Office Applications and Entertainment, Magic Squares

7.6   Analytic Solution, Miscellaneous Inlays

7.6.1 Bordered, Center Square with Diamond or Square Inlay

Magic Squares of the 5^th order with Inlays, as described and constructed in Section 5.6, can be used as Center Squares for 7^th order Bordered Magic Squares.

When the Embedded Magic Squares are based on the consecutive integers 13, 14, ... 37, each 5^th order Magic Square will result in n7 = 21312000 Bordered Magic Squares of order 7 (ref. Section 7.5.1).

The resulting number for each of the described types are summarized below:

Type	Number
Magic/Diamond Inlay	n7 * 8288
Associated/Diamond Inlay	n7 * 496
Concentric/Diamond Inlay	n7 * 2592
Magic/Square Inlay	n7 * 1393920
Magic/Square + Diamond Inlay	n7 * 1440

Attachment 7.7.1 contains for the first occurring border, one example of each of the possible Bordered Magic Squares for Center Squares with Diamond or Square Inlay.

7.6.2 Associated, Diamond Inlays of Order 3 and 4

The 3^th and 4^th order Diamond Inlays of a 7^th order Magic Square can be described by following equations:

which can be added to the equations describing an Associated Square of the 7^th order.

The resulting equations, defining an Associated Magic Square with Diamond Inlays of Order 3 and 4 can be written as:

a(46) =  100 - a(40) - a(34) - a(28)
a(45) =   25 + a(47) - 2 * a(27) + a(26) - a(20) + a(46) + a(40) - a(28)
a(43) =  175 - a(44) - a(45) - a(47) - a(48) - a(49) - a(46)
a(39) =   75 - a(33) - a(27)
a(38) =        a(20) - a(46) + a(28)
a(37) = - 75 + a(41) - a(44) + a(48) + a(27) + a(20) + a(34)
a(36) =  175 - 2 * a(41) - a(42) + a(44) - a(48) + a(33) - 2 * a(20) + a(46) - a(40) - a(34) - a(28)
a(35) = (425 - 2 * a(41) - 2 * a(42) - 2 * a(47) - 2 * a(48) - 2 * a(49) - a(33) + 2 * a(20) - 2 * a(46) +
                                                                                 - 2 * a(40) - 2 * a(34)) / 2
a(32) =  100 - a(26) - 2 * a(20) + a(46) - a(28)
a(29) =   75 - a(35) - 2 * a(33) - 2 * a(27) + a(26) + 3 * a(20) - a(46) - a(34) + a(28)
a(26) =  100 - a(20) -     a(34) -     a(28)
a(25) =   25
a(19) =  100 - a(33) - 2 * a(27)

with the following independent variables:

• a(27), a(33) determining the 3^th order Diamond,
• a(20), a(28), a(34), a(40) determining the 4^th order Diamond and
• a(41), a(42), a(44), a(47), a(48), a(49) determining the remainder of the Associated Magic Square.

An optimized guessing routine (MgcSqr7j1) produced, with

• the 3^th order Diamond based on the consecutive integers {21 ... 29} and
• the 4^th order Diamond based on the integers {13 ... 20; 30 ... 37}
  

2400 Magic Squares with Diamond Inlays of order 3 and 4 within 1,55 hours, of which the first 44 are shown in Attachment 7.7.2.

It should be noted that much more Magic Squares with Diamond Inlays can be generated with routine MgcSqr7j1, when the base for the Diamonds is not limited to the integers defined above.

An alternative method to generate Associated Magic Squares with order 3 and 4 Diamond Inlays will be discussed in Section 18.3.2.

7.6.3 Associated, Square Inlays of Order 3 and 4

The 3^th and 4^th order Square Inlays of a 7^th order Magic Square can be described by following equations:

which can be added to the equations describing a Magic Square of the 7^th order, and result in following set of linear equations:

a(44) = 175 - a(46) - a(48) - a(43) - a(45) - a(47) - a(49)
a(43) = 100 - a(45) - a(47) - a(49)
a(37) =  75 - a(39) - a(41)
a(36) = 100 - a(38) - a(40) - a(42)
a(33) =  50 - a(35) + 0.5 * a(43) + 0.5 * a(45) - 0.5 * a(47) - 0.5 * a(49)
a(32) = 150 - 2 * a(34) - a(46) - 2 * a(48)
a(31) =       a(33) - a(45) + a(47)
a(30) =  75 - a(32) - a(34)
a(29) = 100 - 2 * a(33) - a(35) + a(45) - a(47)
a(28) = 125 - a(38) - a(40) - 2 * a(42)
a(27) = 100 - a(39) - 2 * a(41)
a(26) =  25 + a(38) - a(40)
a(25) =  25

with the following independent variables:

• a(39), a(41) determining the 3^th order Square,
• a(35), a(45), a(47), a(49) determining the 4^th order Square and
• a(34), a(38), a(40), a(42), a(46), a(48) determining the remainder of the Associated Magic Square.

An optimized guessing routine (MgcSqr7j2) produced, with both the 3^th and 4^th order Square Inlays constant, 2560 Associated Magic Squares within 130 seconds, of which the first 48 are shown in Attachment 7.7.3.

An alternative method to find subject squares, based on transformation of Composed Magic Squares, will be discussed in Section 7.8.1.

7.6.4 Ultra Magic, Order 3 Concentric Square and Square Inlay (1)

When the equations describing a 3^th order Concentric Square and a 3^th order Square Inlay are added to the equations describing an Ultra Magic Square of the 7^th order (Section 7.4), following set of linear equations will result:

a(46) = 175 - 2 * a(48) - a(49) - a(32) - a(33) - a(41)
a(44) =-150 + a(45) + a(47) + a(48) + a(49) + a(33) + a(39) + a(41)
a(43) = 150 - 2 * a(45) - 2 * a(47) - a(49) + a(32) - a(39)
a(42) =(-25 + 2 * a(45) + a(32) + 2 * a(33) - 2 * a(41))/2
a(40) =(125 - 2 * a(47) - a(39))/2
a(38) =       a(40) - a(45) + a(47)
a(37) =  75 - a(39) - a(41)
a(36) = -25 - a(42) + a(45) + a(47) + a(39)
a(35) = 150 - a(42) - a(47) - a(48) - a(49) - a(41)
a(34) =  75 - a(35) - a(40) - a(47) + a(41)
a(31) =  75 - a(32) - a(33)
a(30) = -75 - a(36) + a(38) + a(45) + a(47) + a(48) - a(33) + a(39) + a(41)
a(29) = 100 - a(30) - a(34) - a(35)
a(28) = 100 - 2 * a(45) - a(49) - a(33) + a(41)
a(27) = 100 - a(39) - 2 * a(41)
a(26) = 100 - a(32) - 2 * a(33)
a(25) =  25

with the following independent variables:

• a(32), a(33) determining the 3^th order Concentric Square,
• a(39), a(41) determining the 3^th order Square Inlay and
• a(45), a(47), a(48), a(49) determining the remainder of the Ultra Magic Square.

An optimized guessing routine (MgcSqr7j3), produced the 8 possible Ultra Magic Squares with Order 3 Concentric Square and Square Inlay within 15,5 hours, which are shown in Attachment 7.7.4.

7.6.5 Ultra Magic, Order 3 Concentric Square and Square Inlay (2)

Another possible 3^th order Square Inlay (Type 2) can be defined by following equations:

which can be added to the equations describing an Ultra Magic Square of the 7^th order (Section 7.4), and result in following set of linear equations:

a(45) = (175 - 2 * a(48) - a(32) - 2 * a(33)) / 2
a(44) =  100 - a(45) - a(47) - a(48)
a(43) =   75 - a(46) - a(49)
a(40) = (225 - 2 * a(48) - 2 * a(32) - 2 * a(33) - a(46)) / 2
a(39) =   75 - a(41) - 2 * a(47) + a(32) + a(33) - a(49)
a(38) = -175 + a(40) + a(45) + a(47) + 2 * a(48) + a(32) + 2 * a(33)
a(37) = -175 + a(41) + 2 * a(47) + 2 * a(48) + a(46) + 2 * a(49)
a(36) =  225 - a(41) - a(42) - a(45) - a(47) - 2 * a(48) - a(33) - a(49)
a(35) =  150 - a(41) - a(42) - a(47) - a(48) - a(49)
a(34) =   75 - a(40) - a(42) - a(48) + a(49)
a(31) =   75 - a(32) - a(33)
a(30) = -225 + a(40) + a(41) + a(42) + a(45) + a(47) + 3 * a(48) + a(32) + a(33)
a(29) = - 75 + a(42) + a(45) + a(48) + a(33)
a(28) =  100 - a(46) - 2 * a(49)
a(27) = -125 + a(41) + 2 * a(42) + 2 * a(47) + 2 * a(48) - a(32) - a(33) + a(49)
a(26) =  100 - a(32) - 2 * a(33)
a(25) =   25

with the following independent variables:

• a(32), a(33) determining the 3^th order Concentric Square,
• a(46), a(49) determining the 3^th order Square Inlay and
• a(41), a(42), a(47), a(48) determining the remainder of the Ultra Magic Square.

An optimized guessing routine (MgcSqr7j4), produced the 8 possible Ultra Magic Squares with Order 3 Concentric Square and Square Inlay within 495 seconds, which are shown in Attachment 7.7.5.

7.6.6 Ultra Magic, Order 3 Square Inlays

When the equations of each of the two described order 3 Square Inlays are added to the equations describing an Ultra Magic Square of the 7^th order (Section 7.4), following set of linear equations will result:

a(47) = (250 - 2 * a(48) - a(39) - 2 * a(41) - a(46) - 2 * a(49))/2
a(44) =  100 - a(45) - a(47) - a(48)
a(43) =   75 - a(46) - a(49)
a(40) =  (25 - 2 * a(42) + a(46) + 2 * a(49))/2
a(37) =   75 - a(39) - a(41)
a(36) =  100 - a(38) - a(40) - a(42)
a(35) =  150 - a(42) - a(47) - a(48) - a(41) - a(49)
a(34) =   75 - a(40) - a(42) - a(48) + a(49)
a(33) =        a(38) - a(40) - a(45) - a(47) + a(41) + a(46) + a(49)
a(32) =   25 - a(38) + a(40) + 2 * a(42) + a(45) + a(47) - 2 * a(46) - 2 * a(49)
a(31) = -100 - a(38) + a(40) - a(45) + a(47) + 2 * a(48) + a(41) + a(46) + a(49)
a(30) = - 50 + a(40) + a(42) + a(45) + a(47) + a(48) - a(46) - a(49)
a(29) =   75 + a(38) - a(40) - a(42) - a(47) - a(48) - a(41) + a(46) + a(49)
a(28) =  100 - a(46) - 2 * a(49)
a(27) =  100 - a(39) - 2 * a(41)
a(26) =  200 - a(38) - a(40) - 2 * a(42) + a(45) - a(47) - a(39) - 2 * a(41)
a(25) =   25

with the following independent variables:

• a(39), a(41) determining the 3^th order Square Inlay 1,
• a(46), a(49) determining the 3^th order Square Inlay 2 and
• a(38), a(42), a(45), a(48) determining the remainder of the Ultra Magic Square.

An optimized guessing routine (MgcSqr7j5), produced the 8 possible Ultra Magic Squares with Order 3 Square Inlays within 37,5 minutes, which are shown in Attachment 7.7.6.

7.6.7 Ultra Magic, Order 3 Concentric Square and Diamond Inlay

When the equations describing a 3^th order Concentric Square and a 3^th order Diamond Inlay are added to the equations describing an Ultra Magic Square of the 7^th order (Section 7.4), following set of linear equations will result:

a(43) =  175 - a(44) - a(45) - a(46) - a(47) - a(48) - a(49)
a(42) = (-75 + 2 * a(45) + a(33) + 2 * a(39))/2
a(41) =  150 + a(44) - a(45) - a(47) - a(48) - a(49) - a(33) - a(39)
a(40) = (175 - a(44) + a(45) - a(46) - a(47) - a(48) - 2 * a(39))/2
a(38) =        a(40) - a(45) + a(47)
a(37) = -125 + a(46) + 2 * a(48) + a(49) + a(33) + a(39)
a(36) = - 25 - a(42) + a(45) + a(47) + a(39)
a(35) =   75 + a(42) - a(44) - a(45) - a(39)
a(34) = - 50 - a(40) + a(42) + a(48) + a(49) + a(33)
a(32) = - 25 + 2 * a(39)
a(31) =  100 - a(33) - 2 * a(39)
a(30) =  200 - a(40) - a(42) + a(45) - a(46) - a(47) - a(48) - a(49) - a(33) - a(39)
a(29) =   50 - a(42) + a(45) - a(48)
a(28) =  300 - 2 * a(45) - a(46) - 2 * a(48) - 2 * a(49) - 2 * a(33) - 2 * a(39)
a(27) =   75 - a(33) - a(39)
a(26) =  125 - 2 * a(33) - 2 * a(39)
a(25) =   25

with the following independent variables:

• a(33), a(39) determining the 3^th order Concentric Square and Diamond Inlay,
• a(44), a(45), a(46), a(47), a(48), a(49) determining the remainder of the Ultra Magic Square.

An optimized guessing routine (MgcSqr7j6), produced 16 Ultra Magic Squares with Order 3 Concentric Square and Diamond Inlay within 10,5 hours, which are shown in Attachment 7.7.7.

7.6.8 Ultra Magic, Order 3 Square and Diamond Inlay (1)

When the equations describing a 3^th order Square Inlay (Type 1) and a 3^th order Diamond Inlay are added to the equations describing an Ultra Magic Square of the 7^th order (Section 7.4), following set of linear equations will result:

a(44) =   50 + a(45) - a(46) + a(47) - a(48) - a(39)
a(43) =  125 - 2 * a(45) - 2 * a(47) - a(49) + a(39)
a(40) = (475 - 2 * a(42) + 2 * a(45) - 2 * a(46) - 2*a(47) - 4*a(48) - 2*a(49) - 3*a(39) - 4*a(41))/2
a(38) =  (25 - 2 * a(42) + a(39) + 2 * a(41))/2
a(37) =   75 - a(39) - a(41)
a(36) = -150 + a(42) - a(45) + a(46) + a(47) + 2 * a(48) + a(49) + a(39) + a(41)
a(35) =  150 - a(42) - a(47) - a(48) - a(49) - a(41)
a(34) =   50 - a(38) - a(42) + a(45) - a(46) + a(47) - a(48) + a(41)
a(33) = - 25 + 2 * a(41)
a(32) = -150 + 2 * a(42) - 2 * a(45) + a(46) + 2 * a(48) + a(49) + 2 * a(39) + a(41)
a(31) =  125 - 2 * a(39) - 2 * a(41)
a(30) = - 25 + a(38) + a(42) + a(48) - a(41)
a(29) =   50 - a(42) + a(45) - a(48)
a(28) = -100 - 2 * a(45) + a(46) + 2 * a(48) + 2 * a(39) + 2 * a(41)
a(27) =  100 - a(39) - 2 * a(41)
a(26) = - 50 + a(46) + 2 * a(48) + a(49) - a(41)
a(25) =   25

with the following independent variables:

• a(39), a(41) determining the 3^th order Square and Diamond Inlay,
• a(42), a(45), a(46), a(47), a(48), a(49) determining the remainder of the Ultra Magic Square.

An optimized guessing routine (MgcSqr7j7), produced 16 Ultra Magic Squares with Order 3 Square and Diamond Inlay within 1,5 hours, which are shown in Attachment 7.7.8.

7.6.9 Ultra Magic, Order 3 Square and Diamond Inlay (2)

When the equations describing a 3^th order Square Inlay (Type 2) and a 3^th order Diamond Inlay are added to the equations describing an Ultra Magic Square of the 7^th order (Section 7.4), following set of linear equations will result:

a(44) =  100 - a(45) - a(47) - a(48)
a(43) =   75 - a(46) - a(49)
a(41) =  150 - a(45) - 2 * a(47) - a(48) - a(49)
a(40) = (200 - 2 * a(42) + 2 * a(45) - 2 * a(48) - a(33) - 2 * a(39) - a(46))/2
a(37) = - 25 - a(45) + a(48) + a(46) + a(49)
a(38) =  100 - a(40) - 2 * a(42) + 2 * a(45) + a(47) - a(48) - a(39) - a(46)
a(36) = - 50 + a(42) + a(47) + a(48)
a(35) =      - a(42) + a(45) + a(47)
a(34) =   75 - a(40) - a(42) - a(48) + a(49)
a(32) = - 50 + 2 * a(42) - a(45) + a(48) + a(39)
a(31) =  100 - a(33) - 2 * a(39)
a(30) =        a(40) + a(42) - a(45) - a(47) + a(48) + a(39) - a(49)
a(29) =   50 - a(42) + a(45) - a(48)
a(28) =  100 - a(46) - 2 * a(49)
a(27) =   75 - a(33) - a(39)
a(26) =   25 + a(45) + a(48) - a(33) - a(39)
a(25) =   25

with the following independent variables:

• a(46), a(49) determining the 3^th order Square Inlay,
• a(33), a(39) determining the 3^th order Diamond Inlay and
• a(42), a(45), a(47), a(48) determining the remainder of the Ultra Magic Square.

An optimized guessing routine (MgcSqr7j8), produced 80 Ultra Magic Squares with Order 3 Square and Diamond Inlay within 9,5 hours, which are shown in Attachment 7.7.9.

7.6.10 Spreadsheet Solutions

The linear equations deducted above, have been applied in following Excel Spread Sheets:

• CnstrSngl7f1, Diamond Inlays Order 3 and 4
• CnstrSngl7f2, Square  Inlays Order 3 and 4
• CnstrSngl7f3, Order 3 Concentric Square and Square Inlay (1)
• CnstrSngl7f4, Order 3 Concentric Square and Square Inlay (2)
• CnstrSngl7f5, Order 3 Square Inlays
• CnstrSngl7f6, Order 3 Concentric Square and Diamond Inlay
• CnstrSngl7f7, Order 3 Square and Diamond Inlay (1)
• CnstrSngl7f8, Order 3 Square and Diamond Inlay (2)

Only the red figures have to be “guessed” to construct one of the applicable 7^th order Magic Squares (wrong solutions are obvious).