Office Applications and Entertainment, Magic Squares

14.0    Special Magic Squares, Prime Numbers

14.8    Magic Squares (10 x 10), Part II

Miscellaneous types of order 10 Prime Number Magic Squares - composed of order 4 and order 6 Magic Sub Squares - have been discussed in previous section(s).

Following sections will deal with order 10 Prime Number Magic Squares, composed of order 5 (semi) Magic Sub Squares and miscellaneous types of Inlaid Magic Squares.

14.8.13 Composed Magic Squares (10 x 10)
        Pan Magic Sub Squares (5 x 5)

Prime Number Magic Squares of order 10 - with Magic Sum 2 * s1 - can be composed out of 5^th order Prime Number Pan Magic Squares with Magic Sum s1.

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

In section 14.3.1, a procedure was developed to generate 5^th order Prime Number Pan Magic Squares with Magic Sum s1.

With some minor modifications subject procedure can be used to find a set of 4 (or more) Prime Number Pan Magic Squares with Magic Sum s1 - each containing 25 different Prime Numbers.

Attachment 14.8.1 contains a few of such sets which were found by means of procedure Priem5b3.

Each set of four squares can be arranged in 24 ways into a 10^th order Prime Number Magic Square with Magic Sum 2 * s1 as shown in Attachment 14.8.2 for Mc10 = 2 * 2925 = 5850.

Further we should realize that each Pan Magic Square of the 5^th order, based on a set of 25 distinct integers, is a member of a collection of 28800 Pan Magic Square of the 5^th order.

Consequently, based on one single set of 4 Prime Number Pan Magic Squares of the 5^th order as shown in Attachment 14.8.1, 24 * 28800⁴ = 1,65 10¹⁹ Prime Number Magic Squares of the 10^th order can be constructed.

Attachment 14.8.3 contains miscellaneous Prime Number Magic Squares composed of Pan Magic Sub Squares as generated with procedure Priem5b3.

14.8.14 Pan Magic Squares (10 x 10)
        Pan Magic Square Inlays (5 x 5)

Prime Number Magic Squares composed of Pan Magic Sub Squares, as discussed in Section 14.8.13 above, can be transformed into (Inlaid) Pan Magic Squares as illustrated below:

The resulting (Inlaid) Pan Magic Squares are Four Way V type Zig Zag.

Attachment 14.8.5 shows the Pan Magic Squares, which can be obtained by transformation of the Composed Magic Squares as shown in Attachment 14.8.3.

Each square shown corresponds with numerous (Inlaid) Pan Magic Squares with the same Magic Sum.

14.8.15 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 5^th order Anti Symmetric Magic Squares;
  
  
• Attachment 14.8.66 shows for miscellaneous Magic Sums the related 10^th order Associated Magic Squares;
  
  
• Attachment 14.8.67 shows the corresponding Pan Magic and Complete Magic Squares (Eulers Transformation).

Subject Composed Magic Squares can be transformed into (Inlaid) Four Way V type ZigZag Magic Squares by means of the transformation illustrated below for respectively:

Inlaid Four Way V type ZigZag Associated Magic Square B1, Mc10 = 26950:

Inlaid Four Way V type ZigZag Croswise Symmetric Magic Square B2, Mc10 = 26950:

Each square shown above and in the referred attachments corresponds with numerous squares for the same Magic Sum.

Notes:
For the Associated Magic Squares B1 also the Semi Diagonals sum to the Magic Sum.
For the Crosswise Symmetric Magic Squares B2 also half of the Broken Diagonals sum to the Magic Sum.

14.8.16 Composed Magic Squares (10 x 10)
        Order 5 Magic Cube Based

Order 10 Prime Number Magic Squares composed of Order 5 (Semi) Magic Sub Squares can be constructed based on:

• Prime Number Concentric Magic Cubes, as deducted in Section 7.3.3 or
  
  
• Prime Number Associated Magic Cubes, as deducted in Section 7.7.2

of Chapter 'Prime Number Magic Cubes'.

A typical example of an order 10 Prime Number Composed Magic Square, based on the planes of a Prime Number Concentric Magic Cube of half the Magic Sum, is shown below.

Mc10 = 46630

523	7549	2503	5167	7573	9133	8893	5107	139	43
7873	7	3673	6073	5689	2269	9103	1627	3259	7057
8233	7039	5323	2593	127	1723	1567	6679	5743	7603
6247	3907	3499	8599	1063	907	3319	5683	4987	8419
439	4813	8317	883	8863	9283	433	4219	9187	193
4903	277	6607	4999	6529	463	4513	1009	8443	8887
6373	4339	3643	6007	2953	3637	9319	5653	3253	1453
3373	3583	2647	7759	5953	9199	2287	4003	6733	1093
5869	6067	7699	223	3457	8263	5419	5827	727	3079
2797	9049	2719	4327	4423	1753	1777	6823	4159	8803

It can be noticed that also the Semi Diagonals sum to the Magic Sum Mc10.

Subject Composed Magic Squares can be transformed into four Way V type ZigZag Magic Squares by means of the transformation described in Section 14.8.15 above.

Attachment 14.8.4 shows for miscellaneous Magic Sums the first occurring Composed Magic Square described above.

14.8.17 Inlaid Magic Squares (10 x 10)
        Simple Magic Square Inlays (3 x 3)

The 10^th order Prime Number Inlaid Magic Square shown below, is composed out of a Concentric Border, an Associated Border and four each 3^th order Embedded Simple Magic Squares with different Magic Sums.

Mc10 = 6150 29 43 59 67 773 953 1033 1039 1063 1091 199 1213 769 521 353 311 479 727 547 1031 653 991 1151 89 557 1123 109 607 293 577 659 421 5 599 1193 97 613 1129 863 571 787 181 641 1109 47 619 1117 103 1103 443 857 127 1097 263 491 1051 409 433 1049 373 883 367 11 617 1223 13 631 1249 809 347 947 937 743 971 137 829 853 211 239 283 997 683 503 751 919 877 709 461 17 233 139 1187 1171 1163 457 277 197 191 167 1201

s3 1797 1839 1851 1893

The method to generate the order 8 Inlaid Magic Center Square with Order 3 Embedded Simple Magic Squares with different Magic Sums has been discussed in Section 14.6.12.

The order 10 Bordered Magic Square shown above can be transformed into the Window Type Magic Square shown below:

Mc10 = 6150 29 43 59 67 773 953 1033 1039 1063 1091 199 1151 89 557 991 293 1123 109 607 1031 653 5 599 1193 421 863 97 613 1129 577 659 641 1109 47 181 1103 619 1117 103 571 787 769 521 353 1213 547 311 479 727 443 857 503 751 919 683 17 877 709 461 373 883 1097 263 491 127 1049 1051 409 433 347 947 11 617 1223 367 809 13 631 1249 283 997 743 971 137 937 239 829 853 211 233 139 1187 1171 1163 457 277 197 191 167 1201

s3 1797 1839 1851 1893

Attachment 14.8.17 shows for a few Magic Sums the first occurring Bordered - and corresponding Window Type Magic Square.

Each square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the four inlays and variation of the borders (window).

14.8.18 Inlaid Magic Squares (10 x 10)
        (Pan) Magic Square Inlays (4 x 4)

The 10^th order Prime Number Inlaid Magic Square shown below, is composed out of an Associated Border and four each 4^th order Embedded Pan Magic Squares with different Magic Sums.

Mc10 = 7350 1409 1297 719 563 241 181 503 659 1237 541 1277 229 239 1093 1259 431 421 919 1109 373 863 953 1399 89 379 601 1427 113 739 787 449 317 151 1181 1171 521 331 1009 1019 1201 277 1321 1031 457 11 1327 701 839 13 1373 97 467 487 883 1163 43 47 1471 1499 1193 269 613 1433 197 757 1447 1523 19 71 1021 683 617 337 1033 1013 59 31 1487 1483 607 1097 1303 743 887 67 1511 1459 83 7 193 929 233 811 967 1289 1229 907 751 173 61

s4 2820 2880 3000 3060

The method to generate Order 10 Inlaid Magic Squares with Order 4 Embedded (Pan) Magic Squares with different Magic Sums has been discussed in Section 14.21.3.

14.8.20 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 Prime Number Magic Squares of order 11, which will be described in following sections.