Office Applications and Entertainment, Magic Squares  
Index  About the Author 
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.
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).
15.1.1 Pan Magic, Complete with Bimagic Semi Diagonals The presented example has following additional properties (ref. Attachment 15.1.1):
Based on the characteristics described above, two collections of Sudoku Comparable Squares could be generated:
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).
15.1.2 Magic, Symmetric (Based on Sudoku Comparable Squares)
The presented example has following additional properties (ref. Attachment 15.1.1):
Based on the characteristics described above, two collections of Sudoku Comparable Squares could be generated:
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).
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 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:
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 obtained results regarding the miscellaneous types of order 8 Bimagic Squares as deducted and discussed in previous sections are summarized in following table: 
Main Characteristics
Type
Tag
Subroutine
Results
Pan Magic, Complete with Bimagic Semi Diagonals (Sudoku Comparable Squares)
Sudoku
B1
Sudoku
B2
Bimagic
M
Symmetric (Sudoku Comparable Squares)
Sudoku
B1
Sudoku
B2
Bimagic
M
Pan Magic, Complete with Trimagic Main Diagonals (John Hendricks)
Octanary
B1
Octanary
B2
Bimagic
M
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.
The total number of Bimagic Squares of Order 9 is, according to above mentioned publication, expected to be about 10^{22}.
15.2.1 John Hendricks (Bimagic Regular Subsquares)
Based on an example of this subset of Bimagic Squares of order 9,
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,
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:
which result each in a collection of 144 Sudoku Comparable Squares (CnstrSqrs9b):
which finally result in a collection of 20736 Partly Compact Bimagic Squares (CnstrSqrs9a), of which 256 Associated.
The Associated Ternary Squares being contained in:
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.
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
Type
Tag
Subroutine
Results
Bimagic Regular Subsquares (John Hendricks)
Ternary
G1

Sudoku
B1/2
Bimagic
M
31104
Based on Ternary  and Sudoku Comparable Squares (Collissons)
Ternary
G1/2

Sudoku
B1/2
Bimagic
M
31104
Partly Compact, Bimagic Regular Subsquares (Donald Keedwell)
Ternary
G1
3456
Ternary
G2

Sudoku
B1
Sudoku
B2
Bimagic
M
20736
Associated (1)
Bimagic
M1
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.

Index  About the Author 