Office Applications and Entertainment, Magic Squares  
Index  About the Author 
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 1 2 3 4 5 6 7 8
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 B2
0 1 2 3 4 5 6 7 8
0 1 2 3 4 5 6 7 8
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 B2'
0 4 8 7 2 3 5 6 1
0 4 8 7 2 3 5 6 1
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
M1= 9 * B1 + B2' + 1
0 4 8 7 2 3 5 6 1
0 4 8 7 2 3 5 6 1
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
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 TarryCazalas This method is based on a mathematical model (arithmetic series) as illustrated below:
with key series (r1, r2)_{3} and (s1, s2)_{3}. 
Base 3
0000 0111 0222 1021 1102 1210 2012 2120 2201 2021 2102 2210 0012 0120 0201 1000 1111 1222 1012 1120 1201 2000 2111 2222 0021 0102 0210 0122 0200 0011 1110 1221 1002 2101 2212 2020 2110 2221 2002 0101 0212 0020 1122 1200 1011 1101 1212 1020 2122 2200 2011 0110 0221 0002 0211 0022 0100 1202 1010 1121 2220 2001 2112 2202 2010 2121 0220 0001 0112 1211 1022 1100 1220 1001 1112 2211 2022 2100 0202 0010 0121 Bimagic
1 14 27 35 39 49 60 70 74 62 66 76 6 16 20 28 41 54 33 43 47 55 68 81 8 12 22 18 19 5 40 53 30 65 78 61 67 80 57 11 24 7 45 46 32 38 51 34 72 73 59 13 26 3 23 9 10 48 31 44 79 56 69 75 58 71 25 2 15 50 36 37 52 29 42 77 63 64 21 4 17
The four digits a, b, c, d of each number should be understood as numbers base 3, resulting in decimal numbers by the equation:
Notes:
Based on the 81 numbers base 3 (0000 2222) 672 series (a1, a2)_{3} can be found for which the determinant does not equal to zero.
Subject procedure (CnstrSqrs48) generated 1152 (unique) bimagic squares within 21,5 seconds, of which 672 essential different.
All 1152 bimagic squares deducted above can be decomposed in pairs of Orthogonal Sudoku Squares  for which also the nine regular sub squares contain the integers 0 ... 8  as illustrated below: 
B1
0 1 2 3 4 5 6 7 8 6 7 8 0 1 2 3 4 5 3 4 5 6 7 8 0 1 2 1 2 0 4 5 3 7 8 6 7 8 6 1 2 0 4 5 3 4 5 3 7 8 6 1 2 0 2 0 1 5 3 4 8 6 7 8 6 7 2 0 1 5 3 4 5 3 4 8 6 7 2 0 1 B2
0 4 8 7 2 3 5 6 1 7 2 3 5 6 1 0 4 8 5 6 1 0 4 8 7 2 3 8 0 4 3 7 2 1 5 6 3 7 2 1 5 6 8 0 4 1 5 6 8 0 4 3 7 2 4 8 0 2 3 7 6 1 5 2 3 7 6 1 5 4 8 0 6 1 5 4 8 0 2 3 7 M = 9*B1 + B2 + 1
1 14 27 35 39 49 60 70 74 62 66 76 6 16 20 28 41 54 33 43 47 55 68 81 8 12 22 18 19 5 40 53 30 65 78 61 67 80 57 11 24 7 45 46 32 38 51 34 72 73 59 13 26 3 23 9 10 48 31 44 79 56 69 75 58 71 25 2 15 50 36 37 52 29 42 77 63 64 21 4 17
The Sudoku Squares obtained by decomposition of all 1152 bimagic squares can be referred to as
{B1} and {B2}.
While including the transposed squares of a and b above a collection of 8448 Bimagic Squares M can be found.
15.4.9 Donald Keedwell
Donald Keedwell studied the construction of TarryCazalas Bimagic Squares based on pairs of Orthogonal Sudoku Squares in more detail.
The operator α moves the rows of M one row upwards (with wrap around).

M1
M Mαβ Mα^{2}β^{2} Mβ Mαβ^{2} Mα^{2} Mβ^{2} Mα Mα^{2}β M2
M Mα^{2} Mα Mαβ^{2} Mβ^{2} Mα^{2}β^{2} Mα^{2}β Mαβ Mβ B1
0 6 3 4 1 7 8 5 2 7 4 1 2 8 5 3 0 6 5 2 8 6 3 0 1 7 4 6 3 0 1 7 4 5 2 8 4 1 7 8 5 2 0 6 3 2 8 5 3 0 6 7 4 1 3 0 6 7 4 1 2 8 5 1 7 4 5 2 8 6 3 0 8 5 2 0 6 3 4 1 7 B2
0 7 5 1 8 3 2 6 4 2 6 4 0 7 5 1 8 3 1 8 3 2 6 4 0 7 5 4 2 6 5 0 7 3 1 8 3 1 8 4 2 6 5 0 7 5 0 7 3 1 8 4 2 6 8 3 1 6 4 2 7 5 0 7 5 0 8 3 1 6 4 2 6 4 2 7 5 0 8 3 1 M = 9*B1 + B2 + 1
1 62 33 38 18 67 75 52 23 66 43 14 19 80 51 29 9 58 47 27 76 57 34 5 10 71 42 59 30 7 15 64 44 49 20 81 40 11 72 77 48 25 6 55 35 24 73 53 31 2 63 68 39 16 36 4 56 70 41 12 26 78 46 17 69 37 54 22 74 61 32 3 79 50 21 8 60 28 45 13 65
Attachment 15.4.94 shows a collection of 192 different TarryCazalas Squares,
which can be constructed based on the three sets of Keedwell Operator Matrices
(ref. CnstrSqrs9c).
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

Bimagic Squares, Aritmetic Series
TarryCazalas
Bimagic Squares, Sudoku Based
Donald Keedwell
Bimagic Squares
Alternative 1
Next section will provide an alternative construction method for Associated Bimagic Squares of order 9.

Index  About the Author 