Shift method (2)

 

Put in the first row of the first grid the numbers 0 up to 8. Construct the second and the third row of the first grid by shifting the first row each time 3 places to the left.

 

 

                                  First grid, first three rows

           

0

1

2

3

4

5

6

7

8

0

1

2

3

4

5

6

7

8

     

0

1

2

3

4

5

6

7

8

0

1

2

3

4

5

6

7

8

     

0

1

2

3

4

5

6

7

8

0

1

2

3

4

5

6

7

8

           

 

 

The first three rows of the first grid consists of three 3x3 sub-squares. Construct row 4 up to 6 by swapping the sequence of the three columns in the three 3x3 sub-squares into 2-3-1 (instead of 1-2-3). Construct row 7 up to 9 by swapping the sequence of the three columns in the three 3x3 sub-squares into 3-1-2 (instead of 1-2-3).

 

 

 First grid

0

1

2

3

4

5

6

7

8

3

4

5

6

7

8

0

1

2

6

7

8

0

1

2

3

4

5

1

2

0

4

5

3

7

8

6

4

5

3

7

8

6

1

2

0

7

8

6

1

2

0

4

5

3

2

0

1

5

3

4

8

6

7

5

3

4

8

6

7

2

0

1

8

6

7

2

0

1

5

3

4

 

 

The second grid is a reflection (rotated by a quarter) of the first grid. Take 1x number from first grid +1 and add 9x number from the same cell of the second grid.

 

 

 1x number                                  9x number                                  panmagic 9x9 square

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

 

 

This 9x9 magic square is panmagic and 3x3 compact (but not symmetric).


You can use this method to construct magic squares of order is odd square (= 9, 25, 47, 81, ...).

 

This method gives not many possibilities (try it yourself).

 

Download
9x9, Shiftmethod 2.xls
Microsoft Excel werkblad 52.5 KB

 

It is even possible to get a symmetric result.

 

 

1x number +1                           +  9x number                                 ultra magic 9x9 square

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

 

 

This 9x9 magic square is panmagic, 3x3 compact and symmetric, so it is ultra magic.

 

Download
9x9, Shiftmethod 2a.xlsx
Microsoft Excel werkblad 18.9 KB