Construct the second and the third row of the first grid by shifting the first row each time 3 places to the left.

First grid, first three rows

 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8

The first three rows of the first grid consists of three 3x3 sub-squares. Construct row 4 up to 6 by swapping the sequence of the three columns in the three 3x3 sub-squares into 2-3-1 (instead of 1-2-3). Construct row 7 up to 9 by swapping the sequence of the three columns in the three 3x3 sub-squares into 3-1-2 (instead of 1-2-3).

First grid

 0 1 2 3 4 5 6 7 8 3 4 5 6 7 8 0 1 2 6 7 8 0 1 2 3 4 5 1 2 0 4 5 3 7 8 6 4 5 3 7 8 6 1 2 0 7 8 6 1 2 0 4 5 3 2 0 1 5 3 4 8 6 7 5 3 4 8 6 7 2 0 1 8 6 7 2 0 1 5 3 4

The second grid is a reflection (rotated by a quarter) of the first grid. Take 1x number from first grid +1 and add 9x number from the same cell of the second grid.

1x number                                  9x number                                  panmagic 9x9 square

 0 1 2 3 4 5 6 7 8 8 5 2 7 4 1 6 3 0 73 47 21 67 41 15 61 35 9 3 4 5 6 7 8 0 1 2 6 3 0 8 5 2 7 4 1 58 32 6 79 53 27 64 38 12 6 7 8 0 1 2 3 4 5 7 4 1 6 3 0 8 5 2 70 44 18 55 29 3 76 50 24 1 2 0 4 5 3 7 8 6 2 8 5 1 7 4 0 6 3 20 75 46 14 69 40 8 63 34 4 5 3 7 8 6 1 2 0 0 6 3 2 8 5 1 7 4 5 60 31 26 81 52 11 66 37 7 8 6 1 2 0 4 5 3 1 7 4 0 6 3 2 8 5 17 72 43 2 57 28 23 78 49 2 0 1 5 3 4 8 6 7 5 2 8 4 1 7 3 0 6 48 19 74 42 13 68 36 7 62 5 3 4 8 6 7 2 0 1 3 0 6 5 2 8 4 1 7 33 4 59 54 25 80 39 10 65 8 6 7 2 0 1 5 3 4 4 1 7 3 0 6 5 2 8 45 16 71 30 1 56 51 22 77

This 9x9 magic square is panmagic and 3x3 compact (but not symmetric).

Use the shift method to construct magic squares of odd order from 5x5 to infinity.

See

9x9, Shiftmethod 2.xls

It is even possible to get a symmetric result.

1x number +1                           +  9x number                                 ultra magic 9x9 square

 0 2 1 6 8 7 3 5 4 4 7 1 5 8 2 3 6 0 37 66 11 52 81 26 31 60 5 6 8 7 3 5 4 0 2 1 3 6 0 4 7 1 5 8 2 34 63 8 40 69 14 46 75 20 3 5 4 0 2 1 6 8 7 5 8 2 3 6 0 4 7 1 49 78 23 28 57 2 43 72 17 2 1 0 8 7 6 5 4 3 1 4 7 2 5 8 0 3 6 12 38 64 27 53 79 6 32 58 8 7 6 5 4 3 2 1 0 0 3 6 1 4 7 2 5 8 9 35 61 15 41 67 21 47 73 5 4 3 2 1 0 8 7 6 2 5 8 0 3 6 1 4 7 24 50 76 3 29 55 18 44 70 1 0 2 7 6 8 4 3 5 7 1 4 8 2 5 6 0 3 65 10 39 80 25 54 59 4 33 7 6 8 4 3 5 1 0 2 6 0 3 7 1 4 8 2 5 62 7 36 68 13 42 74 19 48 4 3 5 1 0 2 7 6 8 8 2 5 6 0 3 7 1 4 77 22 51 56 1 30 71 16 45

This 9x9 magic square is panmagic, 3x3 compact and symmetric, so it is ultra magic.

9x9, Shiftmethod 2a.xlsx