Office Applications and Entertainment, Magic Squares

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8.3   Medjig Solutions (Main Diagonals, Integers 0, 1, 2 and 3)

8.3.1 General

As described in section 6.8, for any integer n, a magic square C of order 2n can be constructed from any n x n medjig-square A with each row, column, and main diagonal summing to 3n, and any n x n magic square B, by application of the equations:

bi + n2 aj with i = 1, 2, ... n2 and j = 1, 2, ... 4n2.

The Medjig method of constructing a Magic Square of order 8 is as follows:

Construct a 4 x 4 Medjig-Square A (ignoring the original game's limit on the number of times that a given
sequence is used);

Construct a 4 x 4 Magic Square B (7040 possibilities);

Construct a 8 x 8 Magic Square C by applying the equations mentioned above.

B (4 x 4)

b1

b2

b3

b4

b5

b6

b7

b8

b9

b10

b11

b12

b13

b14

b15

b16

Medjig Square A (4 x 4)

a1

a2

a3

a4

a5

a6

a7

a8

a9

a10

a11

a12

a13

a14

a15

a16

a17

a18

a19

a20

a21

a22

a23

a24

a25

a26

a27

a28

a29

a30

a31

a32

a33

a34

a35

a36

a37

a38

a39

a40

a41

a42

a43

a44

a45

a46

a47

a48

a49

a50

a51

a52

a53

a54

a55

a56

a57

a58

a59

a60

a61

a62

a63

a64

Magic Square C (8 x 8)

b1+16*a1

b1+16*a2

b2+16*a3

b2+16*a4

b3+16*a5

b3+16*a6

b4+16*a7

b4+16*a8

b1+16*a9

b1+16*a10

b2+16*a11

b2+16*a12

b3+16*a13

b3+16*a14

b4+16*a15

b4+16*a16

b5+16*a17

b5+16*a18

b6+16*a19

b6+16*a20

b7+16*a21

b7+16*a22

b8+16*a23

b8+16*a24

b5+16*a25

b5+16*a26

b6+16*a27

b6+16*a28

b7+16*a29

b7+16*a30

b8+16*a31

b8+16*a32

b9+16*a33

b9+16*a34

b10+16*a35

b10+16*a36

b11+16*a37

b11+16*a38

b12+16*a39

b12+16*a40

b9+16*a41

b9+16*a42

b10+16*a43

b10+16*a44

b11+16*a45

b11+16*a46

b12+16*a47

b12+16*a48

b13+16*a49

b13+16*a50

b14+16*a51

b14+16*a52

b15+16*a53

b15+16*a54

b16+16*a55

b16+16*a56

b13+16*a57

b13+16*a58

b14+16*a59

b14+16*a60

b15+16*a61

b15+16*a62

b16+16*a63

b16+16*a64

The rows, columns and main diagonals of Square C sum to 2 times the corresponding sum of Magic Square B plus 16 times the corresponding sum of Medjig square A which results in s1 = 2 * 34 + 16 * 12 = 260.

As b(i) ≠ b(j) for i ≠ j with i, j = 1, 2, ... 16 it is obvious that also c(m) ≠ c(n) for n ≠ m with n, m = 1, 2, ... 64.

A numerical example is shown below:

B (4 x 4)

12

13

3

6

7

2

16

9

14

11

5

4

1

8

10

15

Medjig Square A (4 x 4)

1

3

3

0

0

1

1

3

0

2

2

1

3

2

0

2

0

1

2

3

0

3

3

0

3

2

1

0

2

1

2

1

3

0

1

3

2

1

2

0

2

1

0

2

0

3

3

1

2

0

3

2

2

0

0

3

1

3

0

1

3

1

1

2

Magic Square C (8 x 8)

28

60

61

13

3

19

22

54

12

44

45

29

51

35

6

38

7

23

34

50

16

64

57

9

55

39

18

2

48

32

41

25

62

14

27

59

37

21

36

4

46

30

11

43

5

53

52

20

33

1

56

40

42

10

15

63

17

49

8

24

58

26

31

47

8.3.2 Further Analysis

Medjig Squares are described by the same set of linear equations as shown in section 8.1.1, for Magic Squares, however with magic sum 12 and following additional equations:

a( 1) + a( 2) + a( 9) + a(10) = 6
a( 3) + a( 4) + a(11) + a(12) = 6
a( 5) + a( 6) + a(13) + a(14) = 6
a( 7) + a( 8) + a(15) + a(16) = 6

a(17) + a(18) + a(25) + a(26) = 6
a(19) + a(20) + a(27) + a(28) = 6
a(21) + a(22) + a(29) + a(30) = 6
a(23) + a(24) + a(31) + a(32) = 6

a(33) + a(34) + a(41) + a(42) = 6
a(35) + a(36) + a(43) + a(44) = 6
a(37) + a(38) + a(45) + a(46) = 6
a(39) + a(40) + a(47) + a(48) = 6

a(49) + a(50) + a(57) + a(58) = 6
a(51) + a(52) + a(59) + a(60) = 6
a(53) + a(54) + a(61) + a(62) = 6
a(55) + a(56) + a(63) + a(64) = 6

Resulting in the matrix representation:

            
     A * a = s

which can be reduced, by means of row and column manipulations, to the minimum number of linear equations:

a(57) = 12 - a(58) - a(59) - a(60) - a(61) - a(62) - a(63) - a(64)
a(55) =  6 - a(56) - a(63) - a(64)
a(53) =  6 - a(54) - a(61) - a(62)
a(51) =  6 - a(52) - a(59) - a(60)
a(49) =  6 - a(50) - a(57) - a(58)
a(41) = 12 - a(42) - a(43) - a(44) - a(45) - a(46) - a(47) - a(48)
a(39) =  6 - a(40) - a(47) - a(48)
a(37) =  6 - a(38) - a(45) - a(46)
a(35) =  6 - a(36) - a(43) - a(44)
a(33) =  6 - a(34) - a(41) - a(42)
a(25) = 12 - a(26) - a(27) - a(28) - a(29) - a(30) - a(31) - a(32) 
a(23) =  6 - a(24) - a(31) - a(32)
a(21) =  6 - a(22) - a(29) - a(30)
a(19) =  6 - a(20) - a(27) - a(28)
a(17) =  6 - a(18) - a(25) - a(26) 
a(15) = 18 + a(16) - a(22) + a(24) - a(29) + a(32) - a(35) - 2 * a(36) - a(39) - 2 * a(43) - a(44) - a(47) - a(50) +
           - a(55) - a(57) - a(63)
a(10) = 12 - (a(11) + a(12) + a(13) + a(14) + a(15) + a(16) + a(18) - a(20) + a(26) - a(27) + a(34) + (37) + a(42) + 
           + a(46) + a(50) + a(55) + a(58) + a(64))/2
a(9) =  12 - a(10) - a(11) - a(12) - a(13) - a(14) - a(15) - a(16) 
a(8) = -12 - a(16) - a(24) - a(32) + a(35) + a(36) + a(39) + a(43) + a(44) + a(47) + a(51) + a(52) + a(55) + a(59) + 
           + a(60) + a(63)
a(7) = -18 - a(16) + a(22) + a(29) + a(35) + 2*a(36) + 2*a(43) + a(44) + a(50) + a(51) + a(52) + a(57) + a(59) + a(60)
a(6) =  12 - a(14) - a(22) - a(30) - a(38) - a(46) - a(54) - a(62)
a(5) = - 6 - a(13) + a(22) + a(30) + a(38) + a(46) + a(54) + a(62)
a(4) =  12 - a(12) - a(20) - a(28) - a(36) - a(44) - a(52) - a(60)
a(3) =   6 - a(11) + a(20) + a(28) - a(35) - a(43) - a(51) - a(59)
a(2) =  12 - a(10) - a(18) - a(26) - a(34) - a(42) - a(50) - a(58)
a(1) = - 6 - a( 9) + a(18) + a(26) + a(34) + a(42) + a(50) + a(58) 

The linear equations shown above are ready to be solved, for the magic constant 12.

The solutions can be obtained by guessing:

     a(11), a(12), a(13), a(14), a(16), a(18), a(20), a(22), a(24), a(26) ... a(32), a(34), a(36), a(38),
     a(40), a(42) ....... a(48), a(50), a(52), a(54), a(56) and a(58) ... a(64)

and filling out these guesses in the abovementioned equations.

To obtain the integers 0, 1, 2 and 3 also following relations should be applied:

     0 =< a(i) =< 3        for i = 1, 2, ... 10, 15, 17, 19, 21, 23, 25, 33, 35, 37, 39, 41, 49, 51, 53, 55, 57
     Int(a(i)) = a(i)      for i = 10

which can be incorporated in a guessing routine, which can be used to generate a defined number of Medjig Squares within a reasonable time.

With the last 9 of the 16 Medjig pieces constant, an optimized guessing routine (MgcSqr8e), produced 1712 Medjig Squares within 72 seconds, which are shown in Attachment 8.3.1.

The resulting Magic Squares, based on the 4th order Magic Square shown in the numerical solution above are shown in Attachment 8.3.2.

It should be noted that, although much faster, not all Magic Squares of the 8th order can be found by means of the Medjig Solution. The Magic Square shown at the begin of Section 8.2 is a clear example.

8.3.3 Pan Magic Complete Squares

The Medjig method of constructing a Pan Magic Complete Square of order 8 is as follows:

Construct a   4 x 4 Pan Magic and Complete Medjig-Square A;
Construct a   4 x 4 Pan Magic Square B (ref. Attachment 1);
Construct the 8 x 8 Pan Magic Complete Square C by applying the equations mentioned in Section 8.3.1 above.

A numerical example is shown below:

B (4 x 4)

12 13 3 6
7 2 16 9
14 11 5 4
1 8 10 15

Medjig Square A (4 x 4)

3 2 1 0 3 2 1 0
1 0 3 2 1 0 3 2
2 3 0 1 2 3 0 1
0 1 2 3 0 1 2 3
0 1 2 3 0 1 2 3
2 3 0 1 2 3 0 1
1 0 3 2 1 0 3 2
3 2 1 0 3 2 1 0

Magic Square C (8 x 8)

60 44 29 13 51 35 22 6
28 12 61 45 19 3 54 38
39 55 2 18 48 64 9 25
7 23 34 50 16 32 41 57
14 30 43 59 5 21 36 52
46 62 11 27 37 53 4 20
17 1 56 40 26 10 63 47
49 33 24 8 58 42 31 15

As mentioned in Section 8.7.5, a collection of 640000 Quaternary Pan Magic Complete Squares could be generated with procedure Quat869, which can be considered as Medjig Squares.

All Medjig Squares of this collection are suitable for the construction of (Pan) Magic Squares.

This collection contains 1600 Medjig Squares suitable for the construction of Pan Magic Complete Squares, of which 48 are shown in in Attachment 8.3.5. The first occurring square is used in the example above.

The Pan Magic Complete Squares resulting from the Medjig Square shown above and the 384 possible 4th order Pan Magic Squares, are shown in Attachment 8.3.3.

It should be moted that although the Medjig Square A might be Associated as well, the resulting Square C will be Pan Magic Complete as the 4 x 4 Square B is Pan Magic Complete.

8.3.4 Associated Magic Squares

The Medjig method of constructing an Associated Magic Square of order 8 is as follows:

Construct a   4 x 4 Associated (Pan Magic) Medjig-Square A;
Construct a   4 x 4 Associated Magic Square B (ref. Attachment 2.5);
Construct the 8 x 8 Associated Magic Square C by applying the equations mentioned in Section 8.3.1 above.

A numerical example is shown below:

B (4 x 4)

16 9 5 4
3 6 10 15
2 7 11 14
13 12 8 1

Medjig Square A (4 x 4)

3 2 3 2 1 0 1 0
1 0 1 0 3 2 3 2
0 1 0 1 2 3 2 3
2 3 2 3 0 1 0 1
2 3 2 3 0 1 0 1
0 1 0 1 2 3 2 3
1 0 1 0 3 2 3 2
3 2 3 2 1 0 1 0

Magic Square C (8 x 8)

64 48 57 41 21 5 20 4
32 16 25 9 53 37 52 36
3 19 6 22 42 58 47 63
35 51 38 54 10 26 15 31
34 50 39 55 11 27 14 30
2 18 7 23 43 59 46 62
29 13 28 12 56 40 49 33
61 45 60 44 24 8 17 1

As mentioned in Section 8.7.5, a collection of 27752 Quaternary Associated Pan Magic Squares could be generated with procedure Quat867, which can be considered as Medjig Squares.

All Medjig Squares of this collection are suitable for the construction of (Associated) Magic Squares. The first occurring square is used in the example above.

The Associated Magic Squares resulting from the Medjig Square shown above and the 384 possible 4th order Associated Magic Squares, are shown in Attachment 8.3.4.

It should be moted that although the Medjig Square A might be Pan Magic Complete as well, the resulting Square C will be Associated Magic as the 4 x 4 Square B is Associated Magic.

8.3.5 Spreadsheet Solution

The linear equations shown in section 8.3.2 above can be applied in an Excel spreadsheet (Ref. CnstrSngl8e).

The red figures have to be “guessed” to construct a 4 x 4 Medjig Square (wrong solutions are obvious).


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