Office Applications and Entertainment, Magic Squares

14.0    Special Magic Squares, Prime Numbers

14.32   Magic Squares (12 x 12)

14.32.1 Magic Squares, Concentric (12 x 12)

A 12^th order Prime Number Concentric Magic Square consists of a Prime Number Concentric Magic Square of the 10^th order with a border around it.

For Prime Number Concentric Magic Squares of order 12 with Magic Sum s12, it is convenient to split the supplementary rows and columns into parts summing to s4 = s12 / 3:

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	a101	a102	a103	a104	a105	a106	a107	a108
a109	a110	a111	a112	a113	a114	a115	a116	a117	a118	a119	a120
a121	a122	a123	a124	a125	a126	a127	a128	a129	a130	a131	a132
a133	a134	a135	a136	a137	a138	a139	a140	a141	a142	a143	a144

This results in following border equations:

a( 4) = s4 - a(3) - a( 2) - a( 1) a( 8) = s4 - a(7) - a( 6) - a( 5) a(12) = s4 - a(9) - a(10) - a(11) a(144) = s4/2 - a( 1) a(143) = s4/2 - a(11) a(142) = s4/2 - a(10) a(141) = s4/2 - a( 9) a(140) = s4/2 - a( 8) a(139) = s4/2 - a( 7) a(138) = s4/2 - a( 6) a(137) = s4/2 - a( 5) a(136) = s4/2 - a( 4) a(135) = s4/2 - a( 3) a(134) = s4/2 - a( 2) a(133) = s4/2 - a(12)

a(37) = s4 - a(  1) - a( 13) - a( 25) a(85) = s4 - a( 49) - a( 61) - a( 73) a(97) = s4 - a(109) - a(121) - a(133) a( 24) = s4/2 - a( 13) a( 36) = s4/2 - a( 25) a( 48) = s4/2 - a( 37) a( 60) = s4/2 - a( 49) a( 72) = s4/2 - a( 61) a( 84) = s4/2 - a( 73) a( 96) = s4/2 - a( 85) a(108) = s4/2 - a( 97) a(120) = s4/2 - a(109) a(132) = s4/2 - a(121)

which enable the development of a fast procedure to generate Prime Number Concentric Magic Squares of order 12 (ref. Priem12a).

Miscellaneous Prime Number Concentric Magic Squares of order 12, based on 10^th order Concentric Magic Squares as discussed in Section 14.8.7, are shown in following attachments:

• Concentric Magic Center Square                 (ref. Attachment 14.32.11)
  
  
• Concentric Pan Magic Center Square             (ref. Attachment 14.32.12)
  (Embedded Associated Square)

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

14.32.2 Magic Squares, Bordered (12 x 12)

Based on the collections of 10^th order Composed and miscellaneous Bordered Magic Squares, as discussed in Section 14.8.7 and Section 14.8.8 also following 12^th order Bordered Magic Squares can be generated with routine Priem12a:

• Composed Magic Center Squares:
  
  - Associated  , Simple Magic Sub Squares        (ref. Attachment 14.32.31)
  - Pan Magic   , Simple Magic Sub Squares        (ref. Attachment 14.32.32)
  - Composed    , Pan    Magic Sub Squares        (ref. Attachment 14.32.33)
  - Center Cross, Pan    Magic Sub Squares        (ref. Attachment 14.32.34)
  
  
• Bordered Center Square with embedded:
  
  - Ultra Magic Square, Partly Compact            (ref. Attachment 14.32.21)
  - Ultra Magic Square, Compact                   (ref. Attachment 14.32.22)
  - Most Perfect Pan Magic Square                 (ref. Attachment 14.32.23)
  - Composed Square                               (ref. Attachment 14.32.24)
  - Composed Square, Associated                   (ref. Attachment 14.32.25)
  - Franklin Pan Magic Center Square              (ref. Attachment 14.32.29)
             Most Perfect Pan Magic Center Square
  - Ultra Magic Center Square                     (ref. Attachment 14.32.26)
  
  
• Bordered Center Squares:
  
  - Split    Border with Concentric Center Square (ref. Attachment 14.32.27)
  - Composed Border with Associated Center Square (ref. Attachment 14.32.28)

It should be noted that the Attachments listed above contain only those solutions which could be found within 10 seconds.

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

14.32.3 Magic Squares, Inlaid (12 x 12)
        Order 4 Pan Magic Square Inlays

The 12^th order Prime Number Inlaid Magic Square shown below, is composed out of a Concentric Border, an Associated Border and four each 4^th order Embedded Pan Magic Squares with different Magic Sums.

Mc12 = 8820 17 283 1279 1361 41 409 1123 1367 163 383 1103 1291 593 1409 1297 719 563 241 181 503 659 1237 541 877 1117 1277 229 239 1093 1259 431 421 919 1109 373 353 1213 863 953 1399 89 379 601 1427 113 739 787 257 401 449 317 151 1181 1171 521 331 1009 1019 1201 1069 661 277 1321 1031 457 11 1327 701 839 13 1373 809 827 97 467 487 883 1163 43 47 1471 1499 1193 643 1051 269 613 1433 197 757 1447 1523 19 71 1021 419 991 683 617 337 1033 1013 59 31 1487 1483 607 479 823 1097 1303 743 887 67 1511 1459 83 7 193 647 947 929 233 811 967 1289 1229 907 751 173 61 523 179 1187 191 109 1429 1061 347 103 1307 1087 367 1453

s4 2820 2880 3000 3060

The method to generate the order 10 Inlaid Magic Center Square with Order 4 Embedded Pan Magic Squares with different Magic Sums has been discussed in Section 14.21.3.

The order 12 Bordered Magic Square shown above can be transformed into the Window Type Magic Square shown below:

Mc12 = 8820 17 283 1279 1361 41 409 1123 1367 163 383 1103 1291 593 229 239 1093 1259 1277 373 431 421 919 1109 877 1117 953 1399 89 379 863 787 601 1427 113 739 353 1213 317 151 1181 1171 449 1201 521 331 1009 1019 257 401 1321 1031 457 11 277 1373 1327 701 839 13 1069 661 1297 719 563 241 1409 541 181 503 659 1237 809 827 233 811 967 1289 929 61 1229 907 751 173 643 1051 467 487 883 1163 97 1193 43 47 1471 1499 419 991 613 1433 197 757 269 1021 1447 1523 19 71 479 823 617 337 1033 1013 683 607 59 31 1487 1483 647 947 1303 743 887 67 1097 193 1511 1459 83 7 523 179 1187 191 109 1429 1061 347 103 1307 1087 367 1453

s4 2820 2880 3000 3060

Attachment 14.32.3 page 1, shows for a few Magic Sums the first occurring Bordered Magic Square,

Attachment 14.32.3 page 2, shows the 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).

14.32.4 Magic Squares, Inlaid (12 x 12)
        Order 5 Ultra Magic Square Inlays

The 12^th order Prime Number Inlaid Magic Square shown below:

Mc12 = 39600 6581 6569 6491 3019 1009 907 277 379 2389 5861 5939 179 6329 5813 2753 857 4973 1019 5987 2699 1181 3251 2927 1811 4229 683 4799 1523 5153 3257 941 3011 3677 4967 3449 3911 2633 863 5657 3083 509 5303 2657 5717 3209 701 3761 5507 2087 2909 1013 4643 1367 5483 2969 1451 2741 3407 5477 6053 1871 5147 1193 5309 3413 353 3491 3167 5237 3719 431 6269 331 6553 1951 1789 4903 1759 6277 2677 2311 4567 1753 4729 547 1669 4783 3319 3673 3511 1597 3853 3307 4597 4231 4513 1093 439 5233 3391 1549 6343 1627 6151 3517 883 5407 3967 2689 3271 3109 3463 1999 5113 2803 2437 3727 3181 5437 2371 4789 5023 1879 4993 4831 229 5281 2467 4723 4357 757 271 6421 661 739 4211 6221 6323 5693 5591 3581 109 31 19

s5 15415 16045 16955 17585

is composed out of an Associated Border and four each 5^th order Embedded Ultra Magic Squares with different Magic Sums.

The method to generate Order 12 Inlaid Magic Squares with Order 5 Embedded (Ultra) Magic Squares with different Magic Sums has been discussed in Section 14.21.4.

14.32.5 Magic Squares, Eccentric (12 x 12)

Also for Prime Number Eccentric Magic Squares of order 12 it is convenient to split the supplementary rows and columns into parts summing to s4 = s12 / 3.

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	a101	a102	a103	a104	a105	a106	a107	a108
a109	a110	a111	a112	a113	a114	a115	a116	a117	a118	a119	a120
a121	a122	a123	a124	a125	a126	a127	a128	a129	a130	a131	a132
a133	a134	a135	a136	a137	a138	a139	a140	a141	a142	a143	a144

This enables, based on the same principles, the development of a fast procedure (ref. Priem12b):

• to read the previously generated Eccentric Magic Squares of order 10;
• 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 4;
• to complete the Eccentric Magic Square of order 12 with the two remaining 2 x 4 Magic Rectangles.

Attachment 14.32.2 shows, based on the 10^th order Eccentric Magic Squares as discussed in Section 14.8.11, 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.

14.32.6 Magic Squares, Composed (12 x 12)
        Center Cross, Order 5 (Pan) Magic Sub Squares

The 12^th order Prime Number Composed Magic Square shown below:

Mc12 = 31500

19	1993	4783	3307	3023	449	4801	73	4241	5009	3041	761
3607	2957	3019	1999	1543	4987	263	3119	941	773	4273	4019
4999	1549	367	3257	2953	5011	239	4973	4051	2129	1019	953
17	3253	4933	4549	373	257	4993	29	1031	5153	4751	2161
4483	3373	23	13	5233	457	4793	4931	2861	61	41	5231
227	5101	5119	79	389	53	5059	4919	5113	173	229	5039
5023	149	131	5171	4861	191	5197	331	137	5077	5021	211
67	1489	4457	3469	3643	4817	433	271	1699	5099	3877	2179
4007	3343	3559	1459	757	4969	281	3889	3499	2251	1867	1619
4951	727	307	3881	3259	4957	293	3847	1787	409	3511	3571
181	3797	4651	4219	277	521	4729	31	3583	5167	3767	577
3919	3769	151	97	5189	4831	419	5087	2557	199	103	5179

is an example of a Prime Number Magic Square with Magic Sum s12, composed of:

• Four each 5^th order (Pan) Magic Corner Squares (s5 = 5 * s12 / 12),
• Twentytwo supplementary pairs, each summing to 2 * s12 / 12

Subject Composed Magic Squares can be obtained by transformation of Bordered Magic Squares with Composed Magic Center Squares as discussed in Section 14.32.2 above.

Attachment 14.32.35 shows for a few Magic Sums the first occurring 12^th order Prime Number Composed Magic Square.

14.32.7 Magic Squares, Composed (12 x 12)
        Order 4 Pan Magic Sub Squares

The 12^th order Prime Number Composed Magic Square shown below:

Mc12 = 13860

59 181 2113 2267 2083 2297 29 211 197 43 2251 2129 2281 2099 227 13	73 173 2131 2243 2081 2293 23 223 179 67 2237 2137 2287 2087 229 17	167 379 1871 2203 1801 2273 97 449 439 107 2143 1931 2213 1861 509 37
397 317 1877 2029 1637 2269 157 557 433 281 1913 1993 2153 1753 673 41	149 683 1609 2179 1549 2239 89 743 701 131 2161 1627 2221 1567 761 71	521 463 1657 1979 1429 2207 293 691 653 331 1789 1847 2017 1619 881 103
523 499 1697 1901 1487 2111 313 709 613 409 1787 1811 1997 1601 823 199	337 1021 1259 2003 1193 2069 271 1087 1051 307 1973 1289 2039 1223 1117 241	811 827 1319 1663 929 2053 421 1217 991 647 1499 1483 1889 1093 1381 257

Contains 9 ea Order 4 Pan Magic Squares with the same Magic Sum (Mc4 = 4620).

A method to generate Magic Squares composed of order 4 Pan Magic Squares with the same Magic Sum  has been discussed in Section 14.11.6.

A method to generate Magic Squares composed of order 4 Pan Magic Squares with differnt Magic Sums has been discussed in Section 14.21.5.

14.32.8 Magic Squares, Composed (12 x 12)
        Order 3 Semi Magic Sub Squares

The 12^th order Prime Number Composed Magic Square shown below:

Mc12 = 65772

8111 8273 59 7109 641 8693 1223 7529 7691	311 10613 5519 5399 191 10853 10733 5639 71	349 10651 5443 5323 229 10891 10771 5563 109	4261 5701 6481 4441 2221 9781 7741 8521 181
659 8741 7043 6833 449 9161 8951 7253 239	7019 8831 593 3371 3323 9749 6053 4289 6101	4759 5521 6163 4093 2689 9661 7591 8233 619	4831 5659 5953 3853 2731 9859 7759 8053 631
2039 7643 6761 6173 1451 8819 8231 7349 863	1301 10223 4919 4799 1181 10463 10343 5039 1061	6203 9011 1229 3011 4229 9203 7229 3203 6011	6719 3593 6131 3413 4001 9029 6311 8849 1283
5569 6823 4051 2011 3529 10903 8863 6091 1489	1933 8629 5881 5689 1741 9013 8821 6073 1549	4789 6703 4951 3433 3271 9739 8221 6469 1753	7883 5861 2699 4481 3389 8573 4079 7193 5171

Contains 16 ea Order 3 Semi Magic Squares with the same Magic Sum (Mc3 = 16443).

A method to generate Magic Squares composed of order 3 Semi Magic Squares with the same Magic Sum  has been discussed in Section 14.11.3.

A method to generate Magic Squares composed of order 3 Semi Magic Squares with differnt Magic Sums has been discussed in Section 14.21.5.

14.32.9 Composed Magic Squares (12 x 12)
        Pan Magic Sub Squares (6 x 6)

Prime Number Magic Squares of order 12 - with Magic Sum 2 * s1 - can be composed out of 6^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	a101	a102	a103	a104	a105	a106	a107	a108
a109	a110	a111	a112	a113	a114	a115	a116	a117	a118	a119	a120
a121	a122	a123	a124	a125	a126	a127	a128	a129	a130	a131	a132
a133	a134	a135	a136	a137	a138	a139	a140	a141	a142	a143	a144

In section 14.4.12, a procedure was developed to generate 6^th order Prime Number Pan Magic Squares (Magic Sum s1) composed of Semi Magic Sub Squsres.

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 36 different Prime Numbers.

Attachment 14.32.81 contains a few of such sets which could be found by means of procedure Priem6e3.

Attachment 14.32.83 contains miscellaneous Prime Number Magic Squares composed of Pan Magic Sub Squares as generated with procedure Priem6e3.

Each square shown corresponds with numerous Composed Magic Squares with the same Magic Sum.

14.32.10 Pan Magic Squares (12 x 12)
         Pan Magic Square Inlays (6 x 6)

Prime Number Magic Squares composed of Pan Magic Sub Squares, as discussed in Section 14.32.9 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 and Complete.

Attachment 14.32.85 shows the Pan Magic Squares, which can be obtained by transformation of the Composed Magic Squares as shown in Attachment 14.32.83.

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

14.32.11 Associated Magic Squares (12 x 12)
         Composed of Simple Magic Squares (6 x 6)

Associated Magic Squares, composed of four each Simple Magic Squares, contain two sets of Complementary Anti Symmetric Magic Squares.

• Attachment 14.32.41 shows examples of such suitable 6^th order Anti Symmetric Magic Squares,
  which are composed of order 3 Semi Magic Anti Symmetric Sub Squares;
  
  
• Attachment 14.32.42 shows for miscellaneous Magic Sums the related 12^th order Associated Magic Squares,
  which can be generated with procedure Composed12;
  
  
• Attachment 14.32.43 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, Mc12 = 55692:

Inlaid Four Way V type ZigZag Croswise Symmetric Magic Square B2, Mc12 = 55692:

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.

14.32.12 Magic Squares, Composed (12 x 12)
         Order 6 Magic Cube Based

Order 12 Prime Number Magic Squares composed of Order 6 Magic Sub Squares can be constructed based on Prime Number Perfect Concentric Magic Cubes, as deducted in Section 7.8 of Chapter 'Prime Number Magic Cubes'.

A typical example of an order 12 Prime Number Composed Magic Squares (Mc12 = 36036), based on the planes of a Prime Number Perfect Magic Cube of half the Magic Sum, is shown below.

It can be noticed that also the Semi Diagonals and the Main Bent Diagonals sum to the Magic Sum Mc12.

Attachment 14.32.8 shows for miscellaneous Magic Sums a few of the Composed Magic Squares described above.

14.32.13 Summary

The obtained results regarding the 12^th order Prime Number Magic Squares and related sub squares, as deducted and discussed in previous sections, are summarized in following table:

Comparable routines as listed above, can be used to generate miscellaneous Prime Number Composed Magic Squares, which will be described in following sections.