Office Applications and Entertaiment, Magic Squares

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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:

a1 a2 a3 b1 b2 b3 b4
a4 a5 a6 b5 b6 b7 b8
a7 a8 a9 b9 b10 b11 b12
c1 c2 c3 d1 d2 d3 d4
c4 c5 c6 d5 d6 d7 d8
c7 c8 c9 d9 d10 d11 d12
c10 c11 c12 d13 d14 d15 d16
= >
d1 c1 d2 c2 d3 c3 d4
b1 a1 b2 a2 b3 a3 b4
d5 c4 d6 c5 d7 c6 d8
b5 a4 b6 a5 b7 a6 b8
d9 c7 d10 c8 d11 c9 d12
b9 a7 b10 a8 b11 a9 b12
d13 c10 d14 c11 d15 c12 d16

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

  • One 3th order Simple Magic Corner Square A with Magic Sum s3 = 75 (top/left)
  • One 4th 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 3th order Simple Magic Square,
  • a(25), a(26), a(27), a(32) determining the 4th 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 3th 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 7th order Composed Magic Squares, found for some of the possible 3th 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 7th order Composed Magic Square (wrong solutions are obvious).


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