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 digits (= digit x digit), 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 digits 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 digit +1                                               1x digit +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 digit                                                 + 3x digit

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 digit                                                 + 9x digit

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 digit                                               + 27x digit

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 digits 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.

 

 

Download
9x9, Bimagic 9x9 square.xls
Microsoft Excel werkblad 194.5 KB