Office Applications and Entertainment, Magic Squares  
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B1
B C A D d M b a c a c b B A C d D M D M d a b c A B C b a c A C B M d D A B C d M D c b a d D M b c a C A B c b a C B A D M d M d D c a b B C A C A B M D d a c b B2
R n r Q p S q P s p Q S q s P n r R s q P n R r Q S p n r R S Q p P s q q P s r n R S p Q Q S p P q s r R n P s q R r n p Q S r R n p S Q s q P S p Q s P q R n r
The resulting Square M1 = 9 * B1 + B2 + 1 will be Associated Magic if following conditions are fulfilled:

B1
3 6 0 1 7 4 5 8 2 8 2 5 3 0 6 7 1 4 1 4 7 8 5 2 0 3 6 5 8 2 0 6 3 4 7 1 0 3 6 7 4 1 2 5 8 7 1 4 5 2 8 6 0 3 2 5 8 6 3 0 1 4 7 4 7 1 2 8 5 3 6 0 6 0 3 4 1 7 8 2 5 B2
8 4 0 5 6 1 3 2 7 6 5 1 3 7 2 4 0 8 7 3 2 4 8 0 5 1 6 4 0 8 1 5 6 2 7 3 3 2 7 0 4 8 1 6 5 5 1 6 2 3 7 0 8 4 2 7 3 8 0 4 6 5 1 0 8 4 6 1 5 7 3 2 1 6 5 7 2 3 8 4 0 M1 = 9 * B1 + B2 + 1
36 59 1 15 70 38 49 75 26 79 24 47 31 8 57 68 10 45 17 40 66 77 54 19 6 29 61 50 73 27 2 60 34 39 71 13 4 30 62 64 41 18 20 52 78 69 11 43 48 22 80 55 9 32 21 53 76 63 28 5 16 42 65 37 72 14 25 74 51 35 58 3 56 7 33 44 12 67 81 23 46
Based on these characteristics 8 Magic Lines can be constructed for both B1 and
B2 (ref. Attachment 15.3.21).

B1
D b M d c B C a A c B b a A M d C D C D A B d c b M a b C a c B d D A M B d C A M a c D b M a d D b C A c B A M B C D b a d c d c D M a A B b C a A c b C D M B d B2
Q P p S s n R q r S s Q p q r n P R n S R r p q s Q P q r s P R Q p n S P R q s n S Q r p s n P q r p S R Q p q S Q P R r s n r p n R Q P q S s R Q r n S s P p q
for which following conditions apply:
which finally result in a collection of 448 (112 unique) Associated Bimagic Squares with Trimagic Diagonals (ref. Attachment 15.4.25).
15.4.6 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 8 0
0 3 6 2 5 8 1 4 7 7 1 4 6 0 3 8 2 5 5 8 2 4 7 1 3 6 0 4 7 1 3 6 0 5 8 2 2 5 8 1 4 7 0 3 6 6 0 3 8 2 5 7 1 4 8 2 5 7 1 4 6 0 3 3 6 0 5 8 2 4 7 1 1 4 7 0 3 6 2 5 8 1 2 3 4 5 6 7 8 B2
0 1 2 3 4 5 6 7 8 0
0 3 6 2 5 8 1 4 7 7 1 4 6 0 3 8 2 5 5 8 2 4 7 1 3 6 0 4 7 1 3 6 0 5 8 2 2 5 8 1 4 7 0 3 6 6 0 3 8 2 5 7 1 4 8 2 5 7 1 4 6 0 3 3 6 0 5 8 2 4 7 1 1 4 7 0 3 6 2 5 8 1 2 3 4 5 6 7 8 B2'
0 4 8 7 2 3 5 6 1 0
0 5 7 4 6 2 8 1 3 2 4 6 3 8 1 7 0 5 1 3 8 5 7 0 6 2 4 3 8 1 7 0 5 2 4 6 5 7 0 6 2 4 1 3 8 4 6 2 8 1 3 0 5 7 6 2 4 1 3 8 5 7 0 8 1 3 0 5 7 4 6 2 7 0 5 2 4 6 3 8 1 4 8 7 2 3 5 6 1
M1= 9 * B1 + B2' + 1
0 4 8 7 2 3 5 6 1 0
1 33 62 23 52 75 18 38 67 66 14 43 58 9 29 80 19 51 47 76 27 42 71 10 34 57 5 40 72 11 35 55 6 48 77 25 24 53 73 16 39 68 2 31 63 59 7 30 81 20 49 64 15 44 79 21 50 65 13 45 60 8 28 36 56 4 46 78 26 41 70 12 17 37 69 3 32 61 22 54 74 4 8 7 2 3 5 6 1
De Winkel provides on his website 224 Bimagic Squares constructed in accordance with the principle described above,
which are shown in Attachment 15.4.66.
which finally result in a collection of 4384 Bimagic Squares (ref. Attachment 15.4.64).
which finally result in a collection of 17016 Bimagic Squares (not available in HTML).
which finally result in another 896 (256 unique) Associated Bimagic Squares (ref. Attachment 9.6.10).


Attachment
n9
Common with {A}
Common with {B}
Alternative 1
2304
24
288
Alternative 2
4384
224
1824
Alternative 3

17016
224
3200
Alternative 4
896
16
896
15.4.7 George Chen This method is based on the application of two Diagonal Euler Squares (B1/B2) as illustrated in the example below: 
b31
4 3 8 9 5 1 2 7 6 B1
1 9 5 8 4 3 6 2 7 6 2 7 1 9 5 8 4 3 8 4 3 6 2 7 1 9 5 9 5 1 4 3 8 2 7 6 2 7 6 9 5 1 4 3 8 4 3 8 2 7 6 9 5 1 5 1 9 3 8 4 7 6 2 7 6 2 5 1 9 3 8 4 3 8 4 7 6 2 5 1 9 B2
8 7 9 2 1 3 5 4 6 2 1 3 5 4 6 8 7 9 5 4 6 8 7 9 2 1 3 9 8 7 3 2 1 6 5 4 3 2 1 6 5 4 9 8 7 6 5 4 9 8 7 3 2 1 7 9 8 1 3 2 4 6 5 1 3 2 4 6 5 7 9 8 4 6 5 7 9 8 1 3 2 b32
3 2 1 6 5 4 9 8 7 M1a = B1 + 9 * (B2  1)
64 63 77 17 4 21 42 29 52 15 2 25 37 36 50 71 58 75 44 31 48 69 56 79 10 9 23 81 68 55 22 12 8 47 43 33 20 16 6 54 41 28 76 66 62 49 39 35 74 70 60 27 14 1 59 73 72 3 26 13 34 51 38 7 24 11 32 46 45 57 80 67 30 53 40 61 78 65 5 19 18 M1b = B2 + 9 * (B1  1)
8 79 45 65 28 21 50 13 60 47 10 57 5 76 42 71 34 27 68 31 24 53 16 63 2 73 39 81 44 7 30 20 64 15 59 49 12 56 46 78 41 4 36 26 70 33 23 67 18 62 52 75 38 1 43 9 80 19 66 29 58 51 14 55 48 11 40 6 77 25 72 35 22 69 32 61 54 17 37 3 74
It can be noticed that:
The remainder of both squares is constructed by moving the center squares b31 (b32) horizontally, vertcally or diagoanl wise, where the rows resp. columns of the sub squares are moved (cyclic) as described below:
Based on this method 8 * 8 = 64 sets of Diagonal Euler Squares (B1/B2) can be constructed resulting in 128 (80 unique) Bimagic Squares (ref. Attachment 15.4.07).
15.4.8 Walter Trump, Holger Danielsson
Walter Trump and Holger Danielsson have recently executed the full enumeration of Associated Bimagic Squares.
The obtained results regarding the miscellaneous types of order 9 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
Alternative 1
Bimagic Squares, Sudoku Based
Aale De Winkel

Alternative 1
Alternative 2Bimagic Squares, Euler Based
George Chen

Next section will provide both classical and modern construction methods for Magic Squares of Squares.

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