Office Applications and Entertainment, Magic Squares  
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14.0 Special Magic Squares, Prime Numbers
14.4.20 Symmetric Magic Squares (6 x 6)
In previous sections following possible symmetries have been introduced:
and applied in combination with other properties.
14.4.21 Axial Symmetric Pan Magic Squares
When the equations defining the Axial Symmetry as illustrated below:
a(31) = s1  a(32)  a(33)  a(34)  a(35)  a(36) a(25) = s1  a(26)  a(27)  a(28)  a(29)  a(30) a(24) =  s1/2 + a(26) + a(27) + a(28) + a(33) a(23) = s1/2  a(28)  a(29)  a(30) + a(32) a(22) = 3*s1/2  a(27)  a(28)  a(29)  a(32)  a(33)  a(34)  a(35)  a(36) a(21) = s1/2  a(26)  a(27)  a(28) + a(36) a(20) =  s1/2 + a(28) + a(29) + a(30) + a(35) a(19) =  s1/2 + a(27) + a(28) + a(29) + a(34)
which can be incorporated in a routine to generate subject Axial Symmetric Pan Magic Squares
(ref. MgcSqr6122).
14.4.22 Row Symmetric Pan Magic Squares (Type 1)
When the equations defining the Row Symmetry as illustrated below:
a(31) = s1  a(32)  a(33)  a(34)  a(35)  a(36) a(23) =  a(24) + a(35) + a(36) a(22) = a(24) + a(34)  a(36) a(21) =  a(24) + a(33) + a(36) a(20) = a(24) + a(32)  a(36) a(19) = s1  a(24)  a(32)  a(33)  a(34)  a(35) a(12) = s1/2  a(24)  a(32)  a(34) + a(36) a(11) = s1/2 + a(24) + a(32) + a(34) + a(35) a(10) = s1/2  a(24)  a(32) a(9) = s1/2 + a(24) + a(32) + a(33) + a(34) a(8) = s1/2  a(24)  a(34) a(7) = s1/2 + a(24)  a(33)  a(35)  a(36)
which can be incorporated in a routine to generate subject Row Symmetric Pan Magic Squares
(MgcSqr6123).
14.4.23 Row Symmetric Pan Magic Squares (Type 2)
When the equations defining the Row Symmetry as illustrated below:
a(31) = s1  a(32)  a(33)  a(34)  a(35)  a(36) a(27) = 2*s1  2 * a(28)  2 * a(29)  a(30)  a(33)  2 * a(34)  2 * a(35)  a(36) a(26) = 2*s1 + 2 * a(28) + 3 * a(29) + 2 * a(30)  a(32) + 2 * a(34) + 3 * a(35) + 2 * a(36) a(25) = s1  a(28)  2 * a(29)  2 * a(30) + a(32) + a(33)  a(35)  a(36) a(22) =  s1/2  a(23)  a(24) + a(28) + a(29) + a(30)  a(32) + a(34) + 2 * a(35) + a(36) a(21) = s1 + a(24)  2 * a(28)  2 * a(29)  2 * a(30) + 2 * a(32) + a(33)  2 * a(35)  a(36) a(20) =  s1 + a(23) + 2 * a(28) + 2 * a(29) + 2 * a(30)  a(32) + a(35) a(19) = 3*s1/2  a(23)  a(24)  a(28)  a(29)  a(30)  a(33)  a(34)  a(35)
which can be incorporated in a routine to generate subject Row Symmetric Pan Magic Squares
(MgcSqr6124).
14.4.24 Crosswise Symmetric Pan Magic Squares
When the equations defining the Crosswise Symmetry as illustrated below:
a(31) = s1  a(32)  a(33)  a(34)  a(35)  a(36) a(22) =  a(23) + a(33) + a(36) a(20) = s1  a(21)  a(32)  a(33)  a(35)  a(36) a(19) =  a(24) + a(32) + a(35) a(12) = s1/2 + a(23)  a(32)  a(33)  a(36) a(11) = s1/2 + a(24)  a(32)  a(33)  a(35) a(10) = s1/2 + a(21) + a(32) + a(33) + a(35) a( 9) = s1/2  a(23) + a(32) + a(33) + a(34) + a(35) + a(36) a( 8) = s1/2  a(24)  a(34) a( 7) = s1/2  a(21)  a(35)
which can be incorporated in a routine to generate subject Crosswise Symmetric Pan Magic Squares
(MgcSqr6125).
14.4.25 Symmetric Pan Magic Squares (Type 4)
When the equations defining the assorted symmetry as illustrated below:
a(31) = s1  a(32)  a(33)  a(34)  a(35)  a(36) a(27) = 3*s1/2  a(28)  a(29)  a(32)  a(33)  2 * a(34)  a(35)  a(36) a(26) = s1 + a(29) + a(32) + 2 * a(34) + a(35) + 2 * a(36) a(25) = s1/2  a(29)  a(30) + a(33)  a(36) a(23) = s1/2  a(28)  a(29)  a(30) + a(32) a(22) =  a(24) + a(34) + a(36) a(21) = a(24) + a(33)  a(36) a(20) = s1/2 + a(28) + a(29) + a(30) + a(35) a(19) = s1  a(24)  a(32)  a(33)  a(34)  a(35)
which can be incorporated in a routine to generate subject Symmetric Pan Magic Squares
(MgcSqr6127).
14.4.26 Symmetric Magic Squares (Type 3)
When the equations defining the assorted symmetry as illustrated below:
a(31) = s1  a(32)  a(33)  a(34)  a(35)  a(36) a(26) = 2 * s1/3  a(29)  a(32)  a(35) a(19) = s1  a(20)  a(21)  a(22)  a(23)  a(24) a(12) = 2 * s1/3 + a(21)  a(22)  2 * a(29) + a(31)  a(32)  a(35) a(9) = s1  a(10)  a(31)  a(32)  a(35)  a(36) a(7) = 2 * s1/3  a(21) + a(22) + 2 * a(29) + a(32) + a(35) + a(36)
which can be incorporated in a routine to generate subject Symmetric Magic Squares
(MgcSqr6116).
14.4.27 Symmetric Magic Squares (Type 5)
When the equations defining the assorted symmetry as illustrated below:
a(31) = s1  a(32)  a(33)  a(34)  a(35)  a(36) a(25) = 4*s1/3  a(26)  a(29)  a(30)  a(31)  a(32)  a(35)  a(36) a(24) = a(26)  a(29) + a(31) a(21) = s1  a(22)  a(31)  a(32)  a(35)  a(36) a(20) =  a(23) + a(32) + a(35) a(19) =  a(26) + a(29) + a(36) a(10) = (2*s1  2 * a(22)  2 * a(31)  2 * a(32) + a(33)  a(34)  2*a(35)  2*a(36))/2 a( 9) = s1  a(10)  a(31)  a(32)  a(35)  a(36)
which can be incorporated in a routine to generate subject Symmetric Magic Squares
(MgcSqr6118).
14.4.28 Symmetric Magic Squares (Type 6)
When the equations defining the assorted symmetry as illustrated below:
a(31) = s1  a(32)  a(33)  a(34)  a(35)  a(36) a(25) = s1  a(26)  a(27)  a(28)  a(29)  a(30) a(22) = s1 + a(24)  a(26) + a(27)  a(28) + a(29) + a(32) + 3 * a(33) + a(35) + a(36) a(21) = s1 + a(24)  a(26) + a(29) + a(32) + 2 * a(33) + a(34) + a(35) + a(36) a(20) =  4 * a(21)  a(23)  2 * a(27) + 2 * a(28) + a(32) + 4 * a(34) + a(35) a(19) = s1 + a(24)  2 * a(26) + a(27)  a(28) + 2 * a(29) + a(32) + 3 * a(33)  a(34) + a(35) + 2*a(36)
which can be incorporated in a routine to generate subject Symmetric Magic Squares
(MgcSqr6119).
The obtained results regarding the miscellaneous types of order 6 Prime Number Symmetric Magic Squares as deducted and discussed in previous sections are summarized in following table: 
Type
Characteristics
Subroutine
Results
Pan Magic
Axial Symmetric
Row Symmetric, Type 1
Row Symmetric, Type 2
Crosswise Symmetric
Mixed Symmetric, Type 4
Simple
Mixed Symmetric, Type 3
Mixed Symmetric, Type 5
Mixed Symmetric, Type 6
Comparable routines as listed above, can be used to generate Prime Number Rectangular Compact Magic Squares of order 6, which will be described in following sections.

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