### 9x9 bimagic square

See On the website of Harvey Heinz a bimagic 9x9 square of John Hendricks:

http://www.magic-squares.net/multimagic.htm

A bimagic square is a simple magic square, but also if you fill in the squares of the numbers (= number x number), you get a valid (impure) magic square.

The bimagic 9x9 square of John Hendricks consist of 4 regular ternary grids. You can put the grids in random order and you can swap the numbers 0, 1 and 2 in each grid to get more possibilities:

 0 2 0 0 0 1 0 2 1 0 1 2 1 2 1 1 2 1 2 1 2 0 2 0

 1x number +1                                        1x number +1 0 2 1 2 1 0 1 0 2 0 0 0 1 1 1 2 2 2 1 0 2 0 2 1 2 1 0 2 2 2 0 0 0 1 1 1 2 1 0 1 0 2 0 2 1 1 1 1 2 2 2 0 0 0 0 2 1 2 1 0 1 0 2 1 1 1 2 2 2 0 0 0 1 0 2 0 2 1 2 1 0 0 0 0 1 1 1 2 2 2 2 1 0 1 0 2 0 2 1 2 2 2 0 0 0 1 1 1 0 2 1 2 1 0 1 0 2 2 2 2 0 0 0 1 1 1 1 0 2 0 2 1 2 1 0 1 1 1 2 2 2 0 0 0 2 1 0 1 0 2 0 2 1 0 0 0 1 1 1 2 2 2 + 3x number                                          + 3x number 2 1 0 0 2 1 1 0 2 0 1 2 1 2 0 2 0 1 2 1 0 0 2 1 1 0 2 2 0 1 0 1 2 1 2 0 2 1 0 0 2 1 1 0 2 1 2 0 2 0 1 0 1 2 1 0 2 2 1 0 0 2 1 0 1 2 1 2 0 2 0 1 1 0 2 2 1 0 0 2 1 2 0 1 0 1 2 1 2 0 1 0 2 2 1 0 0 2 1 1 2 0 2 0 1 0 1 2 0 2 1 1 0 2 2 1 0 0 1 2 1 2 0 2 0 1 0 2 1 1 0 2 2 1 0 2 0 1 0 1 2 1 2 0 0 2 1 1 0 2 2 1 0 1 2 0 2 0 1 0 1 2 + 9x number                                           + 9x number 1 2 0 1 2 0 1 2 0 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 1 2 0 1 2 0 1 2 0 2 0 1 2 0 1 2 0 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 2 0 1 2 0 1 2 0 1 0 1 2 0 1 2 0 1 2 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 0 1 2 0 1 2 0 1 2 + 27x number                                         + 27x number 1 1 1 2 2 2 0 0 0 0 1 2 2 0 1 1 2 0 0 0 0 1 1 1 2 2 2 0 1 2 2 0 1 1 2 0 2 2 2 0 0 0 1 1 1 0 1 2 2 0 1 1 2 0 2 2 2 0 0 0 1 1 1 1 2 0 0 1 2 2 0 1 1 1 1 2 2 2 0 0 0 1 2 0 0 1 2 2 0 1 0 0 0 1 1 1 2 2 2 1 2 0 0 1 2 2 0 1 0 0 0 1 1 1 2 2 2 2 0 1 1 2 0 0 1 2 2 2 2 0 0 0 1 1 1 2 0 1 1 2 0 0 1 2 1 1 1 2 2 2 0 0 0 2 0 1 1 2 0 0 1 2 = bimagic 9x9 square                             = bimagic 9x9 square 43 51 29 66 80 58 14 19 9 19 31 70 77 8 38 54 57 15 26 4 12 46 36 41 78 56 70 9 39 78 55 13 52 32 71 20 63 68 73 2 16 24 31 39 53 14 53 56 72 21 33 37 76 7 76 57 71 27 5 10 47 34 42 29 68 26 6 45 75 61 10 49 32 37 54 61 69 74 3 17 22 43 73 4 11 50 62 69 27 30 15 20 7 44 49 30 64 81 59 51 63 12 25 28 67 74 5 44 1 18 23 33 38 52 62 67 75 66 24 36 40 79 1 17 47 59 65 79 60 13 21 8 45 50 28 80 2 41 48 60 18 22 34 64 48 35 40 77 55 72 25 6 11 58 16 46 35 65 23 3 42 81

Putting the grids in random order gives (4 x 3 x 2 x 1 = ) 24 possibilities. Swapping the numbers 0, 1 and 2 within each grid gives ([3x2x1]x[3x2x1]x[3x2x1]x[3x2x1] = ) 1296 possibilities. In total you can construct (24 x 1296 = ) 31104 different 9x9 bimagic squares.

9x9, Bimagic 9x9 square.xls