Office Applications and Entertainment, Magic Squares

15.0   Special Magic Squares, Bimagic Squares, Part 1

A Magic Square is Bimagic if it remains magic after each of the numbers have been squared. It has been proven that the smallest order of Bimagic Squares is 8.

The Magic Sum of the squared numbers is n(n² + 1)(2n² + 1)/6 = s1(2n² +1)/3 with s1 the magic sum of the original square.

15.1   Bimagic Squares (8 x 8)

Walter Trump and Francis Gaspalou published in April 2014 a paper which listed the complete sets of all 8^th order Bimagic Squares (192 * 136244 = 26158848).

As a part of subject publication some interesting subsets of 8^th order Bimagic Squares are provided.

Section 15.1.1 and 15.1.2 below are adopted from the description of two of these subsets. Section 15.1.3 is based on a Class of Bimagic Squares published by John Hendricks.

15.1.1 Pan Magic, Complete with Bimagic Semi Diagonals
       (Based on Sudoku Comparable Squares)

The presented example has following additional properties (ref. Attachment 15.1.1):

• The half rows sum to half the Magic Sum;
• The Semi Diagonals (2 ea), Horizontal Main Bent Diagonals (2 ea) and Mirrored Horizontal Main Bent Diagonals (2 ea) sum to the Magic Sum;
• Partly Rectangular Compact, meaning the 2 x 4 rectangles from left to right sum (partly) to the Magic Sum;
• One of the Sudoku Comparable Squares shares these properties (B2);
• For the other Sudoku Comparable Square both the half rows and columns sum to half the Magic Sum (B1).

Based on the characteristics described above, two collections of Sudoku Comparable Squares could be generated:

•  864 Sudoku Comparable Squares with the properties described above (B2);
• 1008 Sudoku Comparable Squares with both the half rows and columns summing to half the Magic Sum (B1).

which finally resulted in a collection of 76032 Pan Magic Squares, with the properties described above, of which 320 Bimagic (ref. Attachment 15.1.2).

The unique squares (80 ea) common with the collection (640 ea) described by Walter Trump as “Unique Bimagic Squares, Complete with bimagic semi-diagonals made of two Latin squares with Latin main diagonals” are highlighted in red.

15.1.2 Magic, Symmetric (Based on Sudoku Comparable Squares)

The presented example has following additional properties (ref. Attachment 15.1.1):

• The half columns sum to half the Magic Sum;
• The Semi Diagonals (2 ea) sum to the Magic Sum;
• Partly Rectangular Compact, meaning the 4 x 2 rectangles from top to bottom sum (partly) to the Magic Sum;
• The Sudoku Comparable Squares share these properties (B1, B2);
• For the first Sudoku Comparable Square also the half rows sum to half the Magic Sum (B1).

Based on the characteristics described above, two collections of Sudoku Comparable Squares could be generated:

• 1536 Sudoku Comparable Squares with the properties described above (B2);
•  144 Sudoku Comparable Squares with both the half rows and columns summing to half the Magic Sum (B1).

which finally resulted in a collection of 18432 Magic Squares, with the properties described above, of which 320 (80 unique) Bimagic (ref. Attachment 15.1.3).

Verification:
The collection of "Essential Different Associated Bimagic Squares" (841 ea) corresponds with a collection of 161472 Unique Associated Bimagic Squares.

For each of the 80 Unique Bimagic Squares M1 an aspect could be found in this collection.

15.1.3 Pan Magic, Complete with Trimagic Main Diagonals

On Holger Danielsson's site examples are provided of Bimagic Pan Magic Squares with Trimagic Main Diagonals as constructed by John Hendricks.

The presented examples have following additional properties (ref. Attachment 15.1.1):

• Compact (4 x 4), meaning all 4 x 4 Sub Squares, including wrap around, sum to two times the Magic Sum;
• Partly Rectangular Compact, meaning the 2 x 4 rectangles from left to right sum (partly) to the Magic Sum;
• Partly Rectangular Compact, meaning the 4 x 2 rectangles from top to bottom sum (partly) to the Magic Sum;
• The corner points of all 5 x 5 Sub Squares sum to half the Magic Sum.

The Octanary Squares (B1, B2), in which subject Bimagic Pan Magic Squares can be decomposed, are not completely Sudoku Comparable but share the properties listed above and have following additional features:

• Octanary Squares B2:
  - Sudoku Comparable Main Diagonals and Rows;
  - The two left and two right Sub Squares (4 x 4) are identical.
• Octanary Squares B1:
  - Sudoku Comparable Main Diagonals and Columns;
  - The two top and two bottom Sub Squares (4 x 4) are identical.
• General:
  - Each set (2) of Diagonal Sub Squares (2 x 2) within a Sub Square (4 x 4) contains the same integers;
  - The collection {B1} consists of the diagonal mirrored elements of {B2}.

Based on the characteristics described above, two collections of Octanary Squares {B2} and {B1} could be generated with 576 elements ea, which finally resulted in a collection of 322560 Pan Magic Squares, with the properties described above, of which 1280 Bimagic (ref. Attachment 15.1.4).

The unique squares (160 ea) common with the collection (10496 ea) described by Walter Trump as “Unique Bimagic Squares, Complete with bimagic semi-diagonals” are highlighted in red.

15.1.4 Summary

The obtained results regarding the miscellaneous types of order 8 Bimagic Squares as deducted and discussed in previous sections are summarized in following table:

More variations, based on classical construction methods as described by Holger Danielsson (ref. Magische Quadrate, Version 19, 2017), will be provided in Section 15.3.

15.2   Bimagic Squares (9 x 9)

The total number of Bimagic Squares of Order 9 is, according to above mentioned publication, expected to be about 10²².

It is however possible to determine certain subsets, based on some well known examples, which will be deducted in following sections.

15.2.1 John Hendricks (Bimagic Regular Subsquares)

Based on an example of this subset of Bimagic Squares of order 9,

• a decomposition in 4 Ternary Squares can be made (ref. Attachment 9.6.5),
• a collection of 24 Ternary Squares can be constructed (ref. Attachment 9.6.2) and
• a collection of 432 Sudoku Comparable Squares can be generated (ref. Attachment 9.6.3)

which finally results in a collection of 31104 Bimagic Squares (CnstrSqrs9a), of which 384 Associated.

15.2.2 Collissons (Based on Ternary Squares and Sudoku Comparable Squares)

Based on an example provided by Harvey Heinz,

• a decomposition in 4 Ternary Squares can be made (ref. Attachment 9.6.5),
• a collection of 24 Ternary Squares can be constructed (ref. Attachment 15.2.1 ) and
• a collection of 432 Sudoku Comparable Squares can be generated (ref. Attachment 15.2.2)

which finally results in a collection of 31104 Bimagic Squares (CnstrSqrs9a), of which 384 Associated.

15.2.3 Partly Compact (Based on Ternary Squares and Sudoku Comparable Squares)

Based on an example provided by Donald Keedwell, and the decomposition in 4 Ternary Squares (ref. Attachment 9.6.5), two collections of Ternary Squares can be constructed:

• Attachment 9.6.0, Ternary Compact Pan Magic Squares (3456 ea, not available in HTML) and
• Attachment 9.6.6, Ternary Partly Compact Magic Squares (24 ea)

which result each in a collection of 144 Sudoku Comparable Squares (CnstrSqrs9b):

• Attachment 9.6.4, Sudoku Comparable Compact Pan Magic Squares and
• Attachment 9.6.7, Sudoku Comparable Partly Compact Magic Squares

which finally result in a collection of 20736 Partly Compact Bimagic Squares (CnstrSqrs9a), of which 256 Associated.

15.2.4 Associated

The Associated Ternary Squares being contained in:

• Attachment  9.6.2, Ternary Magic Squares, Hendricks based  (8 ea)
• Attachment 15.2.1, Ternary Magic Squares, Collissons based (8 ea)
• Attachment  9.6.6, Ternary Magic Squares, Partly Compact   (8 ea)
• Attachment  9.6.8, Ternary Compact Associated Pan Magic Squares (96 ea)

result in 1544 Sudoku Comparable Associated Magic Squares. Based on this collection 149504 Associated Magic Squares of order 9 could be generated, of which 1792 Bimagic (320 unique), which are shown in Attachment 9.6.9.

A more efficient construction method for Associated Bimagic Squares of order 9 will be provided in Section 15.5.

15.2.5 Summary

The obtained results regarding the miscellaneous types of order 9 Bimagic Squares as deducted and discussed in previous sections are summarized in following table:

More variations, based on classical construction methods as described by Holger Danielsson (ref. Magische Quadrate, Version 19, 2017), will be provided in Section 15.4.