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5.6   Magic Squares, Miscellaneous Inlays

5.6.1 Diamond Inlay General

When an embedded Magic Square is rotated 45 degrees, the embedded square is referred to as a Diamond Inlay.

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

The 3th order Diamond Inlay of a 5th order Magic Square can be described by following linear equations:

 a( 3) + a( 7) + a(11) = 39 a( 9) + a(13) + a(17) = 39 a(15) + a(19) + a(23) = 39 a( 3) + a( 9) + a(15) = 39 a( 7) + a(13) + a(19) = 39 a(11) + a(17) + a(23) = 39 a( 3) + a(13) + a(23) = 39 a(11) + a(13) + a(15) = 39

which can be added to the equations describing a Magic Square of the fifth order (Section 3.2), and result in following linear equations:

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

An optimized guessing routine (MgcSqr5g1), produced 8288 (= 8 * 1036) Magic Squares with Diamond Inlay within 240 seconds, of which the first 142 are shown in Attachment 5.6.1.

5.6.2 Diamond Inlay and Associated

When the equations describing the 3th order Diamond Inlay are added to the equations describing an Associated Magic Square of the fifth order (Section 5.4.1), following set of linear equations will result:

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

An optimized guessing routine (MgcSqr5g2), produced 496 Associated Magic Squares with Diamond Inlay within 42 seconds, which are shown in Attachment 5.6.2.

5.6.3 Concentric with Diamond Inlay

When the equations describing the 3th order Diamond Inlay are added to the equations describing an Concentric Magic Square of the fifth order (Section 5.5.1), following set of linear equations will result:

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

An optimized guessing routine (MgcSqr5g3), produced 2592 Concentric Magic Squares with Diamond Inlay within 62 seconds, of which the first 56 are shown in Attachment 5.6.3.

5.6.4 Square Inlay General

When a Magic Square is embedded in another Magic Square as shown below, the embedded square is referred to as a Square Inlay:

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

The 3th order Square Inlay of a 5th order Magic Square can be described by following linear equations:

 a( 1) + a( 3) + a( 5) = 39 a(11) + a(13) + a(15) = 39 a(21) + a(23) + a(25) = 39 a(1) + a(11) + a(21) = 39 a(3) + a(13) + a(23) = 39 a(5) + a(15) + a(25) = 39 a(1) + a(13) + a(25) = 39 a(5) + a(13) + a(21) = 39

which can be added to the equations describing a Magic Square of the fifth order (Section 3.2), and result in following linear equations:

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

An optimized guessing routine (MgcSqr5g4), counted 1393920 (= 8 * 174240) Magic Squares with Square Inlay within 236 seconds, of which the first 552 are shown in Attachment 5.6.4.

5.6.5 Square and Diamond Inlay

When the equations describing the 3th order Square Inlay are added to the equations describing an Magic Square of the fifth order with Diamond Inlay (Section 5.6.1), following set of linear equations will result:

```a(22) =  0.4 * s1 - a(24)
a(21) =  0.6 * s1 - a(23) - a(25)
a(19) = -0.2 * s1 + 2 * a(25)
a(17) =        s1 - 2 * a(23) - 2 * a(25)
a(16) =  0.2 * s1 - a(18) - a(20) + 2 * a(23)
a(15) =  0.8 * s1 - a(23) - 2 * a(25)
a(13) =  0.2 * s1
a(12) =  0.4 * s1 - a(14)
a(11) = -0.4 * s1 + a(23) + 2 * a(25)
a(10) =  0.4 * s1 - a(20)
a( 9) = -0.6 * s1 + 2 * a(23) + 2 * a(25)
a( 8) =  0.4 * s1 - a(18)
a( 7) =  0.6 * s1 - 2 * a(25)
a( 6) =  0.2 * s1 + a(18) + a(20) - 2 * a(23)
a( 5) = -0.2 * s1 + a(23) + a(25)
a( 4) =  1.8 * s1 - a(14) - 2 * a(23) - a(24) - 4 * a(25)
a( 3) =  0.4 * s1 - a(23)
a( 2) = -1.4 * s1 + a(14) + 2 * a(23) + a(24) + 4 * a(25)
a( 1) =  0.4 * s1 - a(25)
```

With the option 'Check Diamond Inlay' enabled, 1440 Magic Squares with Square and Diamond Inlay could be produced within 85 seconds with procedure MgcSqr5g4, of which the first 360 are shown in Attachment 5.6.5.