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Trick with 8x8 bimagic square
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How to do the trick with the bimagic square
 
If replacing each digit by its square in a magic square produces another magic square, the square is
said to be a bimagic square (see for example the smallest possible [= 8x8] bimagic square from the
book of Arno van den Essen).
 
 
 
 
260
260
260
260
260
260
260
260
 
 
 
 
11180
11180
11180
11180
11180
11180
11180
11180
 
 
260
 
 
 
 
 
 
 
 
260
 
 
11180
 
 
 
 
 
 
 
 
11180
260
 
56
34
8
57
18
47
9
31
 
 
11180
 
3136
1156
64
3249
324
2209
81
961
 
260
 
33
20
54
48
7
29
59
10
 
 
11180
 
1089
400
2916
2304
49
841
3481
100
 
260
 
26
43
13
23
64
38
4
49
 
 
11180
 
676
1849
169
529
4096
1444
16
2401
 
260
 
19
5
35
30
53
12
46
60
 
 
11180
 
361
25
1225
900
2809
144
2116
3600
 
260
 
15
25
63
2
41
24
50
40
 
 
11180
 
225
625
3969
4
1681
576
2500
1600
 
260
 
6
55
17
11
36
58
32
45
 
 
11180
 
36
3025
289
121
1296
3364
1024
2025
 
260
 
61
16
42
52
27
1
39
22
 
 
11180
 
3721
256
1764
2704
729
1
1521
484
 
260
 
44
62
28
37
14
51
21
3
 
 
11180
 
1936
3844
784
1369
196
2601
441
9
 
 
 
I tried to discover a method to produce bimagic squares, but I have failed. But I have discovered a trick
to transform a bimagic 8x8 square in another (= not rotated and/or mirrored version of the) bimagic
8x8 square.
 
The trick is to split up the bimagic 8x8 square in binary patterns (see page ‘panmagic 4x4 square, binary’
or page ‘8x8 most perfect magic, binary’) and swap the digits 0 and 1 in all binary patterns.
 
 
Original patterns bimagic 8x8 square                                 Digits 0 and 1 swapped
1xdigit
 
 
 
 
 
 
 
 
1xdigit
 
 
 
 
 
 
0
0
0
1
0
1
1
1
 
 
1
1
1
0
1
0
0
0
1
0
0
0
1
1
1
0
 
 
0
1
1
1
0
0
0
1
0
1
1
1
0
0
0
1
 
 
1
0
0
0
1
1
1
0
1
1
1
0
1
0
0
0
 
 
0
0
0
1
0
1
1
1
1
1
1
0
1
0
0
0
 
 
0
0
0
1
0
1
1
1
0
1
1
1
0
0
0
1
 
 
1
0
0
0
1
1
1
0
1
0
0
0
1
1
1
0
 
 
0
1
1
1
0
0
0
1
0
0
0
1
0
1
1
1
 
 
1
1
1
0
1
0
0
0
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
+ 2x digit
 
 
 
 
 
 
 
+ 2x digit
 
 
 
 
 
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
1
0
1
0
0
1
0
1
 
 
0
1
0
1
1
0
1
0
0
1
0
1
1
0
1
0
 
 
1
0
1
0
0
1
0
1
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
1
0
1
0
0
1
0
1
 
 
0
1
0
1
1
0
1
0
0
1
0
1
1
0
1
0
 
 
1
0
1
0
0
1
0
1
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
+ 4x digit
 
 
 
 
 
 
 
+ 4x digit
 
 
 
 
 
0
1
0
1
1
0
1
0
 
 
1
0
1
0
0
1
0
1
1
1
0
0
0
0
1
1
 
 
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
0
1
0
0
1
0
1
 
 
0
1
0
1
1
0
1
0
0
1
0
1
1
0
1
0
 
 
1
0
1
0
0
1
0
1
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
1
0
1
0
0
1
0
1
 
 
0
1
0
1
1
0
1
0
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
+ 8x digit
 
 
 
 
 
 
 
+ 8x digit
 
 
 
 
 
1
1
1
0
1
0
0
0
 
 
0
0
0
1
0
1
1
1
1
1
1
0
1
0
0
0
 
 
0
0
0
1
0
1
1
1
0
0
0
1
0
1
1
1
 
 
1
1
1
0
1
0
0
0
1
1
1
0
1
0
0
0
 
 
0
0
0
1
0
1
1
1
0
0
0
1
0
1
1
1
 
 
1
1
1
0
1
0
0
0
1
1
1
0
1
0
0
0
 
 
0
0
0
1
0
1
1
1
0
0
0
1
0
1
1
1
 
 
1
1
1
0
1
0
0
0
0
0
0
1
0
1
1
1
 
 
1
1
1
0
1
0
0
0
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
+ 16x digit
 
 
 
 
 
 
 
+ 16x digit
 
 
 
 
 
0
1
1
0
0
1
1
0
 
 
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
 
 
0
1
1
0
0
1
1
0
0
1
1
0
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
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
0
1
1
0
0
1
1
0
0
1
1
0
 
 
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
 
 
0
1
1
0
0
1
1
0
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
+ 32x digit + 1
 
 
 
 
 
 
+ 32x digit + 1
 
 
 
 
0
0
1
0
1
0
1
1
 
 
1
1
0
1
0
1
0
0
0
1
0
0
1
1
0
1
 
 
1
0
1
1
0
0
1
0
1
0
1
1
0
0
1
0
 
 
0
1
0
0
1
1
0
1
1
1
0
1
0
1
0
0
 
 
0
0
1
0
1
0
1
1
1
1
0
1
0
1
0
0
 
 
0
0
1
0
1
0
1
1
1
0
1
1
0
0
1
0
 
 
0
1
0
0
1
1
0
1
0
1
0
0
1
1
0
1
 
 
1
0
1
1
0
0
1
0
0
0
1
0
1
0
1
1
 
 
1
1
0
1
0
1
0
0
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Bimagic 8x8 square of ‘van den Essen’
 
Bimagic 8x8 square new
 
9
31
57
8
47
18
56
34
 
 
56
34
8
57
18
47
9
31
32
45
11
17
58
36
6
55
 
 
33
20
54
48
7
29
59
10
39
22
52
42
1
27
61
16
 
 
26
43
13
23
64
38
4
49
46
60
30
35
12
53
19
5
 
 
19
5
35
30
53
12
46
60
50
40
2
63
24
41
15
25
 
 
15
25
63
2
41
24
50
40
59
10
48
54
29
7
33
20
 
 
6
55
17
11
36
58
32
45
4
49
23
13
38
64
26
43
 
 
61
16
42
52
27
1
39
22
21
3
37
28
51
14
44
62
 
 
44
62
28
37
14
51
21
3
 
 
I have checked that the new bimagic square is right.
 
 
 
 
260
260
260
260
260
260
260
260
 
 
 
 
11180
11180
11180
11180
11180
11180
11180
11180
 
 
260
 
 
 
 
 
 
 
 
260
 
 
11180
 
 
 
 
 
 
 
 
11180
260
 
9
31
57
8
47
18
56
34
 
 
11180
 
81
961
3249
64
2209
324
3136
1156
 
260
 
32
45
11
17
58
36
6
55
 
 
11180
 
1024
2025
121
289
3364
1296
36
3025
 
260
 
39
22
52
42
1
27
61
16
 
 
11180
 
1521
484
2704
1764
1
729
3721
256
 
260
 
46
60
30
35
12
53
19
5
 
 
11180
 
2116
3600
900
1225
144
2809
361
25
 
260
 
50
40
2
63
24
41
15
25
 
 
11180
 
2500
1600
4
3969
576
1681
225
625
 
260
 
59
10
48
54
29
7
33
20
 
 
11180
 
3481
100
2304
2916
841
49
1089
400
 
260
 
4
49
23
13
38
64
26
43
 
 
11180
 
16
2401
529
169
1444
4096
676
1849
 
260
 
21
3
37
28
51
14
44
62
 
 
11180
 
441
9
1369
784
2601
196
1936
3844
 
 
 
This is no coincidence. Does someone know the mathematical clue of the above mentioned trick.
Please visit my guest book and contact me (see page ‘contact’).


N.B.: If you swap 0 and 1 in the binary patterns of a magic square, you swap the highest digit with
the lowest digit, the second highest digit with the second lowest digit, etcetera. And ... the 'inverse'
magic square has the same magic features as the original (even if the original is a panmagic, most
perfect, bimagic, trimagic, concentric, bordered, or inlaid magic square)!!!



See how to use 6 binary grids to produce 108 different bimagic 8x8 squares:

http://www.magichypercubes.com/Encyclopedia/DataBase/BiPanSquares_Order08.html



See for more [binary grids of] bimagic 8x8 squares:

http://www.magichypercubes.com/Encyclopedia/DataBase/Order08BiPandiagonal.html


Download for analysis and construction:  trick with 8x8 bimagic square



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Magic squares and 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|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|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|Most magic cube per order|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|Perfect (Nasik) 16x16x16, step 1&2|Perfect (Nasik) 16x16x16, step 3&4|Perfect (Nasik) 16x16x16, result|Favorite Links