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Pan magic 4x4 square
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How to produce 4x4 pan magic squares
 

All 4x4 magic squares [= all 12 groups]


Excluding rotation and/or mirroring there are 880 different pure magic 4x4 squares. On the website of Harvey Heinz
(http://www.magicsquares.net/order4list.htm) the pure magic 4x4 squares are devided into 12 groups. Magic
squares in the same group have the same structure.



Group 1

Group 2

Group 3

Group 4

Group 5

Group 6

Group 7

Group 8

Group 9

Group 10

Group 11

Group 12


The lines are connecting the digits 1-16, 2-15, 3-14, 4-13, 5-12, 6-11, 7-10 and 8-9

(N.B.: 1+16 = 2+15 = 3+14 = 4+13 = 5+12 = 6+11 =7+10 = 8+9 = 17, that is half of the magic sum of 34).


See below an Excel download with all 880 x 8 is 7040 pure magic 4x4 squares for analysis on binary grids and Sudoku
grids.


Panmagic 4x4 squares [= group 1]

Of the 880 squares 48 are panmagic (= group 1). These panmagic squares have (as well as the bigger most perfect
magic squares) the following structure:


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


The sum of two digits of the same colour is each time (the lowest digit plus the highest digit of the magic square, in
this case 1+16=) 17.


You only need to know 3 panmagic squares to produce all (excluding rotation and/or mirroring) 48 panmagic 4x4
squares:

  
1
8
13
12
 
 
1
8
11
14
 
 
1
8
10
15
15
10
3
6
 
 
15
10
5
4
 
 
14
11
5
4
4
5
16
9
 
 
6
3
16
9
 
 
7
2
16
9
14
11
2
7
 
 
12
13
2
7
 
 
12
13
3
6
 
 
On the 2x2 carpet of one of the three 4x4 squares you can find 16 different 4x4 sub-squares. See the following
example of the third square:
 

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


 
Select a random 4x4 sub-square on the carpet (stay out of the gray area, because of double solutions). The
(for example yellow marked) selected 4x4 sub-square can be rotated and/or mirrored (see below for what
happens to the digits):
 
 
Selected 4x4 square
4
14
11
5
 
Mirroring
 
5
11
14
4
 
 
 
 
 
15
1
8
10
 
 
 
10
8
1
15
 
 
 
 
 
6
12
13
3
 
 
 
3
13
12
6
 
 
 
 
 
9
7
2
16
 
 
 
16
2
7
9
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
A quarter turn
9
6
15
4
 
Mirroring
 
4
15
6
9
 
 
 
 
 
7
12
1
14
 
 
 
14
1
12
7
 
 
 
 
 
2
13
8
11
 
 
 
11
8
13
2
 
 
 
 
 
16
3
10
5
 
 
 
5
10
3
16
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
2x quarter turn
16
2
7
9
 
Mirroring
 
9
7
2
16
 
 
 
 
 
3
13
12
6
 
 
 
6
12
13
3
 
 
 
 
 
10
8
1
15
 
 
 
15
1
8
10
 
 
 
 
 
5
11
14
4
 
 
 
4
14
11
5
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
3x quarter turn
5
10
3
16
 
Mirroring
 
16
3
10
5
 
 
 
 
 
11
8
13
2
 
 
 
2
13
8
11
 
 
 
 
 
14
1
12
7
 
 
 
7
12
1
14
 
 
 
 
 
4
15
6
9
 
 
 
9
6
15
4
 
 
There are 3 basic 4x4 panmagic squares. On the 2x2 carpet of one of the three 4x4 squares you can find 16
different 4x4 sub-squares (3x16=48). When you rotate and/or mirror one of the 48 different 4x4 squares, you
can find 8 different 4x4 squares. So 3 x 16 x 8 results in 384 different panmagic 4x4 squares (including rotation
and/or mirroring).


Transformation group 3 --> 1 --> 2

It is possible to transform a 4x4 magic square of group 3 in two steps into a 4x4 magic square of group 1 and
to transform a 4x4 magic square of group 1 in two steps into a 4x4 magic square of group 2. Swap each time
the yellow row with the red row and the yellow column with the red column.


4x4 magic square [3]        
                     
    34 34 34 34          
  34         34        
34   1 12 14 7          
34   15 6 4 9          
34   8 13 11 2          
34   10 3 5 16          
                     
                     
** first step [3] --> [1] **      
                     
    34 34 34 34          
  14         22        
34   1 12 14 7          
34   15 6 4 9          
34   10 3 5 16          
34   8 13 11 2          
                     
                     
4x4 magic square [1]        
                     
    34 34 34 34          
  34         34        
34   1 12 7 14          
34   15 6 9 4          
34   10 3 16 5          
34   8 13 2 11          
                     
                     
** First step [1] --> [2] **      
                     
    34 34 34 34          
  24         44        
34   1 7 12 14          
34   15 9 6 4          
34   10 16 3 5          
34   8 2 13 11          
                     
                     
4x4 magic square [2]        
                     
    34 34 34 34          
  34         34        
34   1 7 12 14          
34   10 16 3 5          
34   15 9 6 4          
34   8 2 13 11          



Reflected grids

The 4x4 magic squares of group 1, 2 and 3 have a (fully) diagonal symmetric structure (and the other 9
groups have not). Because of the structure it is possible to produce 4x4 magic squares of group 1, 2 and
3 by using reflected grids. Reflection means that the second grid is the same as the first grid, but it is turned
by a quarter and mirrored. See below a 4x4 magic square of group 1, 2 respectively 3, which is produced
by using reflected grids.


Take 1 x digit + 1   Take 1 x digit + 1   Take 1 x digit + 1  
                                   
0 1 2 3     0 0 3 3     0 3 1 2    
3 2 1 0     3 3 0 0     2 1 3 0    
1 0 3 2     1 1 2 2     3 0 2 1    
2 3 0 1     2 2 1 1     1 2 0 3    
                                   
                                   
+ 4 x digit =     + 4 x digit =     + 4 x digit =    
                                   
0 3 1 2     0 3 1 2     0 2 3 1    
1 2 0 3     0 3 1 2     3 1 0 2    
2 1 3 0     3 0 2 1     1 3 2 0    
3 0 2 1     3 0 2 1     2 0 1 3    
                                   
                                   
4x4 MS group 1     4x4 MS group 2     4x4 MS group 3    
                                   
1 14 7 12     1 13 8 12     1 12 14 7    
8 11 2 13     4 16 5 9     15 6 4 9    
10 5 16 3     14 2 11 7     8 13 11 2    
15 4 9 6     15 3 10 6     10 3 5 16    



The Following downloads for analysis and construction of 4x4 (pan)magic squares in EXCEL format are available:


More information on page:  
Pan magic 4x4 square, explanation



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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|Extra magic 12x12 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|18x18 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