Office Applications and Entertainment, Magic Squares

19.0   Special Magic Squares, Prime Numbers, Bent Diagonals

19.1   Introduction

The concept of Prime Number Bent Diagonal Magic Squares as introduced in Section 14.6.16 (Franklin Squares) will be further elaborated in this chapter.

The Franklin Squares are an example of 4 way Bent Diagonal (Semi) Magic Squares.

Only following cases have to be considered:

• One Way   Left to Right with wrap around
• Two Way   Left to Right with wrap around and reverse
• Two Way   Left to Right with wrap around and
            Top to Bottom with wrap around
• Three Way Left to Right with wrap around and reverse and
            Top to Bottom with wrap around
• Four Way  Left to Right with wrap around and reverse and
            Top to Bottom with wrap around and reverse

as all other possible cases can be obtained by means of rotation and (or) reflection.

19.2   Magic Squares (4 x 4)
       Two Way Bent Diagonal

When the equations defining the Left to Right Bent Diagonals of a fourth order Magic Square:

a(1) + a(6) + a(10) + a(13) = s1
a(2) + a(7) + a(11) + a(14) = s1
a(3) + a(8) + a(12) + a(15) = s1
a(4) + a(5) + a( 9) + a(16) = s1

are added to the equations describing a Simple Magic Square of the fourth order, the resulting Bent Diagonal Magic Square is described by following equations:

a(13) =   s1 -   a(14) - a(15) - a(16)	
a(10) =          a(11) - a(13) + a(16)	
a( 9) = 2*s1 - 2*a(11) - a(12) - a(14) - a(15) - 2*a(16)	

a(8) = 0.5 * s1 - a(12)     a(4) = 0.5 * s1 - a(16)
a(7) = 0.5 * s1 - a(11)     a(3) = 0.5 * s1 - a(15)
a(6) = 0.5 * s1 - a(10)     a(2) = 0.5 * s1 - a(14)
a(5) = 0.5 * s1 - a( 9)     a(1) = 0.5 * s1 - a(13)

The consequential symmetry (Axial Symmetrical) is worth to be noticed.

Further it can be deducted that:

a(1) = s1 - a(8) - a(12) - a(13)
a(2) = s1 - a(5) - a( 9) - a(14)
a(3) = s1 - a(6) - a(10) - a(15)
a(4) = s1 - a(7) - a(11) - a(16)

which are the defining equations for the Right to Left Bent Diagonals. Consequently the resulting Magic Square is Two Way Bent Diagonal.

The solutions can be obtained by guessing a(11), a(12), a(14) a(15), a(16) and filling out these guesses in the equations deducted above.

A routine can be written to generate subject Magic Squares of order 4 (ref. BentDia41).

Attachment 19.2.1 shows for a few Magic Sums the first occurring Prime Number Bent Diagonal Magic Square.

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

Note
Due to the Axial Symmetry Three Way and Four Way Bent Diagonal Magic Squares of order 4 can't exist.

19.3   Magic Squares (5 x 5)
       One Way Bent Diagonal

When the equations defining the Left to Right Bent Diagonals of a fifth order Magic Square:

a(1) + a( 7) + a(13) + a(17) + a(21) = s1
a(2) + a( 8) + a(14) + a(18) + a(22) = s1
a(3) + a( 9) + a(15) + a(19) + a(23) = s1
a(4) + a(10) + a(11) + a(20) + a(24) = s1
a(5) + a( 6) + a(12) + a(16) + a(25) = s1

are added to the equations describing a Simple Magic Square of the fifth order, the resulting One Way Bent Diagonal Magic Square is described by following equations:

a( 1) =        s1 - a( 7) - a(13) - a(19) - a(25)
a(17) =        s1 - a(21) - a(13) - a( 7) - a( 1)
a( 9) =  0.2 * s1 - a(17) - 0.5 * a(21) + 0.5 * a(1) + a(25)
a( 5) = -0.4 * s1 + a( 9) + a(17) + a( 7) + a(19) - a(25)
a(12) = -0.2 * s1 + a(21) + a( 1)
a(11) = -0.6 * s1 + a( 9) + a(17) + a( 7) + a(19)
a(23) =        s1 - a(19) - a(15) - a( 9) - a( 3)
a(14) =  0.8 * s1 - a(15) - a( 9) - a(19)
a(18) =           - a( 8) + a(15) + a( 9) - a(13) + a(19)
a(22) =  0.2 * s1 - a( 2) + a(13)
a( 4) =  0.4 * s1 - a( 2) - a( 3) - a( 9) + a(21) + a(13) - a(19) + a(25)
a(24) = -0.2 * s1 + a( 2) + a(15) + a( 3) + a( 9) - a(21) - a(13) + a(19) - a(25)
a(20) =  1.4 * s1 - a(10) - a(15) - a( 9) - a(17) - a( 7) - a(19)
a(16) = -0.4 * s1 + a(10) + a( 8) + a( 7) + a(13) - a(19)
a( 6) =        s1 - a(10) - a( 8) - a( 9) - a( 7)

The solutions can be obtained by guessing the 10 parameters:

      a(i) for i = 2. 3, 7, 8, 10, 13, 15, 19, 21, 25

and filling out these guesses in the equations deducted above.

A routine can be written to generate subject Magic Squares of order 5 (ref. BentDia51).

Attachment 19.3.1 shows for a few Magic Sums the first occurring Prime Number Bent Diagonal Magic Square.

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

Notes
It can be proven that One Way Bent Diagonal Pan Magic Squares of order 5 can't exist.
It can be proven that One Way Bent Diagonal Associated Magic Squares of order 5 can't exist.
It can be proven that Two Way Bent Diagonal Magic Squares of order 5 can't exist.

19.4   Magic Squares (6 x 6)

Comparable routines as deducted above can be written to generate Prime Number One and Two Way Bent Diagonal Magic Squares of order 6.

However such routines are not very feasible due to the high number of independent variables (resp 18 and 15 ea).

In following sections solutions will be found for more strict defined Prime Number Bent Diagonal Magic Squares of the sixth order.

19.4.3 Two Way Bent Diagonal
       Axial Symmetric, Pan Magic

Axial Symmetry Magic Squares are - per definition - Two Way Bent Diagonal.

Two Way Bent Diagonal Axial Symmetric Pan Magic Squares of the sixth order are described by following 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(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)

The solutions can be obtained by guessing the 10 parameters:

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

and filling out these guesses in the equations deducted above.

A routine can be written to generate subject Magic Squares of order 6 (ref. BentDia65).

Attachment 19.4.5 shows for a few Magic Sums the first occurring Prime Number Bent Diagonal Pan Magic Square.

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

19.4.4 One Way Bent Diagonal
       Rectangular Compact

When the equations defining the Left to Right Bent Diagonals of a sixth order Magic Square:

a(1) + a( 8) + a(15) + a(21) + a(26) + a(31) = s1
a(2) + a( 9) + a(16) + a(22) + a(27) + a(32) = s1
a(3) + a(10) + a(17) + a(23) + a(28) + a(33) = s1
a(4) + a(11) + a(18) + a(24) + a(29) + a(34) = s1
a(5) + a(12) + a(13) + a(19) + a(30) + a(35) = s1
a(6) + a( 7) + a(14) + a(20) + a(25) + a(36) = s1

are added to the equations describing a Rectangular Compact Magic Square of the sixth order (ref. Section 6.10), the resulting One Way Bent Diagonal Magic Square is described by following 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(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)
a(17) = 5*s1/3 - a(18) + a(24) - 2 * a(32) - 2 * a(34) - a(35) - 4 * a(36)
a(15) =          a(18) - a(33) + a(36)
a(14) = 5*s1/3 - a(18) + a(24) - 3 * a(32) - 2 * a(34) - 4 * a(36)
a(12) = 2*s1/3 - a(18) - a(24) - a(30)
a(10) = - s1   + a(18) + a(24) + a(29) + a(30) + a(34) + a(35) + a(36)
a( 9) = 2*s1/3 - a(18) - a(24) - a(30) + a(33) - a(36)
a( 7) =          a(18) + a(24) + a(29) + a(30) - a(32) - a(33) - a(34)
a( 5) =          a(18) + a(24) - a(35)
a( 4) = 2*s1/3 - a(18) - a(24) - a(34)
a( 2) =          a(18) + a(24) - a(32)
a( 1) = - s1/3 - a(18) - a(24) + a(32) + a(33) + a(34) + a(35) + a(36)

The consequential symmetry is worth to be noticed and results in 2 additional Bent Diagonals from Right to Left, as:

      a(6) + a(11) + a(16) + a(22) + a(29) + a(36) = s1
      a(3) + a( 8) + a(13) + a(19) + a(26) + a(33) = s1

The solutions can be obtained by guessing the 9 parameters:

      a(i) for i = 18, 24, 29, 30, 32 ... 36

and filling out these guesses in the equations deducted above.

A routine can be written to generate subject Magic Squares of order 6 (ref. BentDia63).

Attachment 19.4.3 shows for a few Magic Sums the first occurring Prime Number Bent Diagonal Magic Square.

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

19.4.5 Two Way Bent Diagonal
       Rectangular Compact

When the equations defining the Right to Left Bent Diagonals of a sixth order Magic Square:

a(6) + a(11) + a(16) + a(22) + a(29) + a(36) = s1
a(5) + a(10) + a(15) + a(21) + a(28) + a(35) = s1
a(4) + a( 9) + a(14) + a(20) + a(27) + a(34) = s1 
a(3) + a( 8) + a(13) + a(19) + a(26) + a(33) = s1
a(2) + a( 7) + a(18) + a(24) + a(25) + a(32) = s1
a(1) + a(12) + a(17) + a(23) + a(30) + a(31) = s1

are added to the equations describing a One Way Bent Diagonal Rectangular Compact Magic Square of the sixth order, as deducted in previous section, the resulting Two Way Bent Diagonal Magic Square is described by following 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(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)

The consequential symmetry (Axial Symmetrical) is worth to be noticed.

The solutions can be obtained by guessing the 8 parameters:

      a(i) for i = 24, 29, 30, 32 ... 36

and filling out these guesses in the equations deducted above.

A routine can be written to generate subject Magic Squares of order 6 (ref. BentDia64).

Attachment 19.4.4 shows for a few Magic Sums the first occurring Prime Number Bent Diagonal Magic Square.

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

19.6   Magic Squares (7 x 7)

19.6.1 Two Way Bent Diagonal
       Ultra Magic

Based on the equations defining seventh order Two Way Bent Diagonal Ultra Magic Squares:

a(43) =   s1     - a(44) - a(45) - a(46) - a(47) - a(48) - a(49)
a(39) = 2*s1     - 2*a(40) - 2*a(41) - 2*a(42) + 2*a(44) + a(45) - a(46) - 3*a(47) - 4*a(48) - 2*a(49)
a(38) =            a(40) - a(44) - a(45) + a(47) + a(48)
a(37) = - s1     + a(41) + a(46) + 2*a(47) + 2*a(48) + 2*a(49)
a(36) =            a(42) - a(44) + a(48)
a(35) = 6*s1/7   - a(41) - a(42) - a(47) - a(48) - a(49)
a(34) = - s1/7   - a(40) - a(42) + a(44) + a(45) + a(47) + a(49)
a(33) = - 8*s1/7 + 2*a(40) + a(41) + 2*a(42) - 2*a(44) - 2*a(45) + a(46) + 2*a(47) + 4*a(48) + a(49)
a(32) =   6*s1/7 - 2*a(40) + a(45) - a(46) - a(47) - 2*a(48)
a(31) = - 8*s1/7 + 2*a(40) + a(41) + 2*a(42) - a(44) - a(45) + a(46) + a(47) + 3*a(48) + a(49)
a(30) =   6*s1/7 - a(40) - a(42) + a(44) + a(45) - a(46) - a(47) - 2*a(48) - a(49)
a(29) =   6*s1/7 - a(41) - a(42) + a(44) - a(47) - 2*a(48) - a(49)
a(28) =   8*s1/7 - a(44) - a(45) - a(46) - a(47) - a(48) - 2*a(49)
a(27) =     s1/7 + a(44) - a(48)
a(26) =     s1/7 + a(45) - a(47)
a(25) =     s1/7 

a routine can be written to generate subject Magic Squares of order 7 (ref. BentDia72).

Attachment 19.6.2 shows for a few Magic Sums the first occurring Two Way Bent Diagonal Ultra Magic Square.

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

19.6.2 Four Way Bent Diagonal
       Symmetrical Axes and Main Diagonals

Based on the equations defining seventh order Four Way Bent Diagonal Magic Squares, with Symmetrical Axes and Main Diagonals:

a(44) =  4*s1/7 - a(45) - a(47) - a(48)
a(43) =  3*s1/7 - a(46) - a(49)
a(39) =  3*s1/7 - a(45) - a(47)
a(37) =  4*s1/7 - a(38) - a(40) - a(41)
a(36) =         - a(42) + a(45) + a(47)
a(33) =  3*s1/7 - a(41) - a(49)
a(32) =  2*s1/7 - a(38) - a(40) + a(45) - a(46) + a(47)
a(31) = -4*s1/7 + a(38) + a(40) + a(41) + a(46) + a(49)
a(30) =  5*s1/7 - a(34) - a(38) - a(40) - a(46)
a(29) =    s1/7 - a(35) + a(38) + a(40) - a(45) + a(46) - a(47)
a(28) =  4*s1/7 - a(46) - 2 * a(49)
a(26) = -5*s1/7 - a(27) + a(38) + a(40) + 2 * a(41) + a(46) + 2 * a(49)
a(25) =    s1/7
a(21) =  3*s1/7 - a(27) - a(35)
a(20) =  6*s1/7 - a(34) - a(38) - a(40) - 2 * a(41)
a(16) = -  s1   + a(34) + 2 * a(38) + 2 * a(40) + 2 * a(41) + a(46)
a(15) =           a(27) + a(35) - a(38) - a(40) + a(45) - a(46) + a(47)
a(14) =    s1/7 + a(27) - a(42)
a(12) = -5*s1/7 + a(38) + 2 * a(41) + a(46) + 2 * a(49)
a(10) =  9*s1/7 - 2 * a(38) - a(40) - 2 * a(41) - a(46) - 2 * a(49)
a( 8) =  3*s1/7 - a(27) + a(42) - a(45) - a(47)
a( 6) =  3*s1/7 - a(27) - a(48)
a( 5) =  8*s1/7 + a(27) - a(38) - a(40) - 2 * a(41) - a(46) - a(47) - 2 * a(49)
a( 3) = -4*s1/7 - a(27) + a(38) + a(40) + 2 * a(41) - a(45) + a(46) + 2 * a(49)
a( 2) =    s1/7 + a(27) - a(44)

a routine can be written to generate subject Magic Squares of order 7 (ref. BentDia71).

Attachment 19.6.1 shows for a few Magic Sums the first occurring Four Way Bent Diagonal Magic Square.

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

19.6.3 Additional Properties

Consequential properties resulting from the defining properties of 7^th order Four Way Bent Diagonal Magic Squares are summarised and illustrated in Attachment 19.6.3.

19.7   Magic Squares (8 x 8)
       Four Way Bent Diagonal

Order 8 Four Way Bent Diagonal (Semi) Magic Squares (Franklin Squares) have been discussed in detail in Section 14.6.16.

19.8   Summary

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

Comparable routines as listed above, can be used to generate Prime Number Magic Squares with Diamond Inlays, which will be described in following sections.