Office Applications and Entertainment, Magic Squares

15.0   Special Magic Squares, Bimagic Squares, Non Consecutive Integers

15.8   Bimagic Squares (7 x 7)

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 7 can't exist for the distinct integers {1 ... 49}.

Christian Boyer provides on his website a short history of the development from Partly Bimagic to Bimagic Squares of order 7.

15.8.1 Historical Background

The historical development, from the first Partly Bimagic Squares to Bimagic Squares of order 7 can be summarized as follows:

Type	Author	Year
Partly Bimagic: 4 Rows, 4 Columns	Pfefferman	1891
Nearly Bimagic: 7 Rows, 5 Columns, 1 Diagonal	Christian Boyer	2001
Nearly Bimagic: 7 Rows, 5 Columns, 1 Diagonal	Walter Trump	2002
Nearly Bimagic: 7 Rows, 5 Columns, 1 Diagonal	Pan Feng Chu	2004
Nearly Bimagic: 7 Rows, 7 Columns, 1 Diagonal	Christian Boyer	2005
Bimagic, Miscellaneous Symmetries	Lee Morgenstern	2006

The (Partly) Bimagic Squares of order 7 listed above are shown in Attachment 15.8.1 and Attachment 15.8.2.

Following sections will describe and illustrate how comparable squares can be constructed or generated.

15.8.2 Construction Method, Consecutive Integers

Nearly Bimagic Squares of order 7 can be constructed with a variation of the 'Generator' method as discussed in detail in Section 15.5.

Nearly Bimagic Squares of order 7 shall be based on the 1844 Bimagic Series (ref. Attachment 16.5.3) as published by Achille Rilly (1909).

The construction can be based on:

• Base Generators, composed of one Bimagic Row (top) and one Bimagic Column (left), selected from the 60 series with 7 odd numbers, with the common element in the corner (top/left).
• Row Generators, obtained by completing the Base Generators with six of the remaining series with 3 odd (and 4 even) numbers.

which ensures that subject Row Generators contain the consecutive integers 1 ... 49 and speeds up the construction process.

The construction of Nearly Bimagic Squares of order 7 can be completed as follows:

• Construct Semi Nearly Bimagic Squares based on the Row Generators obtained above, by permutating the numbers within the rows;
• Permutate the rows and columns within the Semi Nearly Bimagic Squares, in order to obtain Nearly Bimagic Squares (if possible).

An example of a Nearly Bimagic Square obtained by permutation of the rows and columns within a Semi Nearly Bimagic Square, is shown below:

Semi Nearly Bimagic 1 9 31 33 39 41 21 17 42 32 13 16 8 47 19 24 15 44 7 18 48 25 45 12 14 43 30 6 27 23 46 2 40 11 26 37 4 36 35 20 38 5 49 28 3 34 10 29 22

Nearly Bimagic 33 39 21 31 1 9 41 13 16 47 32 17 42 8 14 43 6 12 25 45 30 35 20 5 36 37 4 38 34 10 22 3 49 28 29 44 7 48 15 19 24 18 2 40 26 46 27 23 11

Based on the 60 Bimagic Series with odd numbers 424 Base Generators can be constructed (ref. BaseGen7).

Based on the 424 Base Generators 49820 Row Generators can be constructed (ref. RowGen7).

None of these Row Generators can be transformed into Semi Bimagic Squares, which confirms the findings of Christian Boyer and others (ref. CnstrSqrs7a).

However with the same routine 190 Semi Nearly Bimagic Squares with 7 bimagic rows and 5 bimagic columns can be constructed, which are shown in Attachment 15.8.62.

The Semi Nearly Bimagic Squares resulted in 15 Nearly Bimagic Squares with one bimagic diagonal as shown in Attachment 15.8.63.

Page 2 of the attachments mentioned above show the 181 Semi Nearly Bimagic Squares resp. 11 Nearly Bimagic Squares which can be generated based on the 424 Transposed Base Generators (45605 Row Generators).

15.8.3 Lee Morgenstern, Bimagic, Distinct Integers

The different types of Bimagic Squares, published by Lee Morgenstern in 2006 (ref. Attachment 15.8.1), can be constructed with the same 'Generator' method as discussed in detail in Section 15.5.

This construction method is based on the application of:

• Base Generators, composed of one Balanced Bimagic Row (top) and one Balanced Bimagic Column (left) with the common element in the corner (top/left).
• Row Generators, obtained by completing the Base Generators with 3 Anti Symmetric Bimagic Rows and 3 Complementary Rows.
• Column Generators, composed of 3 Anti Symmetric Bimagic Columns based on the first three elements right of the corner (top/left) of the Base Generators.

The first square is based on following series of non consecutive integers, selected from the range {1 ... 67}:

{ 1,  2,  5,  6,  7, 10, 11, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 
 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47,
 48, 49, 50, 51, 53, 54, 57, 58, 61, 62, 63, 66, 67 }

Bimagic Squares of order 7 can be constructed as follows:

• Based on the selected integers 1112 Bimagic Series can be generated, of which 12 Balanced and 534 Anti Symmetric.
• Based on the 12 Balanced - and the 534 Anti Symmetric Series 742 Row Generators and 396 Column Generators can be obtained.
• Unfortunately only one combination of Generators results in a Semi Bimagic Square, by permutation of the numbers within the rows of the Row Generator.
• By permutation of the rows and columns within the Semi Bimagic Square, Bimagic Squares can be obtained as shown below:

Semi Bimagic 34 5 63 21 47 22 46 6 23 35 61 58 25 30 62 33 45 10 7 38 43 15 66 29 26 18 57 27 53 39 2 50 42 41 11 31 48 44 51 49 1 14 37 24 20 19 17 54 67

Bimagic 35 23 25 6 30 58 61 45 33 38 62 43 7 10 44 48 1 31 14 49 51 63 5 22 34 46 47 21 20 24 54 37 67 17 19 2 39 41 53 11 42 50 29 66 57 15 27 18 26

Each 7^th order Bimagic Square corresponds with 24 (= 4 * 6) transformations as described in Section 6.3.

Comparable results can be obtained for the other applied ranges.

It can be noticed that Square #5 is a transformation of Square #4 (Permutation rows 4/5 and columns 4/5).

15.8.4 Summary

The obtained results regarding the miscellaneous types of order 7 Bimagic Squares as deducted and discussed in previous sections are summarized in following table:

Next section will provide methods for the construction of Bimagic Squares of order 16.