Office Applications and Entertainment, Magic Squares  
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14.0 Special Magic Squares, Prime Numbers
14.4.30 Rectangular Compact Magic Squares (6 x 6)
In previous sections the property ‘Compact’  defined as: each 2 x 2 Sub Square sums to 4 * s1 / n 
has been introduced and applied for Prime Number Magic Squares of order 4 and 6.
14.4.31 Partly Rectangular Compact
A Magic Square can be referred to as Partly Rectangular Compact,
when only the six non overlapping Sub Rectangles (2 x 3) sum to the Magic Sum:
a(31) = s1  a(32)  a(33)  a(34)  a(35)  a(36) a(28) = s1  a(29)  a(30)  a(34)  a(35)  a(36) a(25) = s1  a(26)  a(27)  a(28)  a(29)  a(30) a(19) = s1  a(20)  a(21)  a(22)  a(23)  a(24) a(16) = s1  a(17)  a(18)  a(22)  a(23)  a(24) a(13) = s1  a(14)  a(15)  a(16)  a(17)  a(18) a(11) = 4*s1 + a(12) + a(17) + 2*a(18)  2*a(19)  2*a(20)  3*a(21)  a(22)  a(23)  2*a(25) +  3*a(26)  2*a(27)  a(28)  a(29)  3*a(31)  2*a(32)  2*a(33)  a(34)  a(35) a( 8) = ( a( 9)  a(10)  2*a(12)  a(14)  2*a(15)  a(17)  2*a(18) + 2*a(19) + a(20) + 2*a(21) + + a(23) + 2*a(25) + 2*a(26) + a(27) + a(28) + 2*a(31)  2*a(36))/2 a( 7) = s1  a( 8)  a( 9)  a(10)  a(11)  a(12) a( 6) = s1  a(12)  a(18)  a(24)  a(30)  a(36) a( 5) = s1  a(11)  a(17)  a(23)  a(29)  a(35) a( 4) = s1  a(10)  a(16)  a(22)  a(28)  a(34) a( 3) = s1  a( 9)  a(15)  a(21)  a(27)  a(33) a( 2) = s1  a( 8)  a(14)  a(20)  a(26)  a(32) a( 1) = s1  a( 7)  a(13)  a(19)  a(25)  a(31)
which could be incorporated in a guessing routine to produce subject Magic Squares.
However such a routine is not very feasible due to the high number of independent variables (21 ea).
as deducted and discussed in Sections Section 14.4.10 thru Section 14.4.12.
Rectangular Compact Magic Squares can be described by following set of linear equations: a(31) = s1  a(32)  a(33)  a(34)  a(35)  a(36) a(28) = s1  a(29)  a(30)  a(34)  a(35)  a(36) a(27) = a(30)  a(33) + a(36) a(26) = a(29)  a(32) + a(35) a(25) = a(29)  a(30) + a(32) + a(33) + a(34) a(22) =  a(23)  a(24) + a(34) + a(35) + a(36) a(21) = a(24) + a(33)  a(36) a(20) = a(23) + a(32)  a(35) a(19) = s1  a(23)  a(24)  a(32)  a(33)  a(34) a(16) = s1  a(17)  a(18)  a(34)  a(35)  a(36) a(15) = a(18)  a(33) + a(36) a(14) = a(17)  a(32) + a(35) a(13) = a(17)  a(18) + a(32) + a(33) + a(34) a(12) = 7*s1/3  a(17)  2 * a(18)  a(30)  2 * a(32)  2 * a(34)  a(35)  4 * a(36) a(11) = s1/3  a(29) a(10) = 8*s1/3 + a(17) + 2 * a(18) + a(29) + a(30) + 2 * a(32) + 3 * a(34) + 2 * a(35) + 5 * a(36) a( 9) = 7*s1/3  a(17)  2 * a(18)  a(30)  2 * a(32) + a(33)  2 * a(34)  a(35)  5 * a(36) a( 8) = s1/3  a(29) + a(32)  a(35) a( 7) = 5*s1/3 + a(17) + 2 * a(18) + a(29) + a(30) + a(32)  a(33) + a(34) + a(35) + 4 * a(36) a( 6) = 4*s1/3 + a(17) + a(18)  a(24) + 2 * a(32) + 2 * a(34) + a(35) + 3 * a(36) a( 5) = 2*s1/3  a(17)  a(23)  a(35) a( 4) = 5*s1/3  a(18) + a(23) + a(24)  2 * a(32)  3 * a(34)  a(35)  4 * a(36) a( 3) = 4*s1/3 + a(17) + a(18)  a(24) + 2 * a(32)  a(33) + 2 * a(34) + a(35) + 4 * a(36) a( 2) = 2*s1/3  a(17)  a(23)  a(32) a( 1) = 2*s1/3  a(18) + a(23) + a(24)  a(32) + a(33)  a(34)  3 * a(36)
which can be incorporated in a routine to generate subject Rectangular Compact Magic Squares
(ref. MgcSqr6103).
14.4.33 Rectangular Compact, Axial Symmetric
Although Rectangular Compact Magic Squares can't be Diagonal Symmetric they can be Axial Symmetric.
a(31) = s1  a(32)  a(33)  a(34)  a(35)  a(36) a(28) = s1  a(29)  a(30)  a(34)  a(35)  a(36) a(27) = a(30)  a(33) + a(36) a(26) = a(29)  a(32) + a(35) a(25) =  a(29)  a(30) + a(32) + a(33) + a(34) a(23) =  s1  2 * a(24) + 2 * a(32) + 2 * a(34) + a(35) + 4 * a(36) a(22) = s1 + a(24)  2 * a(32)  a(34)  3 * a(36) a(21) = a(24) + a(33)  a(36) a(20) =  s1  2 * a(24) + 3 * a(32) + 2 * a(34) + 4 * a(36) a(19) = 2*s1 + a(24)  3 * a(32)  a(33)  3 * a(34)  a(35)  4 * a(36)
which can be incorporated in a routine to generate subject Rectangular Compact Axial Symmetric Magic Squares
(ref. MgcSqr6104).
14.4.34 Rectangular Compact, Row Symmetric, Pan Magic
When the equations defining Row Symmetry as illustrated below:
a(32) = s1/2  a(34)  a(36) a(31) = s1/2  a(33)  a(35) a(28) = s1  a(29)  a(30)  a(34)  a(35)  a(36) a(27) = a(30)  a(33) + a(36) a(26) = s1/2 + a(29) + a(34) + a(35) + a(36) a(25) = s1/2  a(29)  a(30) + a(33)  a(36) a(22) =  a(23)  a(24) + a(34) + a(35) + a(36) a(21) = a(24) + a(33)  a(36) a(20) = s1/2 + a(23)  a(34)  a(35)  a(36) a(19) = s1/2  a(23)  a(24)  a(33) + a(36)
and appears to be Pan Magic as well.
The obtained results regarding the miscellaneous types of order 6 Prime Number Rectangular Compact Magic Squares as deducted and discussed in previous sections are summarized in following table: 
Type
Characteristics
Subroutine
Results
Rect. Compact
General
Axial Symmetric
Row Symmetric, Pan Magic
Comparable routines as listed above, can be used to generate Prime Number Magic Squares of order 7, which will be described in following sections.

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