Office Applications and Entertainment, Magic Squares  
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5.6 Magic Squares, Miscellaneous Inlays
When an embedded Magic Square is rotated 45 degrees, the embedded square is referred to as a Diamond Inlay.
The 3^{th} order Diamond Inlay of a 5^{th} order Magic Square can be described by following linear equations:
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 3^{th} 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 3^{th} 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.
When a Magic Square is embedded in another Magic Square as shown below, the embedded square is referred to as a Square Inlay:
The 3^{th} order Square Inlay of a 5^{th} order Magic Square can be described by following linear equations:
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 3^{th} 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.
The linear equations deducted in previous sections for 5^{th} order Magic Squares, have been applied in following Excel Spread Sheets:
Only the red figures have to be “guessed” to construct one of the applicable Magic Squares (wrong solutions are obvious).
The results regarding the miscellaneous types of 5^{th} order Magic Squares as deducted and discussed in previous sections are summarized in following table: 
Type
Subroutine
Results
Total Number
Remarks
Classic Magic
8 * 275305224
Walter Trump
Pan Magic
28800

Associated
388352

Ultra Magic
128

Cocentric
23040
Note 1
Eccentric
3072
Note 2
Magic/Diamond Inlay
8288

Associated/Diamond Inlay
496

Concentric/Diamond Inlay
2592

Magic/Square Inlay
1393920

Magic/Square + Diamond Inlay
1440

Note 1: The Embedded Magic Square is based on the consecutive integers 9, 10, ... 17.

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