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How to make perfect magic squares, (hyper)cubes, etcetera
The sky is the limit!!!
6x6 magic square
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How to produce 6x6 magic squares
  
Panmagic (or extra magic) 6x6 squares do not exist. The structure of 6x6 squares is irregular. To
produce pure 6x6 squares, you need to puzzle. The funniest method is the medjig method of Willem
Barink (source: Wikipedia, Dutch language version). The fastest method, is the method of Strachey.
The most simple method is Paul Michelet's trick.

 

Medjig method

Take as 1st square a pure magic 3x3 square, but ‘blow it up’ by presenting the digits 2x2. Produce a
2nd square existing of 9 (2x2) medjig tiles.
 
Note that an 2x2 medjig tile consists of all the digits 0, 1, 2 en 3, but not always in the same sequence;
you must choose the tiles, taking care that the sum of the digits of each row/column/diagonal is (6 x
1,5 =) 9. Take (1x) a digit from the 1st square and add 9x a digit from the same cell of the 2nd square.
 
 
 1x digit                         + 9x digit                =   pure magic 6x6 square
2
2
9
9
4
4
 
2
3
0
2
0
2
 
20
29
9
27
4
22
2
2
9
9
4
4
 
1
0
3
1
3
1
 
11
2
36
18
31
13
7
7
5
5
3
3
 
3
1
1
2
2
0
 
34
16
14
23
21
3
7
7
5
5
3
3
 
0
2
0
3
3
1
 
7
25
5
32
30
12
6
6
1
1
8
8
 
3
2
2
0
0
2
 
33
24
19
1
8
26
6
6
1
1
8
8
 
0
1
3
1
1
3
 
6
15
28
10
17
35
 
 
You can use this method to produce 'double odd' (= 6x6, 10x10, 14x14, 18x18, …) as well as 'double even'
(= 8x8, 12x12, 16x16, 20x20, ....) magic squares. For example to produce a 10x10 square use as 1st square
a pure (pan)magic 5x5 square, use a 2nd square existing of 25 (2x2) medjig tiles, taking care that the sum
of the digits of each row/column/
diagonal is (10 x 1,5 =) 15 and add 15x a digit from the 2nd square.

N.B.: If you don't like to puzzle, use the Medjig method without puzzling (= LUX method);
see page
'14x14 magic square'.



Method of Strachey

Instead of the mejig method you can use the method of Strachey. Make a 2x2 carpet of a
magic 3x3 square and add 9x the digit from a fixed pattern.
 
 
 1x digit                               +     9x digit                              =       magic 6x6 square
2
9
4
2
9
4
 
 
0
0
3
2
2
2
 
 
2
9
31
20
27
22
7
5
3
7
5
3
 
 
0
3
0
2
2
2
 
 
7
32
3
25
23
21
6
1
8
6
1
8
 
 
0
0
3
2
2
2
 
 
6
1
35
24
19
26
2
9
4
2
9
4
 
 
3
3
0
1
1
1
 
 
29
36
4
11
18
13
7
5
3
7
5
3
 
 
3
0
3
1
1
1
 
 
34
5
30
16
14
12
6
1
8
6
1
8
 
 
3
3
0
1
1
1
 
 
33
28
8
15
10
17
 
 
To use the method of Strachey to produce bigger ‘double odd’ magic squares (= 10x10, 14x14,
18x18, …) it is a disadvantage that you need to swap more and more digits to get a correct magic
square. See the improved method of Strachey to produce a
10x10 magic square (you need to swap
less digits).

N.B.: It is possible to produce a
18x18 magic square with an extra magic feature, because 1/3 row/
column/diagonal gives 1/3 of the magic sum!!!



Paul Michelet's trick

With Paul Michelet's trick you can make a 6x6 magic square as follows:


 Fill in 2x2 Luo shu            add 9x digit                      swap the marked digits         Paul's 6x6 magic square
8
8
1
1
6
6
 
 
0
3
0
3
0
3
 
 
8
35
1
28
6
33
 
 
8
35
1
28
6
33
8
8
1
1
6
6
 
 
2
1
2
1
2
1
 
 
26
17
19
10
24
15
 
 
26
17
19
10
24
15
3
3
5
5
7
7
 
 
0
3
0
3
0
3
 
 
3
30
5
32
7
34
 
 
3
30
32
5
7
34
3
3
5
5
7
7
 
 
2
1
2
1
2
1
 
 
21
12
23
14
25
16
 
 
21
12
23
14
25
16
4
4
9
9
2
2
 
 
0
3
0
3
0
3
 
 
4
31
9
36
2
29
 
 
31
4
9
36
29
2
4
4
9
9
2
2
 
 
2
1
2
1
2
1
 
 
22
13
27
18
20
11
 
 
22
13
27
18
20
11


N.B.: This trick was probably known almost thousand years ago. Look at Yang Hui's 6x6 magic square.

http://www.gap-system.org/~history/Biographies/Yang_Hui.html


Download for construction in EXCEL:  6x6 magic square


Notify that there is an other alternative method of construction: 
Bordered squares (4x4 inside 6x6).



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Magic squares and (hyper)cubes|Contact / guestbook|Most magic square per order|3x3 magic square|3x3 magic square, explanation|Sudoku method (1)|Sudoku method (2)|Sudoku method (3)|Pan magic 4x4 square|Pan magic 4x4 square, explanation|Pan magic 4x4 square, binary|Symmetric transformation|Dürer & Franklin transformation|Transformation method|Transformation method, analysis|[ultra] pan magic 5x5 square|Pan magic 5x5 square, explanation|6x6 magic square|Ultra (pan)magic 8x8 square|Most perfect magic squares, explanation|8x8 most perfect magic squares, binary|Khajuraho method|Khajuraho method, explanation|Basic pattern method (1a)|Basic pattern method (1b)|Basic pattern method (2)|Basic pattern method (3a)|Basic pattern method (3b)|Basic pattern method (3c)|Basic pattern method (4)|Basic pattern method (5)|Basic pattern method (6)|Basic pattern method (7a)|Basic pattern method (7b)|Analysis Franklin panm. 8x8 (1)|Analysis Franklin panm. 8x8 (2)|Basic key method (1)|Basic key method (2)|Quadrant method (Willem Barink)|Quadrant method group 1 up to 5|Quadrant method group 6 up to 10|Quadrant method group 11 up to 19|[ultra] pan magic 9x9 square (1)|pan magic 9x9 square (2)|pan magic 9x9 square (3)|3x extra magic 9x9 square|bimagic 9x9 square|10x10 magic square|Composite 12x12 magic square|14x14 magic square|[Ultra] pan magic 15x15 square|3x extra magic 15x15 square|Composite bimagic 16x16 square|Composite bimagic 32x32 square|The perfect magic square|3x extra magic 18x18 square|Composite 24x24 magic square|Ultra pan magic 25x25 square (1)|Ultra pan magic 25x25 square (2)|Ultra bimagic 25x25 square|[ultra] pan magic 27x27 square|[ultra] pan magic 35x35 square|extra magic 35x35 square|Bordered squares|Inlaid square (1)|Inlaid square (2)|Each magic sum|Water retention challenge|Composite concentric 44x44 square|Most magic cube per order|Most magic 3x3x3 cube|Most magic 4x4x4 cube|symmetric & semi (pan)magic 5x5x5 cube|Symmetric & panmagic 7x7x7 cube|Perfect (Nasik) & compact 8x8x8 cube|Franklin panmagic 16x16x16 cube (1)|Franklin panmagic 16x16x16 cube (2)|Franklin panmagic 16x16x16 cube (3)|[More than] perfect magic 9x9x9 cube|Perfect (Nasik) magic 11x11x11 cube|Perfect (Nasik) magic 15x15x15 cube|Trick with 8x8 bimagic square|Ultra magic 25x25x25 cube|Bimagic 25x25x25 cube|Hyper cube, explanation|3x3x3x3 (hyper) cube|4x4x4x4 cube, panmagic in levels|4x4x4x4 diagonal cube (1)|4x4x4x4 diagonal cube (2)|4x4x4x4 pantriagonal & panquadragonal cube (1)|4x4x4x4 pantriagonal & panquadragonal cube (2)|5x5x5x5 cube, panmagic in levels|5x5x5x5 pantriagonal cube|8x8x8x8 pandiagonal and pantriagonal cube|8x8x8x8 diagonal and panquadragonal cube|8x8x8x8 pantriagonal and panquadragonal cube|9x9x9x9 ultra magic hypercube|16x16x16x16 perfect magic cube|16x16x16x16 most perfect magic cube (1)|16x16x16x16 most perfect magic cube (2)|3x3x3x3x3 magic hypercube|4x4x4x4x4 magic hypercube|5x5x5x5x5 magic hypercube|3x3x3x3x3x3 magic hypercube (1)|3x3x3x3x3x3 magic hypercube (2)|4x4x4x4x4x4 magic hypercube|Magic thema pages (explanation)|Anti-magic squares|Magic prime squares|IXOHOXI magic squares|Domino magic squares|Multiplication magic squares|Magic 3/5/6/7-angle|Magic hexagon|Magic stars and circles|Alfa-magic squares|Geomagic squares|Magic knight tour|Magic ABC|Favorite Links