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How to make perfect magic squares & cubes
The sky is the limit!!!
Pan magic 4x4 square, binary
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How to produce a panmagic 4x4 square by using binary patterns?
 
You can split up the three panmagic 4x4 basic squares (see page ‘panmagic 4x4 square’) in binary patterns:
 
 
  1x digit                     +      2x digit                     +       4x digit                     +       8x digit +1               =      panmagic 4x4 square
0
1
0
1
 
 
0
1
0
1
 
 
0
1
1
0
 
 
0
0
1
1
 
 
1
8
13
12
0
1
0
1
 
 
1
0
1
0
 
 
1
0
0
1
 
 
1
1
0
0
 
 
15
10
3
6
1
0
1
0
 
 
1
0
1
0
 
 
0
1
1
0
 
 
0
0
1
1
 
 
4
5
16
9
1
0
1
0
 
 
0
1
0
1
 
 
1
0
0
1
 
 
1
1
0
0
 
 
14
11
2
7
 
 
  1x digit                     +      2x digit                     +       4x digit                     +       8x digit +1               =      panmagic 4x4 square
0
1
0
1
 
 
0
1
1
0
 
 
0
1
0
1
 
 
0
0
1
1
 
 
1
8
11
14
0
1
0
1
 
 
1
0
0
1
 
 
1
0
1
0
 
 
1
1
0
0
 
 
15
10
5
4
1
0
1
0
 
 
0
1
1
0
 
 
1
0
1
0
 
 
0
0
1
1
 
 
6
3
16
9
1
0
1
0
 
 
1
0
0
1
 
 
0
1
0
1
 
 
1
1
0
0
 
 
12
13
2
7
 
 
  1x digit                     +      2x digit                     +       4x digit                     +       8x digit +1               =      panmagic 4x4 square
0
1
1
0
 
 
0
1
0
1
 
 
0
1
0
1
 
 
0
0
1
1
 
 
1
8
10
15
1
0
0
1
 
 
0
1
0
1
 
 
1
0
1
0
 
 
1
1
0
0
 
 
14
11
5
4
0
1
1
0
 
 
1
0
1
0
 
 
1
0
1
0
 
 
0
0
1
1
 
 
7
2
16
9
1
0
0
1
 
 
1
0
1
0
 
 
0
1
0
1
 
 
1
1
0
0
 
 
12
13
3
6
 
 
 
There are the following 4x 2 binary patterns to produce panmagic 4x4 squares:
 
 
 
 H1a
 
 
 
 
 H1b
 
0
0
1
1
 
 
1
1
0
0
1
1
0
0
 
 
0
0
1
1
0
0
1
1
 
 
1
1
0
0
1
1
0
0
 
 
0
0
1
1
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 H2a
 
 
 
 
 H2b
 
0
1
1
0
 
 
1
0
0
1
1
0
0
1
 
 
0
1
1
0
0
1
1
0
 
 
1
0
0
1
1
0
0
1
 
 
0
1
1
0
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 V1a
 
 
 
 
 V1b
 
0
1
0
1
 
 
1
0
1
0
0
1
0
1
 
 
1
0
1
0
1
0
1
0
 
 
0
1
0
1
1
0
1
0
 
 
0
1
0
1
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 V2a
 
 
 
 
 V2b
 
0
1
0
1
 
 
1
0
1
0
1
0
1
0
 
 
0
1
0
1
1
0
1
0
 
 
0
1
0
1
0
1
0
1
 
 
1
0
1
0
 
 
 
Take the following 3 steps:
 
[1] Choose H1a or H1b and H2a or H2b and V1a or V1b and V2a or V2b (in the example below has been
chosen for H1b, H2b, V1b and V2a). There are 2x2x2x2 = 16 possibilities.
 
[2] Choose the sequence H1H2V1V2 or H1H2V2V1 or H1V1H2V2 or H1V1V2H2 or H1V2H2V1 or H1V2V1H2
or H2H1V1V2 or H2H1V2V1 or H2V1H1V2 or H2V1V2H1 or H2V2H1V1 or H2V2V1H1 or V1H1H2V2 or V1H1V2H2
or V1H2H1V2 or V1H2V2H1 or V1V2H1H2 or V1V2H2H1 or V2H1H2V1 or V2H1V1H2 or V2H2H1V1 or V2H2V1H1
or V2V1H1H2 or V2V1H2H1 (in the example below has been chosen for sequence H1H2V2V1). There are 24 possi-
bilities.
 
[3] Produce the panmagic 4x4 square:
 
 
  1x digit (H1b)          +        2x digit (H2b)          +       4x digit (V2a)           +       8x digit (V1b) +1      =      panmagic 4x4 square
1
1
0
0
 
 
1
0
0
1
 
 
0
1
0
1
 
 
1
0
1
0
 
 
12
6
9
7
0
0
1
1
 
 
0
1
1
0
 
 
1
0
1
0
 
 
1
0
1
0
 
 
13
3
16
2
1
1
0
0
 
 
1
0
0
1
 
 
1
0
1
0
 
 
0
1
0
1
 
 
8
10
5
11
0
0
1
1
 
 
0
1
1
0
 
 
0
1
0
1
 
 
0
1
0
1
 
 
1
15
4
14
 
 
Notify tha you can produce all 16 (see step 1) x 24 (see step 2) = 384 panmagic 4x4 squares (including rotating
and/or mirroring) !!!


You can also use binary patterns to produce most perfect 8x8 magic squares, see page '8x8 most perfect magic
squares, binary'




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Magic squares|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|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|10x10 magic square|Composite 12x12 magic square|14x14 magic square|[Ultra] pan magic 15x15 square|3x extra magic 15x15 square|The perfect magic square|3x extra magic 18x18 square|Ultra pan magic 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|Most magic 4x4x4 cube|symmetric & semi (pan)magic 5x5x5 cube|Symmetric & panmagic 7x7x7 cube|Perfect (Nasik) & compact 8x8x8 cube|[More than] perfect magic 9x9x9 cube|Perfect (Nasik) magic 11x11x11 cube|Perfect (Nasik) magic 15x15x15 cube|Trick with 8x8 bimagic square|Favorite Links