Office Applications and Entertainment, Magic Squares

14.0   Special Magic Squares, Prime Numbers

14.8   Magic Squares (10 x 10), Part I

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 4^th order Pan Magic Sub Squares A, B and C (s4 = 4 * s10 / 10),
• One each 6^th 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 10^th order Prime Number Composed Magic Square for miscellaneous 6^th order Corner Squares and Magic Sums:

• Attachment 14.8.51a, based on Concentric Magic Squares          (ref. Attachment 14.4.3);
• Attachment 14.8.52a, based on Eccentric Magic Squares           (ref. Attachment 14.4.5);
• Attachment 14.8.53a, based on Ultra Magic Squares               (ref. Attachment 14.4.7);
• Attachment 14.8.54a, based on Compact Ultra Magic Squares       (ref. Attachment 14.4.9);
• Attachment 14.8.55 , based on Associated Composed Magic Squares (ref. Attachment 14.8.5).

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 10^th order Prime Number Composed Magic Square for miscellaneous 6^th order Corner Squares and Magic Sums:

• Attachment 14.8.51b, based on Concentric Magic Squares          (ref. Attachment 14.4.3);
• Attachment 14.8.52b, based on Eccentric Magic Squares           (ref. Attachment 14.4.5);
• Attachment 14.8.53b, based on Ultra Magic Squares               (ref. Attachment 14.4.7);
• Attachment 14.8.54b, based on Compact Ultra Magic Squares       (ref. Attachment 14.4.9).

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 4^th order Pan Magic Corner Squares A, B and C (s4 = 4 * s10 / 10),
• One each 6^th 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 10^th order Prime Number Composed Magic Square, based on 6^th 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:

• Attachment 14.4.7a, based on Partly Compact Ultra Magic Squares (ref. Attachment 14.4.7);
• Attachment 14.4.9a, based on Compact Ultra Magic Squares        (ref. Attachment 14.4.9).

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 10^th order Prime Number Composed Magic Square for subject 6^th order Corner Squares and miscellaneous Magic Sums:

• Attachment 14.8.57, based on Partly Compact Pan Magic Squares (ref. Attachment 14.4.7a);
• Attachment 14.8.59, based on Compact Pan Magic Squares        (ref. Attachment 14.4.9a).

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 4^th 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 (10^th 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 10^th 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.

Composed Magic Squares as discussed above, can also be obtained by transformation of Bordered Magic Squares with Composed Magic Center Squares (ref. Attachment 14.8.70).

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 4^th order Pan Magic Corner Square A (s4 = 4 * s10 / 10),
• One 6^th 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 10^th order Prime Number Composed Magic Square for miscellaneous 6^th order Corner Squares and Magic Sums:

• Attachment 14.9.51, based on Concentric Magic Squares          (ref. Attachment 14.4.3);
• Attachment 14.9.52, based on Eccentric Magic Squares           (ref. Attachment 14.4.5);
• Attachment 14.9.53, based on Ultra Magic Squares               (ref. Attachment 14.4.7);
• Attachment 14.9.55, based on Associated Composed Magic Squares (ref. Attachment 14.8.5).

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

14.8.7 Concentric Magic Squares (10 x 10)

A 10^th order Prime Number Concentric Magic Square consists of a Prime Number Concentric Magic Square of the 8^th 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 8^th 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 8^th 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 10^th 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 6^th 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 4^th 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 10^th 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 8^th 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:

Comparable routines as listed above, can be used to generate miscellaneous other types of order 10 Prime Number Magic Squares, which will be described in following sections.