Office Applications and Entertainment, Magic Squares  
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19.0 Special Magic Squares, Bent Diagonals
The concept of Bent Diagonal Magic Squares as introduced in Section 8.4 (Franklin Squares) will be further elaborated in this chapter.
The Franklin Squares are an example of 4 way Bent Diagonal (Semi) Magic Squares.
as all other possible cases can be obtained by means of rotation and (or) reflection.
19.2 Magic Squares (4 x 4)
Although order 4 Bent Diagonal Magic Squares can be filtered from the 7040 Magic Squares as found in Section 2.2,
the defining equations for the applicable case(s) will be deducted below for a better understanding.
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 (ref. Section 2.2), 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 and complies with Dudeney Group 6.
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.
19.3 Magic Squares (5 x 5) 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 (ref. Section 3.2.2), 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:
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 Simple Magic Square of the sixth order (ref. Section 6b.2.2), 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(25) = s1  a(26)  a(27)  a(28)  a(29)  a(30) a(21) = a(22)  a(26) + a(29)  a(31) + a(36) a(19) = s1  a(20)  a(21)  a(22)  a(23)  a(24) a(14) = 2*s1  a(15)  2*a(17)  2*a(18)  a(20)  a(22)  2*a(23)  2*a(24) + a(26)  a(29) + a(31)  a(36) a(13) =  a(16) + a(17) + a(18)  a(19)  a(21) + a(23) + a(24)  a(26) + a(29)  a(31) + a(36) a(12) = s1 + a(16)  a(17)  2 * a(18) + a(21)  a(23)  2 * a(24) + a(26)  a(29)  a(30) + a(31)  a(36) a(11) = s1  a(17)  a(18)  a(23)  a(24)  a(29) a(10) = s1  a(16)  a(17)  a(22)  a(23)  a(28) a( 9) = s1  a(15)  a(16)  a(21)  a(22)  a(27) a( 8) =  s1 + 2 * a(17) + 2*a(18)  a(21) + a(22) + 2*a(23) + 2*a(24)  2*a(26) + a(29)  a(31) + a(36) a( 7) =  s1 + a(15) + a(16) + a(17) + a(18) + a(21) + a(22) + a(23) + a(24)  a(25) a( 6) =  a(16) + a(17) + a(18)  a(21) + a(23) + a(24)  a(26) + a(29)  a(31) a( 5) = a(18) + a(24)  a(35) a( 4) = a(17) + a(23)  a(34) a( 3) = a(16) + a(22)  a(33) a( 2) = a(15) + a(21)  a(32) a( 1) = 2*s1  a(15)  2 * a(17)  2 * a(18)  a(22)  2 * a(23)  2 * a(24) + a(26)  a(29)  a(36)
The linear equations shown above are ready to be solved, for the magic constant s1 = 111.
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 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(25) = s1  a(26)  a(27)  a(28)  a(29)  a(30) a(21) = a(22)  a(26) + a(29)  a(31) + a(36) a(17) = 2*s1/3  a(18)  a(23)  a(24) a(16) = a(18)  a(22) + a(24) a(15) = 2*s1/3  a(18)  a(22)  a(24) + a(26)  a(29) + a(31)  a(36) a(14) = a(18)  a(20) + a(24) a(13) =  s1/3  a(18) + a(20) + 2 * a(22) + a(23)  a(26) + a(29)  a(31) + a(36) a( 6) = 2*s1/3  a(18)  a(24)  a(36) a( 5) = a(18) + a(24)  a(35) a( 4) = 2*s1/3  a(18)  a(24)  a(34) a( 3) = a(18) + a(24)  a(33) a( 2) = 2*s1/3  a(18)  a(24)  a(32) a( 1) = a(18) + a(24)  a(31)
The consequential symmetry  Row 2 and 5 are Axial Symmetrical  is worth to be noticed.
19.4.3 Two Way Bent Diagonal
Axial Symmetry Magic Squares are  per definition  Two Way Bent Diagonal.
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(21) = a(22)  a(26) + a(29)  a(31) + a(36) a(19) = s1  a(20)  2 * a(22)  a(23)  a(24) + a(26)  a(29) + a(31)  a(36)
The solutions can be obtained by guessing the 14 parameters:
19.4.4 One Way Bent Diagonal 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:
19.4.5 Two Way Bent Diagonal 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.
19.6 Magic Squares (7 x 7)
Order 7 Four Way Bent Diagonal Magic Squares  as published by Harry White in 2002  have been discussed in detail in Section 7.9.
19.6.2 Two Way Bent Diagonal
It can be proven that Four Way Bent Diagonal Magic Squares can't be Associated.
a(1) + a( 9) + a(17) + a(25) + a(31) + a(37) + a(43) = s1 a(2) + a(10) + a(18) + a(26) + a(32) + a(38) + a(44) = s1 a(3) + a(11) + a(19) + a(27) + a(33) + a(39) + a(45) = s1 a(4) + a(12) + a(20) + a(28) + a(34) + a(40) + a(46) = s1 a(5) + a(13) + a(21) + a(22) + a(35) + a(41) + a(47) = s1 a(6) + a(14) + a(15) + a(23) + a(29) + a(42) + a(48) = s1 a(7) + a( 8) + a(16) + a(24) + a(30) + a(36) + a(49) = s1 a(1) + a(14) + a(20) + a(26) + a(34) + a(42) + a(43) = s1 a(2) + a( 8) + a(21) + a(27) + a(35) + a(36) + a(44) = s1 a(3) + a( 9) + a(15) + a(28) + a(29) + a(37) + a(45) = s1 a(4) + a(10) + a(16) + a(22) + a(30) + a(38) + a(46) = s1 a(5) + a(11) + a(17) + a(23) + a(31) + a(39) + a(47) = s1 a(6) + a(12) + a(18) + a(24) + a(32) + a(40) + a(48) = s1 a(7) + a(13) + a(19) + a(25) + a(33) + a(41) + a(49) = s1 are added to the equations describing an Ultra Magic Square of the seventh order (ref. FSection 7.3), the resulting Two Way Bent Diagonal Ultra Magic Square is described by following equations: 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
The solutions can be obtained by guessing the 9 parameters:
19.7 Magic Squares (8 x 8)
Order 8 Four Way Bent Diagonal (Semi) Magic Squares (Franklin Squares) have been discussed in detail in Section 8.4.
19.8 Magic Squares (9 x 9) When the equations defining the horizontal and vertical Bent Diagonals of a nineth order Magic Square, are added to the equations describing an Associated Magic Square of the nineth order, the resulting Four Way Bent Diagonal Magic Square is described by following equations: a(73) = s1  a(74)  a(75)  a(76)  a(77)  a(78)  a(79)  a(80)  a(81) a(68) =  4*s1/9 + a(74) + a(75) + a(77) + a(79) + a(80) a(65) = 8*s1/9  a(66)  a(67)  a(69)  a(70)  a(71)  a(75)  a(79) a(64) = 5*s1/9  a(72)  a(74)  a(77)  a(80) a(63) = 4*s1/9  a(71)  a(79)  a(81) a(60) = 21*s1/9  a(61)  a(62)  2*a(69)  3*a(70)  2*a(71)  a(72) + 0.5*a(74) + 0.5*a(75) + 0.5*a(76) +  a(77)  2.5*a(78)  3.5*a(79)  3.5*a(80)  a(81) a(59) = 6*s1/9  a(67)  a(69)  a(75)  a(77)  a(79) a(58) = a(60)  a(66)  a(67) + a(69) + a(70)  a(74)  a(75)  a(76) + a(78) + a(79) + a(80) a(57) = 17*s1/9 + a(61) + 2*a(69) + 2*a(70) + 2*a(71) + a(77) + 2*a(78) + 4*a(79) + 2*a(80) + 2*a(81) a(56) = 13*s1/9 + a(62) + a(67) + a(69) + 2*a(70) + 2*a(71) + 2*a(72)  a(74)  a(76) + a(77) + + a(78) + 2*a(79) + 3*a(80) a(55) = 13*s1/9 + a(66) + a(67) + a(69) + a(70) + a(71) + a(74) + a(75) + a(76) + a(77) + a(78) + + 2*a(79) + a(80) + a(81) a(54) = 6*s1/9  a(62)  a(70)  a(72)  a(78)  a(80) a(53) = s1/9  a(61)  a(69) + a(72) + a(80) a(52) = 5*s1/9  a(60)  a(62)  a(70)  a(72)  a(80) + a(81) a(51) = 4*s1/9  a(61)  a(71)  a(81) a(50) = 14*s1/9  2*a(58)  2*a(66)  a(67) + a(69)  3*a(74)  2*a(75)  2*a(76)  a(77)  a(80) a(49) = 4*s1/9  a(57)  a(65)  a(73) a(48) = (2*s1/9 + 2*a(61) + a(74)  a(75) + a(76)  a(78)  a(79)  a(80))/2 a(47) = 23*s1/9  a(61)  a(67)  2*a(69)  2*a(70)  2*a(71)  a(72)  2*a(77)  2*a(78) +  4*a(79)  3*a(80)  2*a(81) a(46) = 14*s1/9  a(62)  a(66)  a(67)  a(69)  2*a(70)  2*a(71)  a(72) + a(74)  a(78)  2*a(79)  2*a(80) a(45) = 6*s1/9  2*a(72)  a(77)  2*a(80) a(44) = 18*s1/9  a(66)  a(67)  a(69)  a(70)  2*a(71)  a(75)  a(76)  a(77)  a(78) +  3*a(79)  2*a(80)  2*a(81) a(43) = s1/9 + a(66)  a(70) + a(74) + a(75) + a(76)  a(78)  a(79)  a(80) a(42) = s1/9 + a(67)  a(69) + a(75) + a(76)  a(78)  a(79) a(41) = s1/9
The resulting Four Way Bent Diagonal Associated Magic Square appears to be Pan Diagonal as well (Ultra Magic).
19.9 Magic Squares (12 x 12)
Order 12 Four Way Bent Diagonal Magic Squares (Franklin Like) have been discussed in detail in Section 12.2.1.
19.10 Magic Squares (16 x 16)
Order 16 Four Way Bent Diagonal Magic Squares (Franklin Squares) have been discussed in detail in Section 12.3.1.
The obtained results regarding the miscellaneous types of Bent Diagonal Magic Squares as deducted and discussed in previous sections are summarized in following table: 
Order
Characteristics
Subroutine
Results
4
Two Way Bent Diagonal
5
One Way Bent Diagonal
6
One Way Bent Diagonal
Two Way Bent Diagonal
Two Way Bent Diagonal, Axial Symmetric
One Way Bent Diagonal, Rect Compact
Two Way Bent Diagonal, Rect Compact
7
Two Way Bent Diagonal, Ultra Magic
9
Four Way Bent Diagonal, Ultra Magic
Comparable routines as listed above, can be used to generate Lozenge Squares, which will be described in following sections.

Index  About the Author 