15.0 Special Magic Squares, Bimagic Squares, Non Consecutive Integers
15.7 Bimagic Squares (6 x 6)
A Magic Square is Bimagic if it remains magic after each of the numbers have been squared.
It has been proven that Bimagic Squares of order 6 can't exist for the distinct integers {1 ... 36}.
Christian Boyer provides on his website a short history of the development from Partly Bimagic to Bimagic Squares of order 6.
15.7.1 Historical Background
The historical development, from the first Partly Bimagic Squares to Bimagic Squares of order 6
can be summarized as follows:
Type

Author

Year

Partly Bimagic: Rows, Columns

Pfefferman

1894

Partly Bimagic: Rows, Columns, 1 Diagonal

Christian Boyer

2005

Bimagic, Associated

Jaroslaw Wroblewski

2006

Bimagic, Crosswise Symmetric

Lee Morgenstern

2006

The (Partly) Bimagic Squares of order 6 listed above are shown in
Attachment 6.13.1.
Following sections will describe and illustrate how comparable squares can be constructed or generated.
15.7.2 Pfefferman, Partly Bimagic
Pfeffermans Partly Bimagic Square, with only bimagic rows and columns
(ref. Attachment 6.13.1)
is based on a series of non consecutive integers, selected from the range {1 ... 49} as illustrated below:
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
the non occurrring integers are shaded. Partly Bimagic Squares of order 6 can be constructed as follows:

Based on the selected integers 68 Bimagic Series can be generated, which are shown in Attachment 6.13.21.

Based on the 68 Bimagic Series (ref. Attachment 6.13.21) 100 squares with 6 bimagic rows can be obtained,
which are shown in Attachment 6.13.22;

Based on the 100 'Generators' obtained above, 20 (transposed) Semi Bimagic Squares can be constructed by permutation of the numbers within the rows
(ref. Attachment 6.13.23);

By permutation of the rows and columns within the Semi Bimagic Squares, 1520 Partly Bimagic Squares can be obtained.
None of the resulting Partly Bimagic Squares have Bimagic Diagonals.
Attachment 6.13.23 shows the 20 Semi Bimagic Squares  with
the number of related Partly Bimagic Squares (nSq6)  as generated with routine CnstrSqrs6132.
Attachment 6.13.24 shows the first occurrring Partly Bimagic Square for each of the 20 Semi Bimagic Squares.
15.7.3 Boyer, Partly Bimagic
Christian Boyers Partly Bimagic Square  with bimagic rows, columns and one bimagic diagonal 
(ref. Attachment 6.13.1)
is based on following series of non consecutive integers, selected from the range {1 ... 55}:
{ 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23,
33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55}
Partly Bimagic Squares of order 6 can be constructed as follows:

Based on the selected integers 100 Bimagic Series can be generated, which are shown in Attachment 6.13.31.

Based on the 100 Bimagic Series (ref. Attachment 6.13.31) 114 squares with 6 bimagic rows can be obtained,
which are shown in Attachment 6.13.32;

Based on the 114 'Generators' obtained above, 16 (transposed) Semi Bimagic Squares can be constructed by permutation of the numbers within the rows
(ref. Attachment 6.13.33);

By permutation of the rows and columns within the Semi Bimagic Squares, 1556 Partly Bimagic Squares can be obtained.
64 of the resulting Partly Bimagic Squares have one Bimagic Diagonal.
Attachment 6.13.33 shows the 16 Semi Bimagic Squares  with
the number of related Partly Bimagic Squares (nSq6)  as generated with routine CnstrSqrs6132.
Attachment 6.13.34
shows the 64 Partly Bimagic Squares with one Bimagic Diagonal.
15.7.4 Wroblewski, Bimagic
Associated
The first four known Associated Bimagic Squares of order 6 were found by Jaroslaw Wroblewski
(ref. Attachment 6.13.1).
A fast routine  in which after the main diagonals the remaining rows and columns are calculated  can be used to generate Associated Bimagic Squares.
This sequence together with the properties of an Associated Magic Square result, after deduction, in following set of linear equations:
a(32) = s1  a(31)  a(33)  a(34)  a(35)  a(36)
a(17) = s1  a( 5)  a(11)  a(23)  a(29)  a(35)
a(10) = s1  a( 4)  a(16)  a(22)  a(28)  a(34)
a(19) = s1  a(20)  a(21)  a(22)  a(23)  a(24)
a(12) = 4*s1/6  a(24) + a(28)  a(23)  a(33)  a(21) + a(26) + (a(2)  a(35)  a(8)  a(29) + a(1)  a(36))/2
a(30) = s1  a( 6)  a(12)  a(18)  a(24)  a(36)
a(18) = s1/3  a(19)
a(17) = s1/3  a(20)
a(16) = s1/3  a(21)
a(15) = s1/3  a(22)
a(14) = s1/3  a(23)

a(13) = s1/3  a(24)
a(12) = s1/3  a(25)
a(11) = s1/3  a(26)
a(10) = s1/3  a(27)
a( 9) = s1/3  a(28)

a(8) = s1/3  a(29)
a(7) = s1/3  a(30)
a(6) = s1/3  a(31)
a(5) = s1/3  a(32)

a(4) = s1/3  a(33)
a(3) = s1/3  a(34)
a(2) = s1/3  a(35)
a(1) = s1/3  a(36)

The solutions can be obtained by guessing:
Diagonals: a(36), a(29), a(22) and a(31), a(26), a(36)
Other: a(35), a(34), a(33) and a(23), a(28), a(24)
and filling out these guesses in the abovementioned equations.
An optimized guessing routine (MgcSqr62e),
produced 192 Associated Bimagic Squares per (s1,s2), which are shown in
Attachment 6.13.44.
The (aspects of) the Associated Bimagic Squares as found by Jaroslaw Wroblewski are highlighted in red.
15.7.5 Morgenstern, Bimagic Squares
Crosswise Symmetric, Column Symmetric Centerlines
Lee Morgenstern's Bimagic Square
 Crosswise Symmetric with Column Symmetric Centerlines 
(ref. Attachment 6.13.1)
is based on following series of non consecutive integers, selected from the range {1 ... 72}:
{ 1, 4, 5, 7, 10, 13, 15, 16, 17, 18, 20, 21, 23, 24, 26, 34, 35, 36,
37, 38, 39, 47, 49, 50, 52, 53, 55, 56, 57, 58, 60, 63, 66, 68, 69, 72}
The construction of this type Bimagic Squares of order 6 can be based on 'Generators' with:

2 Balanced Bimagic Center Rows

2 Anti Symmetric Bimagic Top Rows, which define the 2 Bimagic Bottom Rows as well (complementary)
Bimagic Squares of order 6 can be constructed as follows:

Based on the range {1 ... 72}
1457 Bimagic Series can be generated, of which
15 Balanced and
1124 Anti Symmetric (562 unique).

Based on the
15 Balanced  and the
562 Unique Anti Symmetric Series
229983 Generators can be obtained.

Unfortunately only one Generator results in a Semi Bimagic Square,
by permutation of the numbers within the rows.

By permutation of the rows and columns within the Semi Bimagic Square, Bimagic Squares can be obtained as shown below:
Semi Bimagic
7 
34 
17 
68 
57 
36 
15 
69 
49 
10 
26 
50 
35 
53 
18 
60 
1 
52 
38 
20 
72 
21 
55 
13 
58 
4 
47 
23 
24 
63 
66 
39 
16 
37 
56 
5 

Bimagic
7 
57 
17 
68 
36 
34 
15 
26 
49 
10 
50 
69 
38 
55 
72 
21 
13 
20 
58 
24 
47 
23 
63 
4 
35 
1 
18 
60 
52 
53 
66 
56 
16 
37 
5 
39 

Each 6^{th} order Bimagic Square corresponds with 24 (= 4 * 6) transformations as described in
Section 6.3.
Attachment 6.13.54
page 1

shows the 24 transformations of the Bimagic Square obtained above.
The (aspect of) the original Morgenstern Square is highlighted in red.

Attachment 6.13.54
page 2

shows the 4 Crosswise Symmetric Bimagic Squares which can be obtained by transformation of the original square.

15.7.6 Morgenstern, Bimagic Squares
Crosswise Symmetric
Lee Morgenstern's Crosswise Symmetric Bimagic Square
(ref. Attachment 6.13.1)
is based on following series of non consecutive integers, selected from the range {1 ... 109}:
{ 1, 4, 5, 7, 9, 11, 13, 15, 17, 19, 25, 26, 27, 28, 32, 34, 43, 54,
56, 67, 76, 78, 82, 83, 84, 85, 91, 93, 95, 97, 99, 101, 103, 105, 106, 109}
The construction of Crosswise Symmetric Bimagic Squares of order 6 can be based on 'Generators' with:

3 Anti Symmetric Bimagic Top Rows, which define the

3 Bimagic Bottom Rows as well (complementary)
Each row has one element in common with each of 2 preselected balanced series (diagonals).
Bimagic Squares of order 6 can be constructed as follows:

Based on the range {1 ... 109}
1792 Bimagic Series can be generated, of which
4 Balanced and
1372 Anti Symmetric (686 unique).

Based on the
4 Balanced Series and the
686 Unique Anti Symmetric Series
4374 Generators can be obtained.

Only two Generators result in a Semi Bimagic Square,
by permutation of the numbers within the rows.

By permutation of the rows and columns within the Semi Bimagic Square, Bimagic Squares can be obtained as illustrated below:
Semi Bimagic
13 
84 
43 
82 
5 
103 
17 
15 
106 
25 
76 
91 
28 
105 
7 
97 
26 
67 
78 
83 
56 
1 
101 
11 
85 
34 
19 
93 
95 
4 
109 
9 
99 
32 
27 
54 

Bimagic
13 
103 
84 
43 
5 
82 
78 
11 
83 
56 
101 
1 
85 
4 
34 
19 
95 
93 
109 
54 
9 
99 
27 
32 
17 
91 
15 
106 
76 
25 
28 
67 
105 
7 
26 
97 

Each 6^{th} order Bimagic Square corresponds with 24 (= 4 * 6) transformations as described in
Section 6.3.
Attachment 6.13.64
page 1

shows the 24 transformations for each of the 2 Bimagic Square obtained above.
The (aspects of) the original Morgenstern Square are highlighted in red.

Attachment 6.13.64
page 2

shows the 8 Crosswise Symmetric Bimagic Squares which can be found within these transformations.

15.7.7a Consecutive Integers
Bimagic Diagonals
Although it has been proven that Bimagic Squares of order 6 can't exist for consecutive integers,
Simple Magic Squares of order 6 can have Bimagic Diagonals.
Simple Magic Squares with Bimagic Diagonals can be constructed based on the 98 Bimagic Series (ref. Attachment 6.2.1) as previously published by Christian Boyer (25 may 2002).
The diagonals can be read from these series, the related variables per diagonal can be permutated as required and the square can be completed.
A routine  in which after the main diagonals the remaining rows and columns are calculated 
can be used to generate Simple Magic Squares with Bimagic Diagonals (ref. MgcSqr62c).
Attachment 6.2.2 shows the first occurring Simple Magic Square with Bimagic Diagonals for each suitable set of Bimagic Series (1694).
15.7.7b Consecutive Integers
Partly Bimagic Squares
Although Bimagic Squares of order 6 can't exist for consecutive integers, it is possible to construct Partly Bimagic Squares of order 6 with following procedure:

Based on the 98 Bimagic Series (ref. Attachment 6.2.1) 8 squares with 6 bimagic rows can be obtained,
which are shown in Attachment 6.2.3;

Based on the 8 'Generators' obtained above, numerous (transposed) Semi Magic Squares with 6 bimagic columns can be constructed by permutating the numbers within the rows;

By permutation of the rows and columns within the Semi Magic Squares, Partly Bimagic Squares with (at least) 6 bimagic columns can be obtained.
The procedure described above is illustrated below for the first occurring
Partly Bimagc Square based on Generator 1:
Generator 1
28 
26 
25 
24 
6 
2 
31 
29 
23 
15 
12 
1 
33 
32 
17 
13 
9 
7 
34 
30 
18 
14 
10 
5 
35 
27 
19 
16 
11 
3 
36 
22 
21 
20 
8 
4 

Semi Magic Square
28 
31 
32 
5 
11 
4 
26 
29 
17 
14 
3 
22 
25 
23 
13 
10 
19 
21 
24 
12 
33 
18 
16 
8 
6 
1 
7 
34 
27 
36 
2 
15 
9 
30 
35 
20 

Simple Magic Square
34 
27 
6 
1 
36 
7 
5 
11 
28 
31 
4 
32 
14 
3 
26 
29 
22 
17 
10 
19 
25 
23 
21 
13 
18 
16 
24 
12 
8 
33 
30 
35 
2 
15 
20 
9 

and is a variation on the methods used by e.g. Achille Rilly (1901) for the construction of Bimagic Squares of order 8 based on limited amounts of Bimagic Series (ref. Section 15.3.1).
A routine  in which the required permutations are executed  can be used to generate Partly Bimagic Squares
(ref. CnstrSqrs6).
Attachment 6.2.4 shows the first occurring Partly Bimagic Square of order 6 for each of the eight Generators.
15.7.7c Summary
The obtained results regarding the miscellaneous types of order 6 Bimagic Squares as deducted and discussed in previous sections
are summarized in following table:
