### Shift method (1)

You can use the shift method (= same method to construct the 5x5 and 7x7 panmagic square) to construct panmagic 9x9 squares, but the number of possibilities is limited. If you choose as first row 0-1-2-3-4-5-6-7-8, than you get only a semi-magic 9x9 square. lf you choose as first row 0-2-1-5-4-3-7-6-8 than you get a valid panmagic 9x9 square, because 0+5+7 = 2+4+6 = 1+3+8 = 12, that is 1/3 of (0+1+2+3+4+5+6+7+8=) 36.

In the example below I have chosen for the  first and second grid a different shifted version of 2-1-5-4-3-7-6-8, so the 9x9 magic square is not only panmagic but symmetric as well (and that is the maximum result you can get with shift method (1) ).

- Shift the first row of the first grid each time two places to the right to get the second up to the ninth row of the first grid.

- Shift the first row of the second grid each time two places to the left to get the second up to the ninth row of the second grid.

Take 1x number from the first grid +1

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

+ 9x number from the second grid =

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

= Panmagic and symmetric 9x9 square

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

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

See

9x9, shift method 1.xls