Office Applications and Entertainment, Magic Squares  
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15.0 Special Magic Squares, Bimagic Squares, Part 2
In his book Magische Quadrate (Version 19, 2017), Holger Danielsson provides a detailed description of classical construction methods of Bimagic Squares of order 8.
15.3.1 Achille Rilly
Rilly’s method (1901) is based on the application of Generators of 8 Bimagic Rows, built up out of two Half Generators of 4 Bimagic Rows.
The lower Half Generators for the last 4 rows are based on:
This reduces the applicable number of Bimagic Series to 136, which are shown in Attachment 15.3.01 (68 for the upper Half Generators and another 68 for the lower Half Generators).
Verification:
Alternative:
Verification:
Summary:


Rilly
Gaspalou 1
Jos 1
Gaspalou 2
Jos 2
Bimagic Series used for Rows
136
136
136
88
84
Half Generators (Sup or Inf)
50
50
50
28
28
Generators
2500
2500
2500
784
784
Semi Magic Squares
2824
2920
6400
2884
5428
SM resulting in Bimagic Squares
230
477
706
450
717
Essential Different Bimagic Squares
405)*
2543
3312
2212
2851
)* Calcualated by Gaspalou based on Rilly's original publication.
The differences are mainly due to the number of Bimagic Series which are considered for the columns while constructing the Semi Bimagic Squares and the diagonals while constructing the Bimagic Squares.
15.3.2 Victor Coccoz
The Coccoz method (1892) is based on the application of Generator Sets, each containing a Generator of 8 Bimagic Rows and a Generator of 8 Bimagic Columns.
The total number of Essential Different Bimag Squares is said to be 2188, based on the limited number of Bimagic Series mentioned above.
Alternative 1:
Verification 1:
Alternative 2:
Verification 2:
Alternative 3:
Summary:

Alternative 1
Group Sm Sq 1 493 2128 2 546 2710 3 590 2948 4 585 3060 5 606 2517 Ess Diff 2601 10806 Alternative 2
Group 2 3 4 5 Sm Sq 1 24 20 16 13 73 883 2  22 20 36 78 805 3   36 16 52 604 4    16 16 265 Total     219 2557 Alternative 3b
Gen Sm Sq 1,2 170 431 1,3 181 610 1,4 197 709 1,5 150 562 2,3 212 407 2,4 197 723 2,5 155 706 3,4 291 1233 3,5 138 567 4,5 184 478 Ess Diff 1615 5376
Alternative 3a
Gen 2,3 2,4 2,5 3,4 3,5 4,5 Sm Sq 1,2    10 12 2 24 64 1,3  22 4   14 40 170 1,4 16  20  22  58 224 1,5 0 20  26   46 214 2,3      28 28 66 2,4     24  24 100 2,5    40   40 212 Total       260 1050
B1
D d c B C b a A c C D a d A B b b B A d a D C c d D C b c B A a A a b C B c d D C c d A D a b B B b a D A d c C a A B c b C D d B2
R q Q s r P p S S p P r s Q q R p S s Q P r R q P s S q p R r Q q R r P Q s S p Q r R p q S s P r Q q S R p P s s P p R S q Q r
The resulting Square M1 = 8 * B1 + B2 + 1 will be Associated Magic
if following conditions are fulfilled:

B1
7 0 4 5 3 2 6 1 4 3 7 6 0 1 5 2 2 5 1 0 6 7 3 4 0 7 3 2 4 5 1 6 1 6 2 3 5 4 0 7 3 4 0 1 7 6 2 5 5 2 6 7 1 0 4 3 6 1 5 4 2 3 7 0 B2
5 7 0 3 2 1 6 4 4 6 1 2 3 0 7 5 6 4 3 0 1 2 5 7 1 3 4 7 6 5 2 0 7 5 2 1 0 3 4 6 0 2 5 6 7 4 3 1 2 0 7 4 5 6 1 3 3 1 6 5 4 7 0 2 M1
62 8 33 44 27 18 55 13 37 31 58 51 4 9 48 22 23 45 12 1 50 59 30 40 2 60 29 24 39 46 11 49 16 54 19 26 41 36 5 63 25 35 6 15 64 53 20 42 43 17 56 61 14 7 34 28 52 10 47 38 21 32 57 3
Based on these characteristics 48 Magic Lines can be constructed for both B1 and
B2 (ref. Attachment 15.3.21).
Verification:
15.3.4 Gaston Tarry This method (1903) is based on the application of two Diagonal Euler Squares (B1/B2) as shown below: 
B1
a bc b+d a+c+d b a+c a+d bc+d b a+c a+d bc+d a bc b+d a+c+d a+c+d b+d bc a bc+d a+d a+c b bc+d a+d a+c b a+c+d b+d bc a a+d bc+d b a+c b+d a+c+d a bc b+d a+c+d a bc a+d bc+d b a+c a+c b bc+d a+d bc a a+c+d b+d bc a a+c+d b+d a+c b bc+d a+d B2
p+r qr+s p q+s p+r+s qr p+s q p q+s p+r qr+s p+s q p+r+s qr p+r+s qr p+s q p+r qr+s p q+s p+s q p+r+s qr p q+s p+r qr+s qr p+r+s q p+s qr+s p+r q+s p q p+s qr p+r+s q+s p qr+s p+r qr+s p+r q+s p qr p+r+s q p+s q+s p qr+s p+r q p+s qr p+r+s
The resulting square M1a = B1 + 8 * (B2  1)
will be Pandiagonal and Bimagic if following condition is fulfilled (Bouteloup):

B1
2 1 7 8 3 4 6 5 3 4 6 5 2 1 7 8 8 7 1 2 5 6 4 3 5 6 4 3 8 7 1 2 6 5 3 4 7 8 2 1 7 8 2 1 6 5 3 4 4 3 5 6 1 2 8 7 1 2 8 7 4 3 5 6 B2
7 2 3 6 8 1 4 5 3 6 7 2 4 5 8 1 8 1 4 5 7 2 3 6 4 5 8 1 3 6 7 2 1 8 5 4 2 7 6 3 5 4 1 8 6 3 2 7 2 7 6 3 1 8 5 4 6 3 2 7 5 4 1 8 M1a = B1 + 8 * (B2  1)
50 9 23 48 59 4 30 37 19 44 54 13 26 33 63 8 64 7 25 34 53 14 20 43 29 38 60 3 24 47 49 10 6 61 35 28 15 56 42 17 39 32 2 57 46 21 11 52 12 51 45 22 1 58 40 31 41 18 16 55 36 27 5 62
The resulting square M1a is Bimagic, Pandiagonal, Complete and 4 x 4 Compact with Trimagic Main Diagonals and Bimagic Semi Diagonals.
Alternatives:
Verification:
15.3.6 André Gérardin
As illustrated in Section 15.1.1, ten thousands of (Pan) Magic Squares can be found based on Sudoku Comparable Squares of which only a small portion Bimagic.

B1
1 5 6 2 7 3 0 4 6 2 1 5 0 4 7 3 4 0 3 7 2 6 5 1 3 7 4 0 5 1 2 6 0 4 7 3 6 2 1 5 7 3 0 4 1 5 6 2 5 1 2 6 3 7 4 0 2 6 5 1 4 0 3 7 B2
5 6 7 4 1 2 3 0 4 7 6 5 0 3 2 1 1 2 3 0 5 6 7 4 0 3 2 1 4 7 6 5 6 5 4 7 2 1 0 3 7 4 5 6 3 0 1 2 2 1 0 3 6 5 4 7 3 0 1 2 7 4 5 6 M1 = 8 * B1 + B2 + 1
14 47 56 21 58 27 4 33 53 24 15 46 1 36 59 26 34 3 28 57 22 55 48 13 25 60 35 2 45 16 23 54 7 38 61 32 51 18 9 44 64 29 6 39 12 41 50 19 43 10 17 52 31 62 37 8 20 49 42 11 40 5 30 63
For B1 can be noticed that:
For B2 can be noticed that:
Based on these characteristics 96 Magic Lines can be constructed for both B1 and
B2 (ref. Attachment 15.3.61).
Verification:
15.3.8 John Hendricks This method is based on the application of two Octanary Squares (B1/B2) as shown below: 
B1
A C D B a c d b D B A C d b a c B D C A b d c a C A B D c a b d A C D B a c d b D B A C d b a c B D C A b d c a C A B D c a b d B2
c b d a c b d a a d b c a d b c b c a d b c a d d a c b d a c b C B D A C B D A A D B C A D B C B C A D B C A D D A C B D A C B
The resulting Panmagic Square M1 = 8 * B1 + B2 + 1 will be composed of four 4 x 4 corner squares
if following condition is fulfilled:
The resulting squares M2a and M2b will no longer be composed but Bimagic, Pandiagonal, Complete and 4 x 4 Compact with Trimagic Main Diagonals and Bimagic Semi Diagonals.

B1
7 4 2 1 0 3 5 6 2 1 7 4 5 6 0 3 1 2 4 7 6 5 3 0 4 7 1 2 3 0 6 5 7 4 2 1 0 3 5 6 2 1 7 4 5 6 0 3 1 2 4 7 6 5 3 0 4 7 1 2 3 0 6 5 B2
3 6 5 0 3 6 5 0 0 5 6 3 0 5 6 3 6 3 0 5 6 3 0 5 5 0 3 6 5 0 3 6 4 1 2 7 4 1 2 7 7 2 1 4 7 2 1 4 1 4 7 2 1 4 7 2 2 7 4 1 2 7 4 1 M1
60 39 22 9 4 31 46 49 17 14 63 36 41 54 7 28 15 20 33 62 55 44 25 6 38 57 12 23 30 1 52 47 61 34 19 16 5 26 43 56 24 11 58 37 48 51 2 29 10 21 40 59 50 45 32 3 35 64 13 18 27 8 53 42 M2a (Rows Swapped)
60 39 22 9 4 31 46 49 24 11 58 37 48 51 2 29 10 21 40 59 50 45 32 3 38 57 12 23 30 1 52 47 61 34 19 16 5 26 43 56 17 14 63 36 41 54 7 28 15 20 33 62 55 44 25 6 35 64 13 18 27 8 53 42 M2b (Columns Swapped)
60 31 46 9 4 39 22 49 17 54 7 36 41 14 63 28 15 44 25 62 55 20 33 6 38 1 52 23 30 57 12 47 61 26 43 16 5 34 19 56 24 51 2 37 48 11 58 29 10 45 32 59 50 21 40 3 35 8 53 18 27 64 13 42
Based on the characteristics described above 48 suitable sets of Octanary Squares can be constructed, resulting in 96 Bimagic Squares (48 unique) with the properties described above (ref. Attachment 15.3.08).
15.3.10 Aale de Winkel
This method is based on the generation of Sudoku Comparable Squares by means of digital equations.

B1
0 1 2 3 4 5 6 7 0
0 4 1 5 3 7 2 6 5 1 4 0 6 2 7 3 3 7 2 6 0 4 1 5 6 2 7 3 5 1 4 0 7 3 6 2 4 0 5 1 2 6 3 7 1 5 0 4 4 0 5 1 7 3 6 2 1 5 0 4 2 6 3 7 1 2 3 4 5 6 7 B2
0 1 2 3 4 5 6 7 0
0 6 7 1 5 3 2 4 7 1 0 6 2 4 5 3 5 3 2 4 0 6 7 1 2 4 5 3 7 1 0 6 1 7 6 0 4 2 3 5 6 0 1 7 3 5 4 2 4 2 3 5 1 7 6 0 3 5 4 2 6 0 1 7 1 2 3 4 5 6 7 B2'
1 5 3 7 2 6 0 4 1
1 4 6 3 0 5 7 2 0 5 7 2 1 4 6 3 4 1 3 6 5 0 2 7 5 0 2 7 4 1 3 6 3 6 4 1 2 7 5 0 2 7 5 0 3 6 4 1 6 3 1 4 7 2 0 5 7 2 0 5 6 3 1 4 5 3 7 2 6 0 4
M1= 8 * B1 + B2' + 1
1 5 3 7 2 6 0 4 1
2 37 15 44 25 62 24 51 41 14 40 3 50 21 63 28 29 58 20 55 6 33 11 48 54 17 59 32 45 10 36 7 60 31 53 18 35 8 46 9 19 56 30 57 12 47 5 34 39 4 42 13 64 27 49 22 16 43 1 38 23 52 26 61 5 3 7 2 6 0 4
With routine CnstrSqrs10 and B2 = Constant 1152 Bimagic Squares (576 unique) could be generated which are shown in Attachment 15.3.10.
Verification:
15.3.11 Aale de Winkel, Pandiagonal This method is based on the construction of Pan Diagonal Bimagic Squares based on six Binary Pan Magic Squares A, B, C, D, E and F as shown below: 
A
0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 B
0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 C
0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 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 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 D
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 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 E
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 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 F
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 0 0 1 1 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 1 0 0
Every row, column and (pan) diagonal of these squares contains four numbers zero and four numbers one.
Alternatives: 
Input
Decomposition
Results
Attachment
Author
Squares
Unique
Attachment
Squares
Attachment
Squares
Unique
Hendricks
96
48
6
672
336
Gil Lamb
48
24
4
448
448
Gerardin
320
80
24
2688
896
Tarry (1903)
640
160
18
2016
896
)* Supplement of Attachment 15.3.11
r1
1 4 6 7 8 5 3 2 B1
1 4 6 7 1 4 6 7 7 6 4 1 7 6 4 1 4 1 7 6 4 1 7 6 6 7 1 4 6 7 1 4 8 5 3 2 8 5 3 2 2 3 5 8 2 3 5 8 5 8 2 3 5 8 2 3 3 2 8 5 3 2 8 5 B2
3 4 7 8 6 5 2 1 2 1 6 5 7 8 3 4 8 7 4 3 1 2 5 6 5 6 1 2 4 3 8 7 3 4 7 8 6 5 2 1 2 1 6 5 7 8 3 4 8 7 4 3 1 2 5 6 5 6 1 2 4 3 8 7 M1 = 8 * (B1  1) + B2
3 28 47 56 6 29 42 49 50 41 30 5 55 48 27 4 32 7 52 43 25 2 53 46 45 54 1 26 44 51 8 31 59 36 23 16 62 37 18 9 10 17 38 61 15 24 35 60 40 63 12 19 33 58 13 22 21 14 57 34 20 11 64 39
For B1 can be noticed that:
For B2 can be noticed that:
The resulting square M1 is Bimagic, Pandiagonal, Complete and 4 x 4 Compact with Trimagic Main Diagonals and Bimagic Semi Diagonals.
The obtained results regarding the miscellaneous types of order 8 Bimagic Squares as deducted and discussed in previous sections are summarized in following table: 
Main Characteristics
Original Author(s)
Subroutine
Results
Bimagic Squares, Sudoku Based
Victor Coccoz
Bimagic Squares, Euler Based
Gaston Tarry
Bimagic Squares, Sudoku Based
André Gérardin
Bimagic Squares
John Hendricks

Bimagic Squares, Sudoku Based
Aale de Winkel
Bimagic Squares, Binary Based
Bimagic Squares
Gil Lamb
Next section will provide an alternative construction method for Bimagic Squares of order 8.

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