Symmetric transformation (Liki)
Paulus Gerdes introduced the Liki magic square (see http://plus.maths.org/content/new-designs-africa). He showed that it is possible to transform a square with consecutive numbers into a magic
square by swapping half of the numbers symmetrically. You can use this method to construct magic squares which are a multiple of 4 (= 4x4, 8x8, 12x12, 16x16, ... magic square).
The 4x4 magic square contains the numbers 1 up to 16. Each number has an inverse number. The inverse number is the largest number of the square plus 1 minus the original number.
See below the results:
1 --> Inverse number is
16 + 1 -/- 1 = 16
2 --> Inverse number is 16 + 1 -/- 2
= 15
3 -->
Inverse number is 16 + 1 -/- 3 = 14
4 --> Inverse number is 16 + 1 -/-
4 = 13
5
--> Inverse number is 16 + 1 -/- 5 = 12
6 --> Inverse number is 16 + 1
-/- 6 = 11
7 --> Inverse number is 16 + 1 -/- 7
= 10
8 --> Inverse number is 16 + 1 -/- 8 = 9
9 --> Inverse number is 16 + 1 -/-
9 = 8
10 --> Inverse number is 16 + 1 -/- 10 =
7
11 --> Inverse number is 16 + 1 -/- 11 = 6
12 --> Inverse number is 16 + 1 -/- 12 =
5
13 --> Inverse number is 16 + 1 -/- 13 = 4
14 --> Inverse number is 16 + 1 -/- 14 =
3
15 --> Inverse number is 16 + 1 -/- 15 = 2
16 --> Inverse number is 16 + 1 -/- 16 =
1
Transform the 4x4 square with consecutive numbers in a symmetric 4x4 magic square:
4x4 square with
consecutive numbers
|
28 |
32 |
36 |
40 |
|||
|
34 |
34 |
|||||
|
10 |
1 |
2 |
3 |
4 |
||
|
26 |
5 |
6 |
7 |
8 |
||
|
42 |
9 |
10 |
11 |
12 |
||
|
58 |
13 |
14 |
15 |
16 |
Note that the square with consecutive numbers is already symmetric and the (main) diagonals give already the magic sum of 34.
Symmetric 4x4 magic square (1)
|
34 |
34 |
34 |
34 |
|||
|
34 |
34 |
|||||
|
34 |
1 |
15 |
14 |
4 |
||
|
34 |
12 |
6 |
7 |
9 |
||
|
34 |
8 |
10 |
11 |
5 |
||
|
34 |
13 |
3 |
2 |
16 |
ór
Symmetric 4x4 magic square (2)
|
34 |
34 |
34 |
34 |
|||
|
34 |
34 |
|||||
|
34 |
16 |
2 |
3 |
13 |
||
|
34 |
5 |
11 |
10 |
8 |
||
|
34 |
9 |
7 |
6 |
12 |
||
|
34 |
4 |
14 |
15 |
1 |
Use this method to construct magic squares of order is multiple of 4 from 4x4 to infinity. See 4x4, 8x8, 12x12, 16x16, 20x20, 24x24, 28x28, 32x32
Take one step extra and you can transform a 4x4 square with consecutive numbers into the famous square of Albrecht Dürer:
Dürer’s magic square:
|
28 |
32 |
36 |
40 |
28 |
36 |
32 |
40 |
34 |
34 |
34 |
34 |
|||||||||||||
|
34 |
34 |
34 |
34 |
34 |
34 |
|||||||||||||||||||
|
10 |
1 |
2 |
3 |
4 |
10 |
1 |
3 |
2 |
4 |
34 |
16 |
3 |
2 |
13 |
||||||||||
|
26 |
5 |
6 |
7 |
8 |
26 |
5 |
7 |
6 |
8 |
34 |
5 |
10 |
11 |
8 |
||||||||||
|
42 |
9 |
10 |
11 |
12 |
42 |
9 |
11 |
10 |
12 |
34 |
9 |
6 |
7 |
12 |
||||||||||
|
58 |
13 |
14 |
15 |
16 |
58 |
13 |
15 |
14 |
16 |
34 |
4 |
15 |
14 |
1 |