### Reversible magic square

Paul Michelet constructed a 9x9 reversible panmagic (and 3x3 compact & symmetric) square.

In the square are the numbers 11/19, 21/29, 31/39, 41/49, 51/59, 61/69, 71/79, 81/89 and 91/99.

If you reverse the digits of all numbers in the 9x9 panmagic square (i.e. 19 reversed is 91) you get another 9x9 panmagic (and 3x3 compact & symmetric) square.

9x9 reversible panmagic (and 3x3 compact & symmetric) square

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

Reversed 9x9 panmagic (and 3x3 compact & symmetric) square

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

Reversible 9x9 magic square.xls

It is possible to construct a reversible 9x9 magic square, which is not only panmagic, 3x3 compact and symmetric, but also 1/3 of each row and 1/3 of each column gives 1/3 of the magic sum.

Take 1x number from first grid

 1 5 9 6 7 2 8 3 4 8 3 4 1 5 9 6 7 2 6 7 2 8 3 4 1 5 9 1 5 9 6 7 2 8 3 4 8 3 4 1 5 9 6 7 2 6 7 2 8 3 4 1 5 9 1 5 9 6 7 2 8 3 4 8 3 4 1 5 9 6 7 2 6 7 2 8 3 4 1 5 9

+ 10x number from second grid

 1 8 6 1 8 6 1 8 6 5 3 7 5 3 7 5 3 7 9 4 2 9 4 2 9 4 2 6 1 8 6 1 8 6 1 8 7 5 3 7 5 3 7 5 3 2 9 4 2 9 4 2 9 4 8 6 1 8 6 1 8 6 1 3 7 5 3 7 5 3 7 5 4 2 9 4 2 9 4 2 9

= Ultra magic 9x9 reversible square

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

If you reverse the digits of each number, you get:

Reversed ultra magic 9x9 square

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

Ultra magic reversible 9x9 square.xls