Office Applications and Entertainment, Magic Squares

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10.3   Composed Magic Squares

10.3.1 Composed Magic Squares
       Corner Squares Order 4 and 6, Associated Rectangles

The 10th order Magic Square shown below, with Magic Sum s10 = 505, is composed out of:

  • One 4th order Pan Magic Corner Square, Magic Sum s4 = 202 (top/left)
  • One 6th order Symmetric Magic Corner Square, Magic Sum s6 = 303 (bottom/right)
  • Two order 4 x 6 Associated Magic Rectangles (s4 = 202 and s6 = 303)
a1 a2 a3 a4 a5 a6 a7 a8 a9 a10
a11 a12 a13 a14 a15 a16 a17 a18 a19 a20
a21 a22 a23 a24 a25 a26 a27 a28 a29 a30
a31 a32 a33 a34 a35 a36 a37 a38 a39 a40
a41 a42 a43 a44 a45 a46 a47 a48 a49 a50
a51 a52 a53 a54 a55 a56 a57 a58 a59 a60
a61 a62 a63 a64 a65 a66 a67 a68 a69 a70
a71 a72 a73 a74 a75 a76 a77 a78 a79 a80
a81 a82 a83 a84 a85 a86 a87 a88 a89 a90
a91 a92 a93 a94 a95 a96 a97 a98 a99 a100

Based on this definition a procedure (ref. Priem10c2) can be developed:

  • to read the previously generated order 6 Symmetric Magic Squares;
  • to generate the order 4 x 6 Associated Magic Rectangles;
  • to complete the 10 x 10 Composed Magic Squares with the order 4 Pan Magic Square.

Following attachments show the first occurring Composed Magic Square for miscellaneous Symmetric Corner Squares:

Each square shown corresponds with numerous Magic Squares for the same Corner Squares.

10.3.2 Composed Magic Squares
       Almost Associated

If a 10th order Composed Magic Square as discussed in Section 10.3.1 above, is composed out of:

  • One 4th order Associated Magic Corner Square, Magic Sum s4 = 202 (top/left)
  • One 6th order Almost Associated Magic Corner Square, Magic Sum s6 = 303 (bottom/right)
  • Two order 4 x 6 Associated Magic Rectangles (s4 = 202 and s6 = 303)

Subject Composed Magic Square can be transformed into an Almost Associated Magic Square with a Composed Border, as illustrated below:

Composed
65 51 42 44 83 82 63 23 28 24
41 45 54 62 22 26 27 67 80 81
39 47 56 60 20 21 34 74 75 79
57 59 50 36 77 73 78 38 19 18
61 37 33 71 100 4 92 3 99 5
70 48 35 49 8 90 6 91 15 93
58 29 46 69 76 14 13 17 89 94
32 55 72 43 7 12 84 88 87 25
52 66 53 31 16 86 10 95 11 85
30 68 64 40 96 97 98 9 2 1
Inlaid (Almost Associated)
65 51 83 82 63 23 28 24 42 44
41 45 22 26 27 67 80 81 54 62
61 37 100 4 92 3 99 5 33 71
70 48 8 90 6 91 15 93 35 49
58 29 76 14 13 17 89 94 46 69
32 55 7 12 84 88 87 25 72 43
52 66 16 86 10 95 11 85 53 31
30 68 96 97 98 9 2 1 64 40
39 47 20 21 34 74 75 79 56 60
57 59 77 73 78 38 19 18 50 36

Following attachments show miscellaneous examples of Composed - and related Almost Associated Magic Squares:

Each square shown corresponds with numerous Magic Squares for the same Corner Square(s).

10.3.3 Composed Magic Squares, Type 1

The 10th order Magic Square shown below, with Magic Sum s10 = 505, is composed out of:

  • Three 4th order Pan Magic Sub Squares, Magic Sum s4 = 202
  • One 6th order Symmetric Magic Corner Square, Magic Sum s6 = 303 (bottom/right)
  • Eight supplementary pairs, each summing to 101
a1 a2 a3 a4 a5 a6 a7 a8 a9 a10
a11 a12 a13 a14 a15 a16 a17 a18 a19 a20
a21 a22 a23 a24 a25 a26 a27 a28 a29 a30
a31 a32 a33 a34 a35 a36 a37 a38 a39 a40
a41 a42 a43 a44 a45 a46 a47 a48 a49 a50
a51 a52 a53 a54 a55 a56 a57 a58 a59 a60
a61 a62 a63 a64 a65 a66 a67 a68 a69 a70
a71 a72 a73 a74 a75 a76 a77 a78 a79 a80
a81 a82 a83 a84 a85 a86 a87 a88 a89 a90
a91 a92 a93 a94 a95 a96 a97 a98 a99 a100

Based on this definition a procedure (ref. Priem10c3) can be developed:

  • to read the previously generated order 6 Symmetric Magic Squares,
  • to generate the three Pan Magic Squares and
  • to complete the 10 x 10 Composed Magic Squares with the supplementary pairs

Following attachments show the first occurring Composed Magic Square for miscellaneous Symmetric Corner Squares:

Each square shown corresponds with numerous squares for the same Corner Square.

10.3.4 Composed Magic Squares, Type 2

Alternatively Magic Squares of order 10 - composed out of Sub Squares as described in Section 10.3.3 - can be arranged as shown below:

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10
a11 a12 a13 a14 a15 a16 a17 a18 a19 a20
a21 a22 a23 a24 a25 a26 a27 a28 a29 a30
a31 a32 a33 a34 a35 a36 a37 a38 a39 a40
a41 a42 a43 a44 a45 a46 a47 a48 a49 a50
a51 a52 a53 a54 a55 a56 a57 a58 a59 a60
a61 a62 a63 a64 a65 a66 a67 a68 a69 a70
a71 a72 a73 a74 a75 a76 a77 a78 a79 a80
a81 a82 a83 a84 a85 a86 a87 a88 a89 a90
a91 a92 a93 a94 a95 a96 a97 a98 a99 a100

Based on the principles described in Section 10.3.3 above, a comparable procedure (Priem10c4) can be developed.

Following attachments show the first occurring Composed Magic Square for miscellaneous Symmetric Corner Squares:

Each square shown corresponds with numerous squares for the same Corner Square.

10.3.5 Composed Magic Squares, Type 3

Alternatively Magic Squares of order 10 with a Magic Sum s10 = 505, can be composed out of:

  • Three 4th order Pan Magic Sub Squares, Magic Sum s4 = 202
  • One 6th order Eccentric Magic Corner Square, Magic Sum s6 = 303 (bottom/right)
  • Eight supplementary pairs, each summing to 101

and arranged as illustrated below:

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10
a11 a12 a13 a14 a15 a16 a17 a18 a19 a20
a21 a22 a23 a24 a25 a26 a27 a28 a29 a30
a31 a32 a33 a34 a35 a36 a37 a38 a39 a40
a41 a42 a43 a44 a45 a46 a47 a48 a49 a50
a51 a52 a53 a54 a55 a56 a57 a58 a59 a60
a61 a62 a63 a64 a65 a66 a67 a68 a69 a70
a71 a72 a73 a74 a75 a76 a77 a78 a79 a80
a81 a82 a83 a84 a85 a86 a87 a88 a89 a90
a91 a92 a93 a94 a95 a96 a97 a98 a99 a100

Based on the principles described in Section 10.3.3 above, a comparable procedure (Priem10c5) can be developed:

  • to read the previously generated order 6 Eccentric Magic Squares,
  • to generate the three Pan Magic Squares and
  • to complete the 10 x 10 Composed Magic Squares with the supplementary pairs

Attachment 10.2.8 shows for miscellaneous Eccentric Magic Corner Squares, the first occurring 10th order Composed Magic Square, based on 6th order Eccentric Magic Squares.

Each square shown corresponds with numerous squares for the same Eccentric Magic Corner Square.

10.3.6 Composed Magic Squares, Type 4

Alternatively Magic Squares of order 10 with a Magic Sum s10 = 505, can be composed out of:

  • Four 4th order Pan Magic Sub Squares, Magic Sum s4 = 202
  • Eighteen supplementary pairs, each summing to 101

and arranged as illustrated below:

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10
a11 a12 a13 a14 a15 a16 a17 a18 a19 a20
a21 a22 a23 a24 a25 a26 a27 a28 a29 a30
a31 a32 a33 a34 a35 a36 a37 a38 a39 a40
a41 a42 a43 a44 a45 a46 a47 a48 a49 a50
a51 a52 a53 a54 a55 a56 a57 a58 a59 a60
a61 a62 a63 a64 a65 a66 a67 a68 a69 a70
a71 a72 a73 a74 a75 a76 a77 a78 a79 a80
a81 a82 a83 a84 a85 a86 a87 a88 a89 a90
a91 a92 a93 a94 a95 a96 a97 a98 a99 a100

If the center lines, composed of the 18 pairs, are based on the consecutive integers 33 ... 68, the 4 Pan Magic Corner Squares can be constructed as described in Section 23.1 (n = 10).

The Center Cross might be constructed with the method of Al Antaki (10th century):

  • Construct a 6 x 6 Center Cross (s6 = 303)
  • Complete the 10 x 10 Center Cross with the remaining pairs (ref. CntrCross10)

Attachment 10.2.9 shows for a(55) = 33 ... 50 the first occurring Center Cross (10 x 10).

Each Center Cross corresponds with (8!) * (8!) = 1,6 109 Center Crosses, which can be obtained by permutation of the horizontal and vertical pairs.

An example of a Magic Square composed of a Center Cross (10 x 10) and four Pan Magic Squares (4 x 4) is shown below:

4 5 95 98 37 64 12 13 87 90
93 100 2 7 40 61 85 92 10 15
6 3 97 96 54 47 14 11 89 88
99 94 8 1 49 52 91 86 16 9
67 66 43 57 50 53 59 41 36 33
34 35 58 44 48 51 42 60 65 68
20 21 79 82 46 55 28 29 71 74
77 84 18 23 56 45 69 76 26 31
22 19 81 80 63 38 30 27 73 72
83 78 24 17 62 39 75 70 32 25

With the Center Cross fixed, each square corresponds with 4! * 3844 = 0,5 1012 squares.

10.3.7 Summary

The obtained results regarding the miscellaneous types of order 10 Composed Magic Squares as deducted and discussed in previous sections are summarized in following table:

Type

Characteristics

Subroutine

Results

Composed

Ass Rect (2 ea), Crnr Sqrs Order 4 and 6

Priem10c2

Attachment 10.2.5a
Attachment 10.2.5b

Composed (1)

Sub Sqrs Order 4 (3 ea) and 6 (1 ea)

Priem10c3

Attachment 10.2.6a
Attachment 10.2.6b

Composed (2)

Sub Sqrs Order 4 (3 ea) and 6 (1 ea)

Priem10c4

Attachment 10.2.7a
Attachment 10.2.7b

Composed (3)

Crnr Sqrs Order 4 (3 ea) and 6 (1 ea)

Priem10c5

Attachment 10.2.8

Composed (4)

Crnr Sqrs Order 4 (4 ea)

CntrCross10

Attachment 10.2.9

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