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14.0   Special Magic Squares, Prime Numbers

14.8   Magic Squares (10 x 10)

14.8.1 Magic Squares (10 x 10), Composed (1)

Prime Number Magic Squares of order 10 with a Magic Sum s10 can be composed out of:

• Three each 4th order Pan Magic Sub Squares A, B and C (s4 = 4 * s10 / 10),
• One each 6th order (Symmetric) Magic Corner Square D (s6 = 6 * s10 / 10) and
• Eight supplementary pairs, each summing to 2 * s10 / 10

as illustrated below:

 b1 b2 b3 b4 c1 c2 c3 c4 m9 m10 b5 b6 b7 b8 c5 c6 c7 c8 m19 m20 b9 b10 b11 b12 c9 c10 c11 c12 m29 m30 b13 b14 b15 b16 c13 c14 c15 c16 m39 m40 a1 a2 a3 a4 d1 d2 d3 d4 d5 d6 a5 a6 a7 a8 d7 d8 d9 d10 d11 d12 a9 a10 a11 a12 d13 d14 d15 d16 d17 d18 a13 a14 a15 a16 d19 d20 d21 d22 d23 d24 m81 m82 m83 m84 d25 d26 d27 d28 d29 d30 m91 m92 m93 m94 d31 d32 d33 d34 d35 d36

Based on the principles described in previous section a comparable procedure (Priem10a1) can be developed:

• to read previously generated (Symmetric) Magic Squares D,
• to generate the Pan Magic Squares A, B and C and
• to complete the 10 x 10 Composed Magic Squares M.

Following attachments show the first occurring 10th order Prime Number Composed Magic Square for miscellaneous 6th order Corner Squares and Magic Sums:

Each square shown corresponds with numerous squares for the same Magic Sum.

14.8.2 Magic Squares (10 x 10), Composed (2)

Alternatively Prime Number Magic Squares of order 10 - composed out of Sub Squares as described in Section 14.8.1 - can be arranged as shown below:

 b1 b2 b3 b4 m9 c1 c2 c3 c4 m10 b5 b6 b7 b8 m19 c5 c6 c7 c8 m20 b9 b10 b11 b12 m29 c9 c10 c11 c12 m30 b13 b14 b15 b16 m39 c13 c14 c15 c16 m40 m81 m82 m83 m84 d1 d2 d3 d4 d5 d6 a1 a2 a3 a4 d7 d8 d9 d10 d11 d12 a5 a6 a7 a8 d13 d14 d15 d16 d17 d18 a9 a10 a11 a12 d19 d20 d21 d22 d23 d24 a13 a14 a15 a16 d25 d26 d27 d28 d29 d30 m91 m92 m93 m94 d31 d32 d33 d34 d35 d36

Based on the principles described in Section 14.8.1 above, a comparable procedure (Priem10a2) can be developed.

Following attachments show the first occurring 10th order Prime Number Composed Magic Square for miscellaneous 6th order Corner Squares and Magic Sums:

Each square shown corresponds with numerous squares for the same Magic Sum.

14.8.3 Magic Squares (10 x 10), Composed (3)

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

• Three each 4th order Pan Magic Corner Squares A, B and C (s4 = 4 * s10 / 10),
• One each 6th order Eccentric Magic Corner Square D (s6 = 6 * s10 / 10) and
• Eight supplementary pairs, each summing to 2 * s10 / 10

and arranged as illustrated below:

 b1 b2 b3 b4 m5 m6 c1 c2 c3 c4 b5 b6 b7 b8 m15 m16 c5 c6 c7 c8 b9 b10 b11 b12 m25 m26 c9 c10 c11 c12 b13 b14 b15 b16 m35 m36 c13 c14 c15 c16 m41 m42 m43 m44 d1 d2 d3 d4 d5 d6 m51 m52 m53 m54 d7 d8 d9 d10 d11 d12 a1 a2 a3 a4 d13 d14 d15 d16 d17 d18 a5 a6 a7 a8 d19 d20 d21 d22 d23 d24 a9 a10 a11 a12 d25 d26 d27 d28 d29 d30 a13 a14 a15 a16 d31 d32 d33 d34 d35 d36

Based on the principles described above a comparable procedure (Priem10b) can be developed:

• to read previously generated Eccentric Magic Squares D,
• to generate the Pan Magic Squares A, B and C and
• to complete the 10 x 10 Composed Magic Squares M.

Attachment 14.8.56 shows for miscellaneous Magic Sums the first occurring 10th order Prime Number Composed Magic Square, based on 6th order Eccentric Magic Squares (ref. Attachment 14.4.5).

Other Collections of Sub Squares D might be used, provided that the key condition d1 + d8 = d2 + d7 = s10 / 5 is fulfilled.

Following attachments show such collections:

The elements of both collections can be obtained by moving the row and columns such that the original 2 x 2 Center Square becomes the Top-Left Corner Square. The resulting Magic Squares are still (Partly) Compact and Pan Diagonal.

Following attachments show the first occurring 10th order Prime Number Composed Magic Square for subject 6th order Corner Squares and miscellaneous Magic Sums:

Each square shown corresponds with numerous squares for the same Magic Sum.

14.8.4 Magic Squares (10 x 10), Composed (4)

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

• Four each 4th order Pan Magic Corner Squares (s4 = 4 * s10 / 10),
• Eighteen supplementary pairs, each summing to 2 * s10 / 10

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

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

• Construct a 6 x 6 Center Cross (s6 = 6 * s10 / 10)
• Complete the 10 x 10 Center Cross with the remaining pairs

A procedure (Priem10g) can be developed to:

• Generate the 4 x 4 Pan Magic Squares
• Generate the 10 x 10 Center Crosses

Attachment 14.8.41 shows the first occurring 10th order Prime Number Composed Magic Square for a few Magic Sums.

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

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

14.8.5 Magic Squares (10 x 10)
Corner Squares Order 4 and 6, Associated Rectangles

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

• One 4th order Pan Magic Corner Square A (s4 = 4 * s10 / 10),
• One 6th order Magic Corner Square D (s6 = 6 * s10 / 10) and
• Two order 4 x 6 Associated Magic Rectangles B and C (s4 = 4 * s10 / 10 and s6 = 6 * s10 / 10)

as illustrated below:

 a1 a2 a3 a4 b1 b2 b3 b4 b5 b6 a5 a6 a7 a8 b7 b8 b9 b10 b11 b12 a9 a10 a11 a12 b13 b14 b15 b16 b17 b18 a13 a14 a15 a16 b19 b20 b21 b22 b23 b24 c1 c2 c3 c4 d1 d2 d3 d4 d5 d6 c5 c6 c7 c8 d7 d8 d9 d10 d11 d12 c9 c10 c11 c12 d13 d14 d15 d16 d17 d18 c13 c14 c15 c16 d19 d20 d21 d22 d23 d24 c17 c18 c19 c20 d25 d26 d27 d28 d29 d30 c21 c22 c23 c24 d31 d32 d33 d34 d35 d36

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

• to read previously generated (Symmetric) Magic Squares D;
• to generate the Associated Magic Rectangles B and C;
• to complete the 10 x 10 Composed Magic Squares M with the Pan Magic Square A.

Following attachments show the first occurring 10th order Prime Number Composed Magic Square for miscellaneous 6th order Corner Squares and Magic Sums:

Each square shown corresponds with numerous squares for the same Magic Sum.

14.8.6 Associated Magic Squares (10 x 10)
Composed of Simple Magic Squares (5 x 5)

Associated Magic Squares, composed of four each Simple Magic Squares, contain two sets of Complementary Anti Symmetric Magic Squares, as discussed in Section 14.3.10.

• Attachment 14.8.65 shows examples of such suitable 5th order Anti Symmetric Magic Squares;

• Attachment 14.8.66 shows for miscellaneous Magic Sums the related 10th order Associated Magic Squares;

• Attachment 14.8.67 shows the corresponding Pan Magic and Complete Magic Squares (Eulers Transformation).

Each square shown corresponds with numerous squares for the same Magic Sum.

14.8.7 Concentric Magic Squares (10 x 10)

A 10th order Prime Number Concentric Magic Square consists of a Prime Number Concentric Magic Square of the 8th order with a border around it.

For Prime Number Concentric Magic Squares of order 10 with Magic Sum s10, it is convenient to split the supplementary rows and columns into parts summing to s5 = s10 / 2:

 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

This results in following border equations:

 a(10) = s5 - a( 9) - a( 8) - a( 7) - a(6) a(50) = s5 - a(40) - a(30) - a(20) -a(10) a(91) = s10/2 - a(10) a(96) = s10/2 - a( 6) a(97) = s10/2 - a( 7) a(98) = s10/2 - a( 8) a(99) = s10/2 - a( 9) a(11) = s10/2 - a(20) a(21) = s10/2 - a(30) a(31) = s10/2 - a(40) a(41) = s10/2 - a(50) a(  1) = s5 - a( 2) - a( 3) - a( 4) - a(5) a( 81) = s5 - a(71) - a(61) - a(51) - a(1) a(100) = s10/2 - a( 1) a( 92) = s10/2 - a( 2) a( 93) = s10/2 - a( 3) a( 94) = s10/2 - a( 4) a( 95) = s10/2 - a( 5) a( 60) = s10/2 - a(51) a( 70) = s10/2 - a(61) a( 80) = s10/2 - a(71) a( 90) = s10/2 - a(81)

which enable the development of a fast procedure to generate Prime Number Concentric Magic Squares of order 10 (ref. Priem10c).

Miscellaneous Prime Number Concentric Magic Squares of order 10, based on the 8th order Concentric Magic Squares as discussed in Section 14.6.4, are shown in following attachments:

• Concentric Magic Center Square               (ref. Attachment 14.8.61)

• Concentric Pan Magic Center Square           (ref. Attachment 14.8.62)
(Embedded Associated Square)

• Concentric Partly Compact Center Square      (ref. Attachment 14.8.63)
(Embedded Pan Magic Square)

Each square shown corresponds with numerous squares for the same Magic Sum.

14.8.8 Bordered Magic Squares (10 x 10), Miscellaneous Inlays

Based on the collections of 8th order Magic Squares, as deducted in Section 14.6.1 thru 14.6.3, Section 14.6.5, and Section 14.6.7 thru 14.6.10, also following Bordered Magic Squares can be generated with routine Priem10c:

• Composed Magic Center Square                  (ref. Attachment 14.8.70)

• Bordered Center Square with:

- Embedded Ultra Magic Square, Partly Compact (ref. Attachment 14.8.71)
- Embedded Ultra Magic Square, Compact        (ref. Attachment 14.8.72)
- Embedded Most Perfect Pan Magic Square      (ref. Attachment 14.8.73)
- Embedded Composed Square                    (ref. Attachment 14.8.74)
- Embedded Composed Square, Associated        (ref. Attachment 14.8.75)

• Franklin Pan Magic Center Square              (ref. Attachment 14.8.76)
Most Perfect Pan Magic Center Square

• Ultra Magic Center Square                     (ref. Attachment 14.8.78)

It should be noted that the Attachments listed above contain only those solutions which could be found within 100 seconds.

Each square shown corresponds with numerous squares for the same Magic Sum.

14.8.9 Bordered Magic Squares (10 x 10), Split Border

Alternatively a 10th order Bordered Magic Square with Magic Sum s10 can be constructed based on:

• a Magic Center Square of order 6 with Magic Sum s6 = 6 * s10 / 10;
• 32 pairs, each summing to 2 * s10 / 10, surrounding the Symmetric (Pan) Magic Center Square;
• a split of the supplementary rows and columns into three parts:
two summing to s3 = 3 * s10 / 10 and one to s4 = 4 * s10 / 10

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

As the first border - as specified above - occurs for s10 = 10850, it is convenient to construct the border first and the 6th order Magic Center Square later (based on the remainder of the available pairs).

Based on the principles described in previous sections, a fast procedure (Priem10e1) can be developed:

• to generate, four Magic Squares of order 3;
• to transform these four Magic Squares into suitable Corner Squares, as shown above;
• to complete the Border of order 10 with the four remaining 2 x 4 Magic Rectangles.

The Magic Center Square can be added with a separate routine e.g. Priem10e2 for Concentric, Partly Compact, Magic Center Squares as discussed in Section 14.4.4.

Attachment 14.8.81 shows one Prime Number Bordered Magic Square for some of the occurring Magic Sums.

Each square shown corresponds with numerous squares for the same Magic Sum.

14.8.10 Bordered Magic Squares (10 x 10), Composed Border

Another type of order 10 Bordered Magic Squares with Magic Sum s10 can be constructed based on:

• a Border composed out of:
- 4 Semi Magic Squares of order 3 with Magic Sum s3 = 3 * s10 / 10;
- 4 Associated Magic Rectangles order 3 x 4 with s3 = 3 * s10 / 10 and s4 = 4 * s10 / 10;
• a Magic Center Square of order 4 with Magic Sum s4 = 4 * s10 / 10

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

It is convenient to construct the Composed Border first and the 4th order Magic Center Square later (based on the remainder of the available pairs).

Based on the principles described in previous sections, a fast procedure (Priem10f1) can be developed:

• to retrieve, the four Semi Magic Squares of order 3 from previously generated Composed Magic Squares of order 6 (ref. Section 14.4.11);
• to complete the Composed Border of order 10 with the four 3 x 4 Magic Rectangles.

The Magic Center Square can be added with a separate routine e.g. Priem10f2 for Associated Magic Center Squares as discussed in Section 14.2.3.

Attachment 14.8.83 shows one Prime Number Composed Bordered Magic Square for some of the occurring Magic Sums.

Each square shown corresponds with numerous squares for the same Magic Sum.

Note:

As a consequence of the applied properties:

• The opposite Semi Magic Corner Squares are Anti Symmetric and Complementary;
• The Magic Center Square is Center Symmetric (Associated)

the 11th order Composed Magic Square will be associated, if the opposite Magic Rectangles (3 x 4) are Anti Symmetric and Complementary as well.

Attachment 14.8.84 shows one Associated Composed Magic Square for some of the occurring Magic Sums (ref. Priem10f3).

14.8.11 Eccentric Magic Squares (10 x 10)

Also for Prime Number Eccentric Magic Squares of order 10 it is convenient to split the supplementary rows and columns into: two parts summing to s3 = 3 * s10 / 10 and one part summing to s4 = 4 * s10 / 10.

 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

This enables, based on the same principles, the development of a set of fast procedures (ref. Priem10d):

• to read the previously generated Eccentric Magic Squares of order 8;
• to complete the Main Diagonal and determine the related Border Pairs;
• to generate, based on the remainder of the available pairs, a suitable Corner Square of order 3;
• to complete the Eccentric Magic Square of order 10 with the two remaining 2 x 4 Magic Rectangles.

Attachment 14.8.82 shows, based on the 8th order Eccentric Magic Squares as discussed in Section 14.6.6, one Prime Number Eccentric Magic Square for some of the occurring Magic Sums.

Each square shown corresponds with numerous squares for the same Magic Sum.

14.8.12 Summary

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

 Type Characteristics Subroutine Results Composed (1) Sub  Sqrs Order 4 (3 ea) and 6 (1 ea) Composed (2) Sub  Sqrs Order 4 (3 ea) and 6 (1 ea) Composed (3) Crnr Sqrs Order 4 (3 ea) and 6 (1 ea) Composed (4) Crnr Sqrs Order 4 (4 ea) Composed Ass Rect (2 ea), Crnr Sqrs Order 4 and 6 Composed Simple Magic Sub Squares (5 x 5) - Pan Magic and Complete Euler Concentric Split Border Lines Bordered Split Border Lines, Miscellaneous Types Split Border Lines, Center Square order 6 Composed Border, Center Square order 4 Associated Composed Border, Center Square order 4 Eccentric Split Border Lines - - - -
 Comparable routines as listed above, can be used to generate Prime Number Magic Squares of order 11, which will be described in following sections.