Office Applications and Entertainment, Magic Squares

14.0    Special Magic Squares, Prime Numbers

14.4.30 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, 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 for miscellaneous Magic Sums (24 ea) the first occurring Prime Number Symmetric Magic Square.

Each square shown corresponds with numerous squares for the applicable Magic Sum and variable values {a_i}.

14.4.31 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, 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 for miscellaneous Magic Sums (24 ea) the first occurring Prime Number Symmetric Magic Square.

Each square shown corresponds with numerous squares for the applicable Magic Sum and variable values {a_i}.

14.4.32 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, 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 for miscellaneous Magic Sums (24 ea) the first occurring Prime Number Symmetric Magic Square.

Each square shown corresponds with numerous squares for the applicable Magic Sum and variable values {a_i}.

14.4.33 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, 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 for miscellaneous Magic Sums (24 ea) the first occurring Prime Number Symmetric Magic Square.

Each square shown corresponds with numerous squares for the applicable Magic Sum and variable values {a_i}.

14.4.34 Square of the Sun

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

are added to the equations describing a Simple Magic Square of the sixth order, 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 for miscellaneous Magic Sums (24 ea) the first occurring Prime Number Symmetric Magic Square.

Each square shown corresponds with numerous squares for the applicable Magic Sum and variable values {a_i}.

14.4.35 Summary

The obtained results regarding the miscellaneous types of order 6 Prime Number 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 Rectangular Compact Magic Squares, which will be described in following sections.