Bimagic 8x8 square
A bimagic square is a simple magic square, but it is also a simple magic square if you fill in the squares of the numbers (= number x number) in each cell. The 8x8 bimagic square is the smallest bimagic square.
|
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 have constructed a new 8x8 bimagic square by swapping 0 an 1 in the binary grids. I have constructed the inverse 8x8 bimagic square.
Original 8x8 bimagic square Inverse 8x8 bimagic square
|
1x digit |
1x digit |
||||||||||||||||
|
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 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 |
||
See that the new bimagic square is valid.
|
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 |
Notify that an inverse magic square has always the same magic features as the original even if it is a multimagic, concentric or inlaid magic square.
See how to use 6 binary grids to construct 108 different bimagic 8x8
squares on website:
http://www.magichypercubes.com/Encyclopedia/DataBase/BiPanSquares_Order08.html
See many more [binary grids of] bimagic 8x8 squares on website:
http://www.magichypercubes.com/Encyclopedia/DataBase/Order08BiPandiagonal.html
