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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 6th 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 6th 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:

 Type Characteristics Subroutine Results Partly Bimagic Range {1 - 36} Bimagic Diagonals Bimagic Columns Partly Bimagic Bimagic rows and columns Bimagic rows and columns One bimagic diagonal Bimagic Crosswise Symmetric (1) Crosswise Symmetric (2) Associated
 Next section will provide some examples of (classical) construction methods for (Partly) Bimagic Squares of order 7.