Office Applications and Entertainment, Magic Squares

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8.2   Magic Squares, Composed of Sub Squares

8.2.4 Analytic Solution, Magic Squares composed of Associated Magic Sub Squares
      Edward Falkener

In 1892 Edward Falkener published following 8th order Magic Square, composed out of four 4th order Associated Magic Sub Squares:

1 59 56 14 2 60 53 15
46 24 27 33 47 21 28 34
32 38 41 19 31 37 44 18
51 9 6 64 50 12 5 63
3 57 54 16 4 58 55 13
48 22 25 35 45 23 26 36
30 40 43 17 29 39 42 20
49 11 8 62 52 10 7 61

The square has following additional properties:

1. All regular 2 x 2 square (16 ea) sum to half the Magic Sum;
2. Each quarter contains four 3 x 3 sub squares whose corner points sum to half the Magic Sum;
3. The corner points of all 5 x 5 sub squares (16 ea) contain consecutive integers.

Properties 1 and 2 are a consequence of the application of Associated Magic Sub Squares.

The properties mentioned above result, after deduction, in following set of linear equations: 

a(61) = 0.5  * s1 - a(62) - a(63) - a(64)
a(57) = 0.5  * s1 - a(58) - a(59) - a(60)
a(55) = 0.5  * s1 - a(56) - a(63) - a(64)
a(54) = 0.5  * s1 - a(56) - a(62) - a(64)
a(53) =-0.5  * s1 + a(56) + a(62) + a(63) + 2 * a(64)
a(51) = 0.5  * s1 - a(52) - a(59) - a(60)
a(50) = 0.5  * s1 - a(52) - a(58) - a(60)
a(49) =-0.5  * s1 + a(52) + a(58) + a(59) + 2 * a(60)
a(48) = 0.75 * s1 - a(56) - a(62) - a(63) - 2 * a(64)
a(47) =-0.25 * s1 + a(56) + a(62) + a(64)
a(46) =-0.25 * s1 + a(56) + a(63) + a(64)
a(45) = 0.25 * s1 - a(56)
a(44) = 0.75 * s1 - a(52) - a(58) - a(59) - 2 * a(60)
a(43) =-0.25 * s1 + a(52) + a(58) + a(60)
a(42) =-0.25 * s1 + a(52) + a(59) + a(60)
a(41) = 0.25 * s1 - a(52)
a(40) =-0.25 * s1 + a(62) + a(63) + a(64)
a(39) = 0.25 * s1 - a(62)
a(38) = 0.25 * s1 - a(63)
a(37) = 0.25 * s1 - a(64)
a(36) =-0.25 * s1 + a(58) + a(59) + a(60)
a(35) = 0.25 * s1 - a(58)
a(34) = 0.25 * s1 - a(59)
a(33) = 0.25 * s1 - a(60)
a(29) = 0.5  * s1 - a(30) - a(31) - a(32)
a(25) = 0.5  * s1 - a(26) - a(27) - a(28)
a(23) = 0.5  * s1 - a(24) - a(31) - a(32)
a(22) = 0.5  * s1 - a(24) - a(30) - a(32)
a(21) =-0.5  * s1 + a(24) + a(30) + a(31) + 2 * a(32)
a(19) = 0.5  * s1 - a(20) - a(27) - a(28)
a(18) = 0.5  * s1 - a(20) - a(26) - a(28)
a(17) =-0.5  * s1 + a(20) + a(26) + a(27) + 2 * a(28)
a(16) = 0.75 * s1 - a(24) - a(30) - a(31) - 2 * a(32)
a(15) =-0.25 * s1 + a(24) + a(30) + a(32)
a(14) =-0.25 * s1 + a(24) + a(31) + a(32)
a(13) = 0.25 * s1 - a(24)
a(12) = 0.75 * s1 - a(20) - a(26) - a(27) - 2 * a(28)
a(11) =-0.25 * s1 + a(20) + a(26) + a(28)
a(10) =-0.25 * s1 + a(20) + a(27) + a(28)
a( 9) = 0.25 * s1 - a(20)
a( 8) =-0.25 * s1 + a(30) + a(31) + a(32)
a( 7) = 0.25 * s1 - a(30)
a( 6) = 0.25 * s1 - a(31)
a( 5) = 0.25 * s1 - a(32)
a( 4) =-0.25 * s1 + a(26) + a(27) + a(28)
a( 3) = 0.25 * s1 - a(26)
a( 2) = 0.25 * s1 - a(27)
a( 1) = 0.25 * s1 - a(28)

The 16 independent variables are corner points of four of the 5 x 5 Sub Squares, for which the third property applies.

This makes a guessing routine relatively fast, as the ranges per independent variable are limited to 4 values.

An optimized guessing routine (MgcSqr8c2), produced - based on the variable sets as originally used by Edward Falkener - 6720 Magic Squares within 375 seconds, of which 384 are shown in Attachment 8.2.6.

Attachment 8.2.7 shows the first occurring solutions for miscellaneous valid variable sets.

8.2.5 Analytic Solution, Pan Magic Squares composed of Pan Magic Sub Squares
      L.S. Frierson

In 1810 L.S. Frierson published following 8th order Pan Magic Square, composed out of four 4th order Pan Magic Sub Squares (left):

L.S. Frierson Indices (ref. property 3)
64 57 4 5 56 49 12 13
3 6 63 58 11 14 55 50
61 60 1 8 53 52 9 16
2 7 62 59 10 15 54 51
48 41 20 21 40 33 28 29
19 22 47 42 27 30 39 34
45 44 17 24 37 36 25 32
18 23 46 43 26 31 38 35
1 2 3 4 5 6 7 8
9 10 11 12 13 14 15 16
17 18 19 20 21 22 23 24
25 26 27 28 29 30 31 32
33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48
49 50 51 52 53 54 55 56
57 58 59 60 61 62 63 64

The square has following additional properties:

  1. The horizontal and vertical bent diagonals sum to the Magic Sum;
  2. The horizontal reflected bent diagonals sum to the Magic Sum;
  3. The 2 x 2 squares, with exception of squares 25, 27, 29, 31 and 57, 59, 61, 63, sum to half the Magic Sum;
  4. All 4 x 4 squares (64 ea) sum to two times the Magic Sum (4 x 4 compact);
  5. The corner points of all 3 x 3 sub squares, left and right of the vertical axes, sum to half the Magic Sum;
  6. The corner points of all 2 x 4 -, 2 x 6 - and 2 x 8 rectangles, symmetrical around the vertical axes,
    sum to half the Magic Sum;
  7. The corner points of all 2 x 7 - und 3 x 6 - rectangles, symmetrical around the main diagonals,
    sum to half the Magic Sum;
  8. The corner points of all 5 x 5 sub squares (16 ea) contain integers in arithmetic progression (d = 8)
    e.g. {35, 43, 51, 59}.

The properties mentioned above result, after deduction, in following set of linear equations: 

a(61) = 0.5  * s1 - a(62) - a(63) - a(64)
a(59) =             a(60) + a(63) - a(64)
a(58) =           - a(60) + a(62) + a(64)
a(57) = 0.5  * s1 - a(60) - a(62) - a(63)
a(55) = 0.5  * s1 - a(56) - a(63) - a(64)
a(54) =             a(56) - a(62) + a(64)
a(53) =           - a(56) + a(62) + a(63)
a(52) =             a(56) - a(60) + a(64)
a(51) = 0.5  * s1 - a(56) - a(60) - a(63)
a(50) =             a(56) + a(60) - a(62)
a(49) =           - a(56) + a(60) + a(62) + a(63) - a(64)
a(48) = 0.25 * s1 - a(62)
a(47) =-0.25 * s1 + a(62) + a(63) + a(64)
a(46) = 0.25 * s1 - a(64)
a(45) = 0.25 * s1 - a(63)
a(44) = 0.25 * s1 + a(60) - a(62) - a(64)
a(43) =-0.25 * s1 + a(60) + a(62) + a(63)
a(42) = 0.25 * s1 - a(60)
a(41) = 0.25 * s1 - a(60) - a(63) + a(64)
a(40) = 0.25 * s1 - a(56) + a(62) - a(64)
a(39) = 0.25 * s1 + a(56) - a(62) - a(63)
a(38) = 0.25 * s1 - a(56)
a(37) =-0.25 * s1 + a(56) + a(63) + a(64)
a(36) = 0.25 * s1 - a(56) - a(60) + a(62)
a(35) = 0.25 * s1 + a(56) - a(60) - a(62) - a(63) + a(64)
a(34) = 0.25 * s1 - a(56) + a(60) - a(64)
a(33) =-0.25 * s1 + a(56) + a(60) + a(63)
a(31) =             a(32) + a(63) - a(64)
a(30) =           - a(32) + a(62) + a(64)
a(29) = 0.5  * s1 - a(32) - a(62) - a(63)
a(28) =             a(32) + a(60) - a(64)
a(27) =             a(32) + a(60) + a(63) - 2 * a(64)
a(26) =           - a(32) - a(60) + a(62) + 2 * a(64)
a(25) = 0.5  * s1 - a(32) - a(60) - a(62) - a(63) + a(64)
a(24) =           - a(32) + a(56) + a(64)
a(23) = 0.5  * s1 - a(32) - a(56) - a(63)
a(22) =             a(32) + a(56) - a(62)
a(21) =             a(32) - a(56) + a(62) + a(63) - a(64)
a(20) =           - a(32) + a(56) - a(60) + 2 * a(64)
a(19) = 0.5  * s1 - a(32) - a(56) - a(60) - a(63) + a(64)
a(18) =             a(32) + a(56) + a(60) - a(62) - a(64)
a(17) =             a(32) - a(56) + a(60) + a(62) + a(63) - 2 * a(64)
a(16) = 0.25 * s1 + a(32) - a(62) - a(64)
a(15) =-0.25 * s1 + a(32) + a(62) + a(63)
a(14) = 0.25 * s1 - a(32)
a(13) = 0.25 * s1 - a(32) - a(63) + a(64)
a(12) = 0.25 * s1 + a(32) + a(60) - a(62) - 2 * a(64)
a(11) =-0.25 * s1 + a(32) + a(60) + a(62) + a(63) - a(64)
a(10) = 0.25 * s1 - a(32) - a(60) + a(64)
a( 9) = 0.25 * s1 - a(32) - a(60) - a(63) + 2 * a(64)
a( 8) = 0.25 * s1 - a(32) - a(56) + a(62)
a( 7) = 0.25 * s1 - a(32) + a(56) - a(62) - a(63) + a(64)
a( 6) = 0.25 * s1 + a(32) - a(56) - a(64)
a( 5) =-0.25 * s1 + a(32) + a(56) + a(63)
a( 4) = 0.25 * s1 - a(32) - a(56) - a(60) + a(62) + a(64)
a( 3) = 0.25 * s1 - a(32) + a(56) - a(60) - a(62) - a(63) + 2 * a(64)
a( 2) = 0.25 * s1 + a(32) - a(56) + a(60) - 2 * a(64)
a( 1) =-0.25 * s1 + a(32) + a(56) + a(60) + a(63) - a(64)

The 6 independent variables are corner points of four of the 5 x 5 Sub Squares, for which the last property applies.

This makes a guessing routine relatively fast, as the ranges per independent variable are limited to 4 values.

An optimized guessing routine (MgcSqr825), produced - based on the variable sets as originally used by L.S. Frierson - 8 Magic Squares within 0,85 seconds, which are shown in Attachment 8.2.51.

Attachment 8.2.52 shows the first occurring solutions for miscellaneous valid variable sets.

8.2.6 Analytic Solution, Magic Squares composed of Magic Sub Squares
      L.S. Frierson

Another 8th order Magic Square, composed out of four 4th order Magic Sub Squares, published by L.S. Frierson is shown below (left):

L.S. Frierson Indices (ref. property 5, 6, 7)
1 25 56 48 2 26 55 47
40 64 17 9 39 63 18 10
57 33 16 24 58 34 15 23
32 8 41 49 31 7 42 50
3 27 54 46 4 28 53 45
38 62 19 11 37 61 20 12
59 35 14 22 60 36 13 21
30 6 43 51 29 5 44 52

The square has following additional properties:

  1. The four center rows contain two additional 4th order magic sub squares;
  2. The corner points of all 3 x 3 sub squares, left and right of the vertical axes, sum to half the Magic Sum;
  3. All regular 2 x 2 square (16 ea) sum to half the Magic Sum;
  4. All 4 x 4 squares (64 ea) sum to two times the Magic Sum (4 x 4 compact);
  5. The corner points of all rectangles, concentric around the nine points marked in the right square above,
    sum to half the Magic Sum;
  6. The corner points of all octagons, concentric around subject nine points, sum to the Magic Sum;
  7. The sum of the 12 elements of subject octagons sum to 1.5 * the Magic Sum (= 390);
  8. The corner points of all 5 x 5 sub squares (16 ea) contain consecutive integers.

Properties 6 and 7 are a consequence of property 5.

The properties mentioned above result, after deduction, in following set of linear equations: 

a(61) = 0.5 * s1 - a(62) - a(63) - a(64)
a(57) = 0.5 * s1 - a(58) - a(59) - a(60)
a(55) = 0.5 * s1 - a(56) - a(63) - a(64)
a(53) =          - a(54) + a(63) + a(64)
a(52) =            a(56) - a(59) + a(63)
a(51) = 0.5 * s1 - a(56) - a(60) - a(63)
a(50) =            a(54) + a(58) + a(59) + a(60) - a(62) - a(63) - a(64)
a(49) =          - a(54) - a(58) + a(62) + a(63) + a(64)
a(47) =-0.5 * s1 + a(48) - a(54) + a(56) + a(62) + a(63) + 2 * a(64)
a(46) = 0.5 * s1 - a(48) - a(62) - a(64)
a(45) = 0.5 * s1 - a(48) + a(54) - a(56) - a(63) - a(64)
a(44) =            a(48) + a(59) - a(63)
a(43) =-0.5 * s1 + a(48) - a(54) + a(56) + a(60) + a(62) + a(63) + a(64)
a(42) = 0.5 * s1 - a(48) - a(58) - a(59) - a(60) + a(63)
a(41) = 0.5 * s1 - a(48) + a(54) - a(56) + a(58) - a(62) - a(63) - a(64)
a(40) = 0.5 * s1 - a(48) - a(56) - a(64)
a(39) = 0.5 * s1 - a(48) + a(54) - a(62) - a(63) - a(64)
a(38) =            a(48) - a(54) + a(64)
a(37) =-0.5 * s1 + a(48) + a(56) + a(62) + a(63) + a(64)
a(36) = 0.5 * s1 - a(48) - a(56) - a(60)
a(35) = 0.5 * s1 - a(48) + a(54) - a(59) - a(62) - a(64)
a(34) =            a(48) - a(54) - a(58) + a(62) + a(64)
a(33) =-0.5 * s1 + a(48) + a(56) + a(58) + a(59) + a(60)
a(31) =            a(32) + a(63) - a(64)
a(30) =          - a(32) + a(62) + a(64)
a(29) = 0.5 * s1 - a(32) - a(62) - a(63)
a(28) =            a(32) + a(60) - a(64)
a(27) =            a(32) + a(59) - a(64)
a(26) =          - a(32) + a(58) + a(64)
a(25) = 0.5 * s1 - a(32) - a(58) - a(59) - a(60) + a(64)
a(24) =          - a(32) + a(56) + a(64)
a(23) = 0.5 * s1 - a(32) - a(56) - a(63)
a(22) =            a(32) + a(54) - a(64)
a(21) =            a(32) - a(54) + a(63)
a(20) =          - a(32) + a(56) - a(59) + a(63) + a(64)
a(19) = 0.5 * s1 - a(32) - a(56) - a(60) - a(63) + a(64)
a(18) =            a(32) + a(54) + a(58) + a(59) + a(60) - a(62) - a(63) - 2 * a(64)
a(17) =            a(32) - a(54) - a(58) + a(62) + a(63)
a(15) =-0.5 * s1 + a(16) - a(54) + a(56) + a(62) + a(63) + 2 * a(64)
a(14) = 0.5 * s1 - a(16) - a(62) - a(64)
a(13) = 0.5 * s1 - a(16) + a(54) - a(56) - a(63) - a(64)
a(12) =            a(16) + a(59) - a(63)
a(11) =-0.5 * s1 + a(16) - a(54) + a(56) + a(60) + a(62) + a(63) + a(64)
a(10) = 0.5 * s1 - a(16) - a(58) - a(59) - a(60) + a(63)
a( 9) = 0.5 * s1 - a(16) + a(54) - a(56) + a(58) - a(62) - a(63) - a(64)
a( 8) = 0.5 * s1 - a(16) - a(56) - a(64)
a( 7) = 0.5 * s1 - a(16) + a(54) - a(62) - a(63) - a(64)
a( 6) =            a(16) - a(54) + a(64)
a( 5) =-0.5 * s1 + a(16) + a(56) + a(62) + a(63) + a(64)
a( 4) = 0.5 * s1 - a(16) - a(56) - a(60)
a( 3) = 0.5 * s1 - a(16) + a(54) - a(59) - a(62) - a(64)
a( 2) =            a(16) - a(54) - a(58) + a(62) + a(64)
a( 1) =-0.5 * s1 + a(16) + a(56) + a(58) + a(59) + a(60)

The 11 independent variables are corner points of six of the 5 x 5 Sub Squares, for which the last property applies.

This makes a guessing routine relatively fast, as the ranges per independent variable are limited to 4 values.

An optimized guessing routine (MgcSqr826), produced - based on the variable sets as originally used by L.S. Frierson - 48 Magic Squares within 4,75 seconds, which are shown in Attachment 8.2.61.

Attachment 8.2.62 shows the first occurring solutions for miscellaneous valid variable sets.

8.2.7 Analytic Solution, Concentric Pan Magic Squares, Pan Magic Center Square, Composed Border
      Harry A. Sayles

In 1912 Harry A. Sayles published following 8th order Concentric Pan Magic Square, composed out of a 4th order Pan Magic Center Square and a Composed Border:

Harry A. Sayles Indices (ref. property 4)
17 40 31 42 19 38 29 44
13 60 3 54 15 58 1 56
35 22 45 28 33 24 47 26
63 10 49 8 61 12 51 6
18 39 32 41 20 37 30 43
14 59 4 53 16 57 2 55
36 21 46 27 34 23 48 25
64 9 50 7 62 11 52 5
1 2 3 4 5 6 7 8
9 10 11 12 13 14 15 16
17 18 19 20 21 22 23 24
25 26 27 28 29 30 31 32
33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48
49 50 51 52 53 54 55 56
57 58 59 60 61 62 63 64

The Concentric Pan Magic Square contains two embedded order 4 Simple Magic Squares (Middle Rows).
The Border is composed out of three Pan Magic Squares.

The square has following additional properties:

  1. The half rows sum to half the Magic Sum;
  2. The horizontal and vertical bent diagonals sum to the Magic Sum;
  3. The horizontal and vertical reflected bent diagonals sum to the Magic Sum;
  4. The 2 x 2 squares, with exception of squares 2, 18, 34, 50 and 6, 22, 38, 54 sum to half the Magic Sum;
  5. All 4 x 4 squares (64 ea) sum to two times the Magic Sum (4 x 4 compact);
  6. The corner points of half of the 3 x 3 sub squares sum to half the Magic Sum
    (within the 4 middle rows and within the 4 border rows);
  7. The corner points of all 2 x 4 -, 2 x 6 - and 2 x 8 rectangles, symmetrical around the vertical axes,
    sum to half the Magic Sum;
  8. The corner points of all 2 x 7 rectangles, symmetrical around the main diagonals,
    sum to half the Magic Sum;
  9. The corner points of all 5 x 5 sub squares (16 ea) contain consecutive integers

The properties mentioned above result, after deduction, in following set of linear equations: 

a(61) =  0.5  * s1 - a(62) - a(63) - a(64)
a(59) =            - a(60) + a(63) + a(64)
a(58) =            - a(60) + a(62) + a(64)
a(57) =  0.5  * s1 + a(60) - a(62) - a(63) - 2 * a(64)
a(55) =  0.5  * s1 - a(56) - a(63) - a(64)
a(54) =              a(56) + 2 * a(60) - a(62) - a(64)
a(53) =            - a(56) - 2 * a(60) + a(62) + a(63) + 2 * a(64)
a(52) =              a(56) + a(60) - a(64)
a(51) =  0.5  * s1 - a(56) - a(60) - a(63)
a(50) =              a(56) + a(60) - a(62)
a(49) =            - a(56) - a(60) + a(62) + a(63) + a(64)
a(47) =            - a(48) + a(63) + a(64)
a(46) =              a(48) - 2 * a(60) + a(62) + a(64)
a(45) =  0.5  * s1 - a(48) + 2 * a(60) - a(62) - a(63) - 2 * a(64)
a(44) =              a(48) - a(60) + a(64)
a(43) =            - a(48) + a(60) + a(63)
a(42) =              a(48) - a(60) + a(62)
a(41) =  0.5  * s1 - a(48) + a(60) - a(62) - a(63) - a(64)
a(40) =              a(48) - a(56) - 2 * a(60) + 2 * a(62) + a(64)
a(39) =  0.5  * s1 - a(48) + a(56) + 2 * a(60) - 2 * a(62) - a(63) - 2 * a(64)
a(38) =              a(48) - a(56) - 2 * a(60) + a(62) + 2 * a(64)
a(37) =            - a(48) + a(56) + 2 * a(60) - a(62) + a(63) - a(64)
a(36) =              a(48) - a(56) - 3 * a(60) + 2 * a(62) + 2 * a(64)
a(35) =  0.5  * s1 - a(48) + a(56) + 3 * a(60) - 2 * a(62) - a(63) - 3 * a(64)
a(34) =              a(48) - a(56) - a(60) + a(62) + a(64)
a(33) =            - a(48) + a(56) + a(60) - a(62) + a(63)
a(32) =  0.25 * s1 - a(48) + a(60) - a(62)
a(31) = -0.25 * s1 + a(48) - a(60) + a(62) + a(63) + a(64)
a(30) =  0.25 * s1 - a(48) + a(60) - a(64)
a(29) =  0.25 * s1 + a(48) - a(60) - a(63)
a(28) =  0.25 * s1 - a(48) + 2 * a(60) - a(62) - a(64)
a(27) = -0.25 * s1 + a(48) - 2 * a(60) + a(62) + a(63) + 2 * a(64)
a(26) =  0.25 * s1 - a(48)
a(25) =  0.25 * s1 + a(48) - a(63) - a(64)
a(24) =  0.25 * s1 - a(48) + a(56) + a(60) - a(62) - a(64)
a(23) =  0.25 * s1 + a(48) - a(56) - a(60) + a(62) - a(63)
a(22) =  0.25 * s1 - a(48) + a(56) + 3 * a(60) - 2 * a(62) - 2 * a(64)
a(21) = -0.25 * s1 + a(48) - a(56) - 3 * a(60) + 2 * a(62) + a(63) + 3 * a(64)
a(20) =  0.25 * s1 - a(48) + a(56) + 2 * a(60) - a(62) - 2 * a(64)
a(19) =  0.25 * s1 + a(48) - a(56) - 2 * a(60) + a(62) - a(63) + a(64)
a(18) =  0.25 * s1 - a(48) + a(56) + 2 * a(60) - 2 * a(62) - a(64)
a(17) = -0.25 * s1 + a(48) - a(56) - 2 * a(60) + 2 * a(62) + a(63) + 2 * a(64)
a(16) =  0.25 * s1 + a(60) - a(62) - a(64)
a(15) = -0.25 * s1 - a(60) + a(62) + a(63) + 2 * a(64)
a(14) =  0.25 * s1 - a(60)
a(13) =  0.25 * s1 + a(60) - a(63) - a(64)
a(12) =  0.25 * s1 - a(62)
a(11) = -0.25 * s1 + a(62) + a(63) + a(64)
a(10) =  0.25 * s1 - a(64)
a( 9) =  0.25 * s1 - a(63)
a( 8) =  0.25 * s1 - a(56) - a(60) + a(62)
a( 7) =  0.25 * s1 + a(56) + a(60) - a(62) - a(63) - a(64)
a( 6) =  0.25 * s1 - a(56) - a(60) + a(64)
a( 5) = -0.25 * s1 + a(56) + a(60) + a(63)
a( 4) =  0.25 * s1 - a(56) - 2 * a(60) + a(62) + a(64)
a( 3) =  0.25 * s1 + a(56) + 2 * a(60) - a(62) - a(63) - 2 * a(64)
a( 2) =  0.25 * s1 - a(56)
a( 1) = -0.25 * s1 + a(56) + a(63) + a(64)

The 6 independent variables are corner points of five of the 5 x 5 Sub Squares, for which the last property applies.

This makes a guessing routine relatively fast, as the ranges per independent variable are limited to 4 values.

An optimized guessing routine (MgcSqr827), produced - based on the variable sets as originally used by Harry A. Sayles - 8 Magic Squares within 0.92 seconds, which are shown in Attachment 8.2.71.

Attachment 8.2.72 shows the first occurring solutions for miscellaneous valid variable sets.

8.2.8 Spreadsheet Solutions

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

  • CnstrSngl8c2 Magic Squares composed of Associated Magic Sub Squares,
                 Edward Falkener
  • CnstrSngl8d3 Magic Squares composed of Magic Sub Squares,
                 L.S. Frierson
  • CnstrSngl8b3 Pan Magic Squares composed of Pan Magic Sub Squares,
                 L.S. Frierson
  • CnstrSngl8b4 Pan Magic Squares, Concentric, Composed Border
                 Harry A. Sayles

Only the red figures have to be “guessed” to construct one of the applicable (Pan) Magic Squares of the 8th order (wrong solutions are obvious).

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