Office Applications and Entertainment, Magic Squares

14.0    Special Magic Squares, Prime Numbers

14.11   Magic Squares, Higher Order, Composed

14.11.1 Semi Magic Sub Squares (3 x 3)

In Section 14.4.10 was discussed how Prime Number Magic Squares of order 6 - with Magic Sum 2 * s1 - can be composed out of 3^th order Prime Number Semi Magic Squares with Magic Sum s1.

Next sections show sets of Prime Number Semi Magic Squares of the 3^th order, enabling the construction of 9^th, 12^th, 15^th and 18^th order Magic Squares.

14.11.2 Magic Squares (9 x 9)

An example of a Magic Sum for which the required set of 1 Prime Number Magic Center Square and 8 Prime Number Semi Magic Squares can be found is 22947:

11027 11909 11 4229 3851 14867 7691 7187 8069	3719 11279 7949 6269 2039 14639 12959 9629 359	8111 7607 7229 3389 4271 15287 11447 11069 431
4019 11579 7349 5669 2339 14939 13259 9029 659	14489 977 7481 641 7649 14657 7817 14321 809	3191 10259 9497 8387 2081 12479 11369 10607 971
4871 9857 8219 7127 3779 12041 10949 9311 2687	5039 12107 5801 4691 3929 14327 13217 6911 2819	11519 8171 3257 5987 4349 12611 5441 10427 7079

The 8 border squares can be arranged in 8! ways around the center square, resulting in 8! * 8 * 12⁴ * 24⁴ = 2,22 10¹⁵ Magic Squares of the 9^th order with Magic Sum 68841.

Attachment 14.7.43 shows for miscellaneous Magic Sums the first occurring Prime Number Simple Magic Square composed of one Magic Center and eight Semi Magic Border Squares (7 Magic Lines).

Attachment 14.7.45 shows for miscellaneous Magic Sums the first occurring Prime Number Simple Magic Square composed of one Magic Center, four Semi Magic Corner (7 Magic Lines) and four Semi Magic Border Squares(6 Magic Lines).

Attachment 14.7.46 shows for miscellaneous Magic Sums the first occurring Prime Number Associated Magic Square composed of one Magic Center, four Semi Magic Anti Symmetric Corner and four Semi Magic Anti Symmetric Border Squares.

14.11.3 Magic Squares (12 x 12)

An example of a Magic Sum for which a set of 16 Prime Number Semi Magic Squares can be found is 16443:

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

These 16 squares can be arranged in 16! ways, resulting in 16! * 12⁸ * 24⁸ = 9,90 10³² Magic Squares of the 12^th order with Magic Sum 65772.

Attachment 14.7.44 shows for miscellaneous Magic Sums the first occurring Prime Number Simple Magic Square composed of sixteen Semi Magic Sub Squares (7 Magic Lines).

The Semi Magic Sub Squares in Attachment 14.7.44 have been arranged such that the Main Square is also composed of 4 Simple Magic Squares of order 6, as discussed in Section 14.4.10.

Attachment 14.7.44a shows for miscellaneous Magic Sums the first occurring Prime Number Simple Magic Square composed of sixteen Semi Magic Sub Squares (6 and 7 Magic Lines), generated with procedure Priem3f1.

Attachment 14.7.44b shows for miscellaneous Magic Sums the first occurring Prime Number Associated Magic Square composed of sixteen Semi Magic Anti Symmetric Sub Squares, generated with procedure Priem3f2.

14.11.4 Associated Magic Squares (15 x 15)

An example of a Magic Sum for which a set of 1 Prime Number Magic Center Square and 24 Prime Number Semi Magic Squares can be found is s1 = 48561:

The Magic Square shown above (MC = 242805) is Associated and composed of:

• One Magic Center Square,
• Four Complementary Pairs of Semi Magic Anti Symmetric Corner Squares (7 Magic Lines) and
• Eight Complementary Pairs of Semi Magic Anti Symmetric Border Squares (6 Magic Lines)

of order 3 with Magic Sum s1.

With some minor modifications to the procedure applied in Section 14.7.10 (Priem3d3) a procedure can be obtained to generate subject Associated Composed Magic Squares (Priem3f3).

Attachment 14.7.47 shows for miscellaneous Magic Sums the first occurring Prime Number Associated Magic Square composed of one Magic Center, eight Semi Magic Anti Symmetric Diagonal and sixteen Semi Magic Anti Symmetric Border Squares.

14.11.5 Associated Magic Squares (18 x 18)

With some minor modifications to the procedure applied in Section 14.11.9 above (Priem3f2) a procedure can be obtained to generate Associated Composed Magic Squares of order 18 (Priem3f4).

Attachment 14.7.48 shows for miscellaneous Magic Sums the first occurring Prime Number Associated Magic Square composed of 12 Semi Magic Anti Symmetric Diagonal and 24 Semi Magic Anti Symmetric Border Squares.

14.11.6 Pan Magic Sub Squares (4 x 4)

In Section 14.6.1 was discussed how Prime Number Magic Squares of order 8 - with Magic Sum 2 * s1 - can be composed out of 4^th order Prime Number Pan Magic Squares with Magic Sum s1.

Next sections show sets of Prime Number Pan Magic Squares of the 4^th order, enabling the construction of 12^th, 16^th, 20^th, 24^th and 28^th order Magic Squares.

14.11.7 Magic Squares (12 x 12)

An example of a Magic Sum for which a set of 9 Prime Number Pan Magic Squares can be found is 4620:

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

These 9 squares can be arranged in 9! ways, resulting in 9! * 384⁹ = 6,6 10²⁸ Magic Squares of the 12^th order with Magic Sum 13860.

14.11.8 Magic Squares (16 x 16)

An example of a Magic Sum for which a set of 16 Prime Number Pan Magic Squares can be found is 7980:

83 257 3709 3931 3673 3967 47 293 281 59 3907 3733 3943 3697 317 23	101 109 3853 3917 3823 3947 71 139 137 73 3889 3881 3919 3851 167 43	347 271 3623 3739 3433 3929 157 461 367 251 3643 3719 3833 3529 557 61	409 431 3547 3593 3217 3923 79 761 443 397 3581 3559 3911 3229 773 67
433 419 3491 3637 3251 3877 193 659 499 353 3557 3571 3797 3331 739 113	457 577 3329 3617 3083 3863 211 823 661 373 3533 3413 3779 3167 907 127	683 953 2971 3373 2551 3793 263 1373 1019 617 3307 3037 3727 2617 1439 197	1033 787 2903 3257 2393 3767 523 1297 1087 733 2957 3203 3467 2693 1597 223
821 691 3079 3389 2707 3761 449 1063 911 601 3169 3299 3541 2927 1283 229	827 1093 2689 3371 2383 3677 521 1399 1301 619 3163 2897 3469 2591 1607 313	1153 1171 2687 2969 1997 3659 463 1861 1303 1021 2837 2819 3527 2129 1993 331	991 1451 2267 3271 1907 3631 631 1811 1723 719 2999 2539 3359 2179 2083 359
941 929 2797 3313 2503 3607 647 1223 1193 677 3049 3061 3343 2767 1487 383	1433 1327 2333 2887 1709 3511 809 1951 1657 1103 2557 2663 3181 2039 2281 479	1249 1619 2089 3023 1721 3391 881 1987 1901 967 2741 2371 3109 2003 2269 599	1279 1381 2531 2789 2381 2939 1129 1531 1459 1201 2711 2609 2861 2459 1609 1051

These 16 squares can be arranged in 16! ways, resulting in 16! * 384¹⁶ = 4,7 10⁵⁴ Magic Squares of the 16^th order with Magic Sum 31920.

14.11.9 Magic Squares (20 x 20)

An example of a Magic Sum for which a set of 25 Prime Number Pan Magic Squares can be found is 13440:

These 25 squares can be arranged in 25! ways, resulting in 25! * 384²⁵ = 6,3 10⁸⁹ Magic Squares of the 20^th order with Magic Sum 67200.

14.11.10 Magic Squares (24 x 24)

An example of a Magic Sum for which a set of 36 Prime Number Pan Magic Squares can be found is 21420 (ref. Attachment 14.7.5).

These 36 squares can be arranged in 36! ways, resulting in 36! * 384³⁶ = 4,04 10¹³⁴ Magic Squares of the 24^th order with Magic Sum 128520.

14.11.11 Magic Squares (28 x 28)

An example of a Magic Sum for which a set of 49 Prime Number Pan Magic Squares can be found is 27720 (ref. Attachment 14.7.6).

These 49 squares can be arranged in 49! ways, resulting in 49! * 384⁴⁹ = 2,608 10¹⁸⁹ Magic Squares of the 28^th order with Magic Sum 194040.

14.11.12 Pan Magic Sub Squares (5 x 5)

In Section 14.8.14 was discussed how 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.

Next sections show sets of Prime Number Pan Magic Squares of the 5^th order, enabling the construction of 15^th and 20^th order Magic Squares.

14.11.13 Magic Squares (15 x 15)

An example of a Magic Sum for which a set of 9 Prime Number Pan Magic Squares can be found is MC5 = 13125:

19 1993 4783 3307 3023 3607 2957 3019 1999 1543 4999 1549 367 3257 2953 17 3253 4933 4549 373 4483 3373 23 13 5233	73 4241 5009 3041 761 3119 941 773 4273 4019 4973 4051 2129 1019 953 29 1031 5153 4751 2161 4931 2861 61 41 5231	89 1723 4621 3613 3079 3631 3529 3089 1759 1117 4759 1153 127 3547 3539 43 3557 5209 4153 163 4603 3163 79 53 5227
67 1489 4457 3469 3643 4007 3343 3559 1459 757 4951 727 307 3881 3259 181 3797 4651 4219 277 3919 3769 151 97 5189	271 1699 5099 3877 2179 3889 3499 2251 1867 1619 3847 1787 409 3511 3571 31 3583 5167 3767 577 5087 2557 199 103 5179	113 2399 4283 3673 2657 3823 2789 2621 2381 1511 4889 1493 1051 2939 2753 167 2903 5021 4001 1033 4133 3541 149 131 5171
443 2459 4813 2801 2609 3217 2633 2393 2729 2153 4679 2423 557 3049 2417 389 2833 4703 4373 827 4397 2777 659 173 5119	251 4201 4357 3187 1129 3691 619 1097 4261 3457 5107 3517 2791 1123 587 223 1091 4597 4363 2851 3853 3697 283 191 5101	1523 2767 4987 2467 1381 2551 2647 1171 4079 2677 3727 3989 241 2731 2437 421 2521 4993 3637 1553 4903 1201 1733 211 5077

These 9 squares can be arranged in 9! ways, resulting in 9! * 28800⁹ = 4,95 10⁴⁵ Magic Squares of the 15^th order with Magic Sum Mc15 = 39375.

14.11.14 Magic Squares (20 x 20)

An example of a Magic Sum for which a set of 16 Prime Number Pan Magic Squares can be found is Mc5 = 14175:

These 16 squares can be arranged in 16! ways, resulting in 16! * 28800¹⁶ = 4,687 10⁸⁴ Magic Squares of the 20^th order with Magic Sum MC20 = 56700.

The Composed Magic Square shown above, can be transformed into:

• A Magic Square composed of four each order 10 Pan Magic Squares (Inlaid), which can be transformed into
• A Pan Magic Square with four each order 10 Pan Magic Square Inlays

as illustrated in Attachment 14.11.14.

14.11.15 Associated Magic Squares (15 x 15)

Associated Magic Squares of order 15 can be composed out of 9 Prime Number Magic Squares of order 5 as illustrated below:

The Prime Number Associated Magic Square shown above (Mc15 = 300315) is composed of:

• One Ultra Magic Center Square and
• Four Complementary Pairs of Anti Symmetric Magic Squares (ref. Section 14.3.10)

of order 5 with Magic Sum Mc5 = 100105.

The 4 independent Border Squares can be arranged in 1680 ways (= 8 * 7 * 6 * 5), which results in 1680 * 128 * n1 * n2 * n3 * n4 Magic Squares of the 15^th order with Magic Sum Mc15 = 300315.

Based on the transformations, as described in Section 3.5, ni = 32 (i = 1 ... 4) which results already in 1680 * 128 * 32⁴ = 2,25 10¹¹ Magic Squares which can be constructed based on the distinct integers contained in the Center Square and the 4 independent Border Squares with Mc5 = 100105.

The actual order of magnitude of n_i (i = 1 ... 4) is however much higher.

14.11.16 Pan Magic Sub Squares (6 x 6)

In Section 14.32.8 was discussed how 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.

Next section shows sets of Prime Number Pan Magic Squares of the 6^th order, enabling the construction of 18^th order Magic Squares.

14.11.17 Magic Squares (18 x 18)

An example of a Magic Sum for which a set of 9 Prime Number Pan Magic Squares can be found is MC6 = 44730:

Attachment 14.11.17 contains miscellaneous Prime Number Magic Squares composed of 9 Pan Magic Sub Squares as generated with procedure Priem6e3.

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

14.11.18 Summary

The obtained results regarding miscellaneous types of higher order Composed 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 Prime Number Inlaid Magic Squares, which will be described in following sections.