Office Applications and Entertainment, Magic Squares

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14.0    Special Magic Squares, Prime Numbers

14.11   Magic Squares, Higher Order, Composed

14.11.1 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 4th order Prime Number Pan Magic Squares with Magic Sum s1.

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

14.11.2 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 12th order with Magic Sum 13860.

14.11.3 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 16th order with Magic Sum 31920.

14.11.4 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:

139 101 6563 6637
6491 6709 67 173
157 83 6581 6619
6653 6547 229 11
61 353 6353 6673
6323 6703 31 383
367 47 6659 6367
6689 6337 397 17
491 457 6043 6449
5791 6701 239 709
677 271 6229 6263
6481 6011 929 19
251 419 6217 6553
6079 6691 113 557
503 167 6469 6301
6607 6163 641 29
587 691 5801 6361
5483 6679 269 1009
919 359 6133 6029
6451 5711 1237 41
653 607 5981 6199
5519 6661 191 1069
739 521 6067 6113
6529 5651 1201 59
577 1019 5557 6287
5273 6571 293 1303
1163 433 6143 5701
6427 5417 1447 149
733 569 5881 6257
5569 6569 421 881
839 463 5987 6151
6299 5839 1151 151
823 977 5443 6197
5119 6521 499 1301
1277 523 5897 5743
6221 5419 1601 199
971 1459 4973 6037
4621 6389 619 1811
1747 683 5749 5261
6101 4909 2099 331
1097 1249 5171 5923
4721 6373 647 1699
1549 797 5623 5471
6073 5021 1999 347
1327 1439 4831 5843
4363 6311 859 1907
1889 877 5393 5281
5861 4813 2357 409
1423 1523 4967 5527
4217 6277 673 2273
1753 1193 5297 5197
6047 4447 2503 443
1697 1373 4933 5437
4099 6271 863 2207
1787 1283 5023 5347
5857 4513 2621 449
1997 1531 4481 5431
3701 6211 1217 2311
2239 1289 4723 5189
5503 4409 3019 509
937 2153 4483 5867
4177 6173 631 2459
2237 853 5783 4567
6089 4261 2543 547
1801 2129 4283 5227
3389 6121 907 3023
2437 1493 4919 4591
5813 3697 3331 599
1831 2381 4297 4931
3449 5779 983 3229
2423 1789 4889 4339
5737 3491 3271 941
2203 2081 4057 5099
3467 5689 1613 2671
2663 1621 4517 4639
5107 4049 3253 1031
1619 2791 3917 5113
3361 5669 1063 3347
2803 1607 5101 3929
5657 3373 3359 1051
1567 2447 4027 5399
3767 5659 1307 2707
2693 1321 5153 4273
5413 4013 2953 1061
2857 2467 3533 4583
2609 5507 1933 3391
3187 2137 3863 4253
4787 3329 4111 1213
2477 2383 3943 4637
3529 5051 2063 2797
2777 2083 4243 4337
4657 3923 3191 1669
1987 3203 3461 4789
3251 4999 1777 3413
3259 1931 4733 3517
4943 3307 3469 1721
2789 2887 3637 4127
3061 4703 2213 3463
3083 2593 3931 3833
4507 3257 3659 2017

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

14.11.5 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 24th order with Magic Sum 128520.

14.11.6 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 28th order with Magic Sum 194040.

14.11.7 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 3th order Prime Number Semi Magic Squares with Magic Sum s1.

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

14.11.8 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 9th 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.

14.11.9 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 12th 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.

14.11.10 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:

16703 12821 19037
31721 9587 7253
137 26153 22271
5861 14537 28163
13577 14591 20393
29123 19433 5
8237 10691 29633
9521 20123 18917
30803 17747 11
863 22307 25391
16607 8837 23117
31091 17417 53
9677 17483 21401
16691 4967 26903
22193 26111 257
12611 7457 28493
4937 23627 19997
31013 17477 71
21347 21521 5693
26981 2963 18617
233 24077 24251
9803 13217 25541
7211 18443 22907
31547 16901 113
29027 3041 16493
2213 14747 31601
17321 30773 467
19211 1697 27653
1523 26321 20717
27827 20543 191
2837 20477 25247
16931 8627 23003
28793 19457 311
11411 10007 27143
9413 18047 21101
27737 20507 317
32027 647 15887
47 16187 32327
16487 31727 347
32057 11867 4637
11273 14327 22961
5231 22367 20963
32063 12917 3581
9371 23747 15443
7127 11897 29537
32183 11831 4547
11657 6053 30851
4721 30677 13163
31907 1601 15053
773 17627 30161
15881 29333 3347
32261 15473 827
9467 13931 25163
6833 19157 22571
8123 8297 32141
13757 29411 5393
26681 10853 11027
32303 14897 1361
12377 8747 27437
3881 24917 19763
32117 6263 10181
5471 27407 15683
10973 14891 22697
32321 14957 1283
9257 23537 15767
6983 10067 31511
32363 14627 1571
13457 12251 22853
2741 21683 24137
32369 12941 3251
11981 17783 18797
4211 17837 26513
10103 6221 32237
25121 22787 653
13337 19553 15671

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.

14.11.11 Associated Magic Squares (15 x 15)

Alternatively Magic Squares of order 15 can be composed out of 9 Prime Number Magic Squares of order 5:

9173 27059 27893 1721 34259
20639 14633 3911 22853 38069
28631 22811 23981 24083 599
25229 2579 31079 33179 8039
16433 33023 13241 18269 19139
28493 38861 14969 1811 15971
29873 2789 6959 27791 32693
1709 20063 26141 39953 12239
36821 26993 13313 13229 9749
3209 11399 38723 17321 29453
27329 32141 3881 10163 26591
8291 15671 21323 28109 26711
15761 26321 16883 6269 34871
24473 4271 20129 39761 11471
24251 21701 37889 15803 461
809 38933 293 21851 38219
33461 34961 8849 16703 6131
26183 1049 36671 13259 22943
38153 7013 32789 8501 13649
1499 18149 21503 39791 19163
40013 19559 719 28001 11813
491 27773 12503 39089 20249
11579 39779 20021 263 28463
19793 953 27539 12269 39551
28229 12041 39323 20483 29
20879 251 18539 21893 38543
26393 31541 7253 33029 1889
17099 26783 3371 38993 13859
33911 23339 31193 5081 6581
1823 18191 39749 1109 39233
39581 24239 2153 18341 15791
28571 281 19913 35771 15569
5171 33773 23159 13721 24281
13331 11933 18719 24371 31751
13451 29879 36161 7901 12713
10589 22721 1319 28643 36833
30293 26813 26729 13049 3221
27803 89 13901 19979 38333
7349 12251 33083 37253 10169
24071 38231 25073 1181 11549
20903 21773 26801 7019 23609
32003 6863 8963 37463 14813
39443 15959 16061 17231 11411
1973 17189 36131 25409 19403
5783 38321 12149 12983 30869

The Prime Number Magic Square shown above (MC = 300315) is Associated and 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 s1 = 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 16th order with Magic Sum 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 MC = 100105.

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

14.11.12 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.

14.11.13 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:

Type

Characteristics

Subroutine

Results

Order 9

Eight Semi Magic Border Squares
One Magic Center Square

Priem3d1

Attachment 14.7.43

Four Semi Magic Border Squares
Four Semi Magic Corner Squares
One Magic Center Square

Priem3d2

Attachment 14.7.45

Four Semi Magic Anti Symmetric Border Squares
Four Semi Magic Anti Symmetric Corner Squares
One Magic Center Square

Priem3d3

Attachment 14.7.46

Order 12

Sixteen Semi Magic Squares

-

Attachment 14.7.44

Eight Semi Magic Diagonal Squares
Eight Semi Magic Border Squares

Priem3f1

Attachment 14.7.44a

Eight Semi Magic Anti Symmetric Diagonal Squares
Eight Semi Magic Anti Symmetric Border Squares

Priem3f2

Attachment 14.7.44b

Order 15

Eight Semi Magic Anti Symmetric Diagonal Squares
Sixteen Semi Magic Anti Symmetric Border Squares
One Magic Center Square

Priem3f3

Attachment 14.7.47

Order 18

12 Semi Magic Anti Symmetric Diagonal Squares
24 Semi Magic Anti Symmetric Border Squares

Priem3f4

Attachment 14.7.48

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


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