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8.7 More Solutions, Sudoku Comparable
8.7.1 Introduction
Any number m = 0 ... 63 can be written as m = b1 + 8 * b2 with
bi = 0, 1, ... 7 for i = 1, 2.
Consequently any Magic Square M of order 8 with the numbers 1 ... 64 can be written as
M =
B1 +
8 * B2 +
[1]
where the matrices B1 and
B2
- further referred to as Octanary Squares -
contain only the integers 0, 1, ... 7.
8.7.2 Sudoku Comparable Squares
Rows, columns and diagonals of the matrices B1 and B2
containing the eight integers 0, 1, ... 7 will sum to the Magic Sum 28. Subject matrices will be further referred to as Sudoku Comparable Squares.
Sudoku Comparable Squares can be generated by applying the deducted equations, however for a Magic Sum 28 and with the less strict restriction that only the elements of each row, column and diagonal should be different.
8.7.3 Application and Results (1)
Numerous Pan Magic Squares, as discussed in previous sections, can be generated by selecting combinations of Octanary Squares
(B1, B2),
while ensuring that the resulting square M contains all integers 1 thru 64 (CnstrSqrs8b):
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Sudoku Comparable Associated Pan Magic Squares with Non Overlapping Sub Squares,
can be obtained by applying the equations deducted in Section 8.6.7, however with s1 = 28.
An optimized guessing routine (SudSqr8d) produced 1536 Sudoku Comparable Magic Squares within 13,5 minutes, which are shown in Attachment 8.7.3.
Routine CnstrSqrs8b checked the 2357760 (= 1536*1535) possibilities and produced 18432 (= 384 * 48) Associated Pan Magic Squares of the 8th order within 3,5 hours, of which the first 48 are shown in Attachment 8.6.7.
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Sudoku Comparable Complete Pan Magic Squares with Non Overlapping Sub Squares,
can be obtained by applying the equations deducted in Section 8.6.9, however with s1 = 28.
An optimized guessing routine (SudSqr8e) produced with a(64) = 0, 6456 Sudoku Comparable Magic Squares within 13,5 minutes.
Routine CnstrSqrs8b produced with B1 = constant, 228 Complete Pan Magic Squares of the 8th order within 60 seconds, which are shown in Attachment 8.6.9.
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Sudoku Comparable Rectangular Compact Complete Pan Magic Squares,
can be obtained by applying the equations deducted in Section 8.6.11, however with s1 = 28.
An optimized guessing routine (SudSqr8g) produced 9600 Sudoku Comparable Magic Squares within 40 minutes.
Routine CnstrSqrs8b produced with B1 = constant, 480 Complete Pan Magic Squares of the 8th order within 110 seconds, which are shown in Attachment 8.6.11.
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Sudoku Comparable Partly Rectangular Compact Associated Magic Squares,
can be obtained by applying the equations deducted in Section 8.6.12, however with s1 = 28.
An optimized guessing routine (SudSqr8b2) produced 12912 Sudoku Comparable Magic Squares within 1,4 hours.
Routine CnstrSqrs8b produced with B1 = constant, 192 Associated Magic Squares of the 8th order within 100 seconds, which are shown in Attachment 8.6.12.
The same routine produced, with the option 'Sudoku Comparable Rectangles' enabled, 1536 Sudoku Comparable Magic Squares within 1,25 hours
(ref. Attachment 15.1.3b).
Based on this collection {B2} and collection {B1} as defined in Section 15.1.2,
routine CnstrSqrs8a produced
18432 Associated Magic Squares of the 8th order within about one hour, of which 320 Bimagic.
The Bimagic Squares are shown in Attachment 15.1.3.
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Sudoku Comparable Partly Rectangular Compact Complete Pan Magic Squares,
can be obtained by applying the equations deducted in Section 8.6.13, however with s1 = 28.
An optimized guessing routine (SudSqr8a) produced 8064 Sudoku Comparable Magic Squares within 7 minutes.
Routine CnstrSqrs8b produced with B1 = constant, 240 Complete Pan Magic Squares of the 8th order within 72 seconds, which are shown in Attachment 8.6.13.
The same routine produced, with the option 'Sudoku Comparable Rectangles' enabled, 1920 Sudoku Comparable Magic Squares within 4 minutes.
Based on this collection {B2} and collection {B1} as defined in Section 15.1.1,
routine CnstrSqrs8a produced
237312 Complete Pan Magic Squares of the 8th order within about 2 hours, of which 320 Bimagic.
The Bimagic Squares are shown in Attachment 15.1.2.
Attachment 15.1.1 shows a few other examples of two Sudoku Comparable Squares and resulting Pan Magic Square.
8.7.4 Quaternary Solutions
Any number m = 0 ... 63 can be written as m = b1 + 4 * b2
+ 16 * b3
with bi = 0, 1, 2, 3 for i = 1, 2, 3 (Quaternary representation).
Consequently any Magic Square M of order 8 with the numbers 1 ... 64 can be written as
M =
B1 +
4 * B2 +
16 * B3 +
[1]
where the matrices B1,
B2 and
B3
- further referred to as Quaternary Squares - contain only the integers 0, 1, 2 and 3.
Quaternary Squares can be generated by applying the deducted equations, however for a Magic Sum 12.
8.7.5 Application and Results (2)
Numerous Pan Magic Squares, as discussed in previous sections, can be generated by selecting combinations of Quaternary Squares (B1, B2, B2),
while ensuring that the resulting square M contains all integers 1 thru 64 (CnstrSqrs8c).
Further Quaternary Squares with Sudoku Comparable Non Overlapping Subsquares (2 x 2) can be considered as Medjig Squares and Used for Medjig Solutions as discussed in Section 8.3.
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Quaternary Pan Magic Complete Squares with Sudoku Comparable Non Overlapping Subsquares, can be obtained by applying the equations deducted in Section 8.6.9, however with s1 = 12.
An optimized guessing routine (Quat869) produced 640000 Quaternary Squares within 5,4 hours, of which 128 are shown in Attachment 8.7.52.
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Quaternary Associated Pan Magic Squares with Sudoku Comparable Non Overlapping Subsquares, can be obtained by applying the equations deducted in Section 8.6.7, however with s1 = 12.
An optimized guessing routine (Quat867) produced 27752 Quaternary Squares within 15 minutes, of which 128 are shown in Attachment 8.7.51.
Attachment 8.7.5 shows a few examples of three Quaternary Squares and resulting Magic Square.
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