Office Applications and Entertainment, Magic Squares

6.11    Magic Squares, Misc. Symmetries (2)

6.11.11 Symmetric Magic Squares (Type 7)

When the equations defining the assorted symmetry as illustrated below:

are added to the equations describing a Simple Magic Square of the sixth order (ref. Section 6.2), the resulting Symmetric Magic Square can be described by following set of linear equations:

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) =  4 * s1/6 + ( a(26) + a(27) - a(28) - a(29) - a(32) - 2 * a(34) - a(35) - 2 * a(36))/2
a(16) = -2 * s1/6 + (-a(26) + a(27) - a(28) + a(29) + a(32) + 2 * a(33) + a(35) + 2 * a(36))/2
a(18) =  2 * s1/6 -   a(24) + a(29) - a(30) + a(35) - a(36)
a(14) =  2 * s1/6 -   a(20) + a(25) - a(26) + a(31) - a(32)

The solutions can be obtained by guessing the 12 parameters:

      a(i) for i = 20, 24, 26 ... 30, 28, 32 ... 36

and filling out these guesses in the abovementioned equations.

With an optimized guessing routine (MgcSqr61110) numerous Symmetric Magic Squares can be produced.

Attachment 6.11.10 shows the first occuring Symmetric Magic Squares for a(36) = i (i = 1 ... 36).

6.11.12 Symmetric Magic Squares (Type 8)

When the equations defining the assorted symmetry as illustrated below:

are added to the equations describing a Simple Magic Square of the sixth order (ref. Section 6.2), the resulting Symmetric Magic Square can be described by following set of linear equations:

a(32) = ( s1 - a(33) - a(34) - 2 * a(35))/2
a(31) =        a(32) + a(35) - a(36)
a(26) = (-s1 + 2 * a(29) + a(33) + a(34) + 2 * a(35) + 2 * a(36))/2
a(25) =   s1 - a(26) - a(27) - a(28) - a(29) - a(30)
a(23) =        a(24) - a(35) + a(36)
a(21) =        a(22) - a(33) + a(34)
a(20) = ( s1 - 2 * a(22) - 2 * a(24) + a(33) - a(34) + 2 * a(35) - 2 * a(36))/2
a(19) =        a(20) - a(35) + a(36)

The solutions can be obtained by guessing the 10 parameters:

      a(i) for i = 22, 24, 27 ... 30, 33 ... 36

and filling out these guesses in the abovementioned equations.

With an optimized guessing routine (MgcSqr61111) numerous Symmetric Magic Squares can be produced.

Attachment 6.11.11 shows the first occuring Symmetric Magic Squares for a(36) = i (i = 1 ... 36).

6.11.13 Symmetric Magic Squares (Type 9)

When the equations defining the assorted symmetry as illustrated below:

are added to the equations describing a Simple Magic Square of the sixth order (ref. Section 6.2), the resulting Symmetric Magic Square can be described by following set of linear equations:

a(31) =  s1   -     a(32) - a(33) - a(34) - a(35) - a(36)
a(19) =  s1   -     a(20) - a(21) - a(22) - a(23) - a(24)
a(11) =             a(12) - a(35) + a(36)
a( 8) = (s1   - 2 * a(12) + a(31) - a(32) - a(33) - a(34) + a(35) - a(36))/2
a( 7) =  s1   -     a( 8) - a( 9) - a(10) - a(11) - a(12)
a( 4) =  s1/3 -     a(10) + a(21) - a(22) - a(28) + a(33)

The solutions can be obtained by guessing the 12 parameters:

      a(i) for i = 10, 12, 20 ... 24, 28, 32, 34 ... 36

and filling out these guesses in the abovementioned equations.

With an optimized guessing routine (MgcSqr61112) numerous Symmetric Magic Squares can be produced.

Attachment 6.11.12 shows the first occuring Symmetric Magic Squares for a(36) = i (i = 1 ... 36).

6.11.14 Almost Associated Magic Squares

When the equations defining the Almost Associated Property as illustrated below:

are added to the equations describing a Simple Magic Square of the sixth order (ref. Section 6.2), the resulting Symmetric Magic Square can be described by following set of linear equations:

a(31) =      s1   - a(32) - a(33) - a(34) - a(35) - a(36)
a(26) =  4 * s1/6 - a(27) - a(28) - a(29)
a(21) =             a(22) - a(27) + a(28) - a(33) + a(34)
a(20) =- 4 * s1/6 + a(23) + a(27) + a(28) + 2 * a(29)
a(19) = 10 * s1/6 - 2 * a(22) - 2 * a(23) - a(24) - 2 * a(28) - 2 * a(29) + a(33) - a(34)
a(12) =  2 * s1/6 + a(19) - a(24) - a(30) + a(31) - a(36)

The solutions can be obtained by guessing the 12 parameters:

      a(i) for i = 22 ... 24, 27 ... 30, 32 ... 36

and filling out these guesses in the abovementioned equations.

With an optimized guessing routine (MgcSqr61113) numerous Symmetric Magic Squares can be produced.

Attachment 6.11.13 shows the first occuring Symmetric Magic Squares for a(36) = i (i = 1 ... 36).

6.11.15 Square of the Sun

When the equations defining the symmetry of the 'Square of the Sun' as shown below:

are added to the equations describing a Simple Magic Square of the sixth order (ref. Section 6.2), the resulting Symmetric Magic Square can be described by following set of linear equations:

a(31) =      s1   - a(32) - a(33) - a(34) - a(35) - a(36)
a(11) =      s1   - a( 5) - a(17) - a(23) - a(29) - a(35)
a( 4) =  2 * s1/6 + a(21) - a(22) - a(34)
a(27) = 10 * s1/6 - a(28) - a(17) - a(23) - 2 * a(29) - a(5) - a(35) - a(1) - a(36)
a(19) =  8 * s1/6 - a(24) - a(21) - a(22) - 2 * a( 1) - 2 * a(36)
a(12) =  2 * s1/6 - a(30) - a(32) - a(35) + 2 * a( 1)

The solutions can be obtained by guessing the 12 parameters:

      a(i) for i = 17, 21 ... 24, 28 ... 30, 32, 34 ... 36

and filling out these guesses in the abovementioned equations.

With an optimized guessing routine (MgcSqr61114) numerous Symmetric Magic Squares can be produced.

Attachment 6.11.14 shows the first occuring Symmetric Magic Squares for a(36) = i (i = 1 ... 36).

A full enumeration has been executed by Francis Gaspalou, who found 60.207.144.960 'Squares of the Sun'.

6.11.16 Summary

The obtained results regarding the miscellaneous types of order 6 Magic Squares as deducted and discussed in previous sections are summarized in following table:

Comparable routines as listed above, can be used to generate order 6 Pan Magic Squares for non consecutive integers, which will be described in following sections.