Office Applications and Entertainment, Magic Squares

7.8   Non Overlapping Sub Squares

7.8.1 Introduction

Associated Magic Squares with Square Inlays, as described in Section 7.6.3, can be obtained by means of transformation of order 7 Composed Magic Squares as illustrated below:

The Magic Square shown at the left side above is composed out of:

• One 3^th order Simple Magic Corner Square A with Magic Sum s3 = 75 (top/left)
• One 4^th order Associated Magic Corner Square D with Magic Sum s4 = 100 (bottom/right)
• Two Associated Magic Rectangles B/C order 3 x 4 with s3 = 75 and s4 = 100

It should be noted that the reversed transformation is not necessarily possible because of the bottom-left / top-right Main Diagonal.

7.8.2 Analysis

The Composed Magic Square of order 7 can be represented as follows:

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)	a(26)	a(27)	a(28)
a(29)	a(30)	a(31)	a(32)	a(33)	a(34)	a(35)
a(36)	a(37)	a(38)	a(39)	a(40)	a(41)	a(42)
a(43)	a(44)	a(45)	a(46)	a(47)	a(48)	a(49)

The linear equations of the Composed Magic Square can be written as:

a(15) =  3 * s1 / 7 - a(16) -     a(17)
a(10) =  4 * s1 / 7 - a(16) - 2 * a(17)
a( 9) =      s1 / 7
a( 8) =  2 * s1 / 7 - a(10)
a( 3) =  2 * s1 / 7 - a(15)
a( 2) =  2 * s1 / 7 - a(16)
a( 1) =  2 * s1 / 7 - a(17)

a(28) =  4 * s1 / 7 - a(27) - a(26) - a(25)
a(33) =  4 * s1 / 7 - a(32) - a(26) - a(25)
a(34) =  4 * s1 / 7 - a(32) - a(27) - a(25)
a(35) =  4 * s1 / 7 - a(34) - a(33) - a(32)
a(39) =  2 * s1 / 7 - a(35)
a(40) =  2 * s1 / 7 - a(34)
a(41) =  2 * s1 / 7 - a(33)
a(42) =  2 * s1 / 7 - a(32)
a(46) =  2 * s1 / 7 - a(28)
a(47) =  2 * s1 / 7 - a(27)
a(48) =  2 * s1 / 7 - a(26)
a(49) =  2 * s1 / 7 - a(25)

a(43) =  3 * s1 / 7 - a(44)     - a(45)
a(37) =  6 * s1 / 7 - 2 * a(38) - a(44) - 2 * a(45)
a(36) = -3 * s1 / 7 + a(38)     + a(44) + 2 * a(45)
a(31) =  2 * s1 / 7 - a(36)
a(30) =  2 * s1 / 7 - a(37)
a(29) =  2 * s1 / 7 - a(38)
a(24) =  2 * s1 / 7 - a(43)
a(23) =  2 * s1 / 7 - a(44)
a(22) =  2 * s1 / 7 - a(45)

a(19) =(-6 * s1 / 7 - a(21) + 3 * a(38) + 3 * a(44) + 5 * a(45) - a(25))/3
a(18) =  4 * s1 / 7 - a(19) -     a(20) -     a(21)
a(14) =      s1 / 7 + a(18) -     a(21)
a(13) =  6 * s1 / 7 - a(14) - 2 * a(20) - 2 * a(21)
a(12) =  2 * s1 / 7 - a(13)
a(11) =  2 * s1 / 7 - a(14)
a( 7) =  2 * s1 / 7 - a(18)
a( 6) =  2 * s1 / 7 - a(19)
a( 5) =  2 * s1 / 7 - a(20)
a( 4) =  2 * s1 / 7 - a(21)

with the following independent variables:

• a(16), a(17) determining the 3^th order Simple Magic Square,
• a(25), a(26), a(27), a(32) determining the 4^th order Associated Magic Square and
• a(20), a(21), a(38), a(44), a(45) determining the Associated Magic Rectangles and Main Diagonal.

With an optimized guessing routine (MgcSqr7g3), based on the equations above, following cases where considered:

• Case 1: The total number of 3^th order Magic Corner Squares A has been determined, resulting in 124 (= 992 / 8) unique Magic Corner Squares (ref. Attachment 7.8.6)
  
  
• Case 2: With the first occuring Magic Corner Squares A and D constant, 48 Composed Magic Squares could be generated within 3,8 seconds (ref. Attachment 7.8.7)

Attachment 7.8.8 shows the first 7^th order Composed Magic Squares, found for some of the possible 3^th order Magic Corner Square (60 ea).

7.8.3 Spreadsheet Solution

The linear equations deducted in previous sections, can be applied in an Excel Spread Sheet (ref. CnstrSngl7d3).

Only the red figures have to be “guessed” to construct a 7^th order Composed Magic Square (wrong solutions are obvious).