Office Applications and Entertainment, Magic Squares

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14.0    Special Magic Squares, Prime Numbers

14.34   Magic Squares (14 x 14)

14.34.1 Magic Squares, Concentric (14 x 14)

A 14th order Prime Number Concentric Magic Square consists of a Prime Number Concentric Magic Square of the 12th order with a border around it.

For Prime Number Concentric Magic Squares of order 14 with Magic Sum s14, it is convenient to split the supplementary rows and columns into parts summing to s5 = 5 * s14 / 14 and s4 = 4 * s14 / 14:

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 a145 a146 a147 a148 a149 a150 a151 a152 a153 a154
a155 a156 a157 a158 a159 a160 a161 a162 a163 a164 a165 a166 a167 a168
a169 a170 a171 a172 a173 a174 a175 a176 a177 a178 a179 a180 a181 a182
a183 a184 a185 a186 a187 a188 a189 a190 a191 a192 a193 a194 a195 a196

This results in following border equations:

a( 4) = s4 - a( 3) - a( 2) - a( 1)
a( 9) = s5 - a( 8) - a( 7) - a( 6) - a( 5)
a(14) = s5 - a(13) - a(12) - a(11) - a(10)

a(196) = p14 - a( 1)
a(195) = p14 - a(13)
a(194) = p14 - a(12)
a(193) = p14 - a(11)
a(192) = p14 - a(10)
a(191) = p14 - a( 9)
a(190) = p14 - a( 8)
a(189) = p14 - a( 7)
a(188) = p14 - a( 6)
a(187) = p14 - a( 5)
a(186) = p14 - a( 4)
a(185) = p14 - a( 3)
a(184) = p14 - a( 2)
a(183) = p14 - a(14)

a( 43) = s4 - a(  1) - a( 15) - a( 29)
a(113) = s5 - a( 99) - a( 85) - a( 71) - a( 57)
a(127) = s5 - a(183) - a(169) - a(155) - a(141)

a( 28) = p14 - a( 15)
a( 42) = p14 - a( 29)
a( 56) = p14 - a( 43)
a( 70) = p14 - a( 57)
a( 84) = p14 - a( 71)
a( 98) = p14 - a( 85)
a(112) = p14 - a( 99)
a(126) = p14 - a(113)
a(140) = p14 - a(127)
a(154) = p14 - a(141)
a(168) = p14 - a(155)
a(182) = p14 - a(169)

which enable the development of a fast procedure to generate Prime Number Concentric Magic Squares of order 14 (ref. Priem14a).

Miscellaneous Prime Number Concentric Magic Squares of order 14, based on 12th order Concentric Magic Squares as discussed in Section 14.32.1, are shown in Attachment 14.34.1.

14.34.2 Magic Squares, Bordered (14 x 14)

Based on the collections of 12th order Composed and miscellaneous Bordered Magic Squares, as discussed in Section 14.32.2 also following 14th order Bordered Magic Squares can be generated with routine Priem14a:

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.

14.34.3 Magic Squares, Inlaid (14 x 14)
        Order 5 Ultra Magic Square Inlays

The 14th order Prime Number Inlaid Magic Square shown below, is composed out of a Concentric Border, an Associated Border and four each 5th order Embedded Ultra Magic Squares with different Magic Sums.

Mc14 = 46200
23 127 6521 6529 29 173 6451 6547 37 131 457 6337 6389 6449
487 6581 6569 6491 3019 1009 907 277 379 2389 5861 5939 179 6113
6317 6329 5813 2753 857 4973 1019 5987 2699 1181 3251 2927 1811 283
6373 4229 683 4799 1523 5153 3257 941 3011 3677 4967 3449 3911 227
233 2633 863 5657 3083 509 5303 2657 5717 3209 701 3761 5507 6367
389 2087 2909 1013 4643 1367 5483 2969 1451 2741 3407 5477 6053 6211
6217 1871 5147 1193 5309 3413 353 3491 3167 5237 3719 431 6269 383
6361 331 6553 1951 1789 4903 1759 6277 2677 2311 4567 1753 4729 239
241 547 1669 4783 3319 3673 3511 1597 3853 3307 4597 4231 4513 6359
719 1093 439 5233 3391 1549 6343 1627 6151 3517 883 5407 3967 5881
6199 2689 3271 3109 3463 1999 5113 2803 2437 3727 3181 5437 2371 401
6203 4789 5023 1879 4993 4831 229 5281 2467 4723 4357 757 271 397
6287 6421 661 739 4211 6221 6323 5693 5591 3581 109 31 19 313
151 6473 79 71 6571 6427 149 53 6563 6469 6143 263 211 6577
s5
15415 16045
16955 17585

The method to generate the order 12 Inlaid Magic Center Square with Order 5 Embedded Ultra Magic Squares with different Magic Sums has been discussed in Section 14.21.4.

The order 14 Bordered Magic Square shown above can be transformed into the Window Type Magic Square shown below:

Mc14 = 46200
23 127 6521 6529 29 173 6451 6547 37 131 457 6337 6389 6449
487 5813 2753 857 4973 1019 6329 1811 5987 2699 1181 3251 2927 6113
6317 683 4799 1523 5153 3257 4229 3911 941 3011 3677 4967 3449 283
6373 863 5657 3083 509 5303 2633 5507 2657 5717 3209 701 3761 227
233 2909 1013 4643 1367 5483 2087 6053 2969 1451 2741 3407 5477 6367
389 5147 1193 5309 3413 353 1871 6269 3491 3167 5237 3719 431 6211
6217 6569 6491 3019 1009 907 6581 179 277 379 2389 5861 5939 383
6361 661 739 4211 6221 6323 6421 19 5693 5591 3581 109 31 239
241 6553 1951 1789 4903 1759 331 4729 6277 2677 2311 4567 1753 6359
719 1669 4783 3319 3673 3511 547 4513 1597 3853 3307 4597 4231 5881
6199 439 5233 3391 1549 6343 1093 3967 1627 6151 3517 883 5407 401
6203 3271 3109 3463 1999 5113 2689 2371 2803 2437 3727 3181 5437 397
6287 5023 1879 4993 4831 229 4789 271 5281 2467 4723 4357 757 313
151 6473 79 71 6571 6427 149 53 6563 6469 6143 263 211 6577
s5
15415 16045
16955 17585

Attachment 14.34.3 page 1, shows for a few Magic Sums the first occurring Bordered Magic Square,

Attachment 14.34.3 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).

14.34.4 Magic Squares, Eccentric (14 x 14)

Also for Prime Number Eccentric Magic Squares of order 14 it is convenient to split the supplementary rows and columns into: parts summing to s5 = 5 * s14 / 14 and s4 = 4 * s14 / 14:

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 a145 a146 a147 a148 a149 a150 a151 a152 a153 a154
a155 a156 a157 a158 a159 a160 a161 a162 a163 a164 a165 a166 a167 a168
a169 a170 a171 a172 a173 a174 a175 a176 a177 a178 a179 a180 a181 a182
a183 a184 a185 a186 a187 a188 a189 a190 a191 a192 a193 a194 a195 a196

This enables, based on the same principles, the development of a fast procedure (ref. Priem14b):

  • to read the previously generated Eccentric Magic Squares of order 12;
  • 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 14 with the two remaining 2 x 5 Magic Rectangles.

Attachment 14.34.2 shows, based on the 12th order Eccentric Magic Squares as discussed in Section 14.32.5, 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.

14.34.5 Magic Squares, Composed (14 x 14)
        Center Cross

The 14th order Prime Number Composed Magic Square shown below:

Mc14 = 24990
3049 2269 37 1987 2797 571 433 3137 2017 3067 271 2801 2357 197
2293 1303 1759 3259 859 1237 3307 263 3181 631 1543 2447 1289 1619
13 1783 3559 109 1699 3547 3329 241 157 1657 3541 107 1709 3539
1583 773 2999 521 1301 3533 227 3343 769 1213 3373 1553 503 3299
311 2711 2333 1277 2267 1811 239 3331 1123 2281 1951 389 2939 2027
3461 1871 23 3557 1787 11 2311 1259 3463 1861 31 3413 1913 29
179 3433 3457 79 101 2039 71 3313 3347 3359 163 199 2081 3169
3391 137 113 3491 3469 1531 257 3499 223 211 3407 3371 1489 401
3253 1429 673 2003 2843 509 2927 643 3319 1327 709 2287 2521 547
1999 2203 1153 3083 953 1319 3221 349 1669 2557 1129 2887 1171 1297
103 1723 3529 269 1559 3527 313 3257 367 1471 3517 181 1663 3511
1567 727 3061 317 2141 2897 2089 1481 1283 1049 3023 251 2243 2861
487 2617 2251 1571 1367 2417 3079 491 683 2399 2273 1901 1013 2441
3301 2011 43 3467 1847 41 3187 383 3389 1907 59 3203 2099 53

is an example of a Prime Number Magic Square with Magic Sum s14, composed of:

  • Four each 6th order (Pan) Magic Corner Squares (s6 = 6 * s14 / 14),
  • Twentysix supplementary pairs, each summing to 2 * s14 / 14

Subject Composed Magic Squares can be obtained by transformation of Bordered Magic Squares with Composed Magic Center Squares as discussed in Section 14.34.2 above.

Attachment 14.34.36 shows for a few Magic Sums the first occurring 14th order Prime Number Composed Magic Square.

14.34.6 Magic Squares, Inlaid (14 x 14)

The 14th order Prime Number Inlaid Magic Square shown below:

Mc14 = 24990
3319 1327 709 433 3253 1429 673 2003 2843 509 3137 2287 2521 547
1669 2557 1129 3307 1999 2203 1153 3083 953 1319 263 2887 1171 1297
367 1471 3517 3329 103 1723 3529 269 1559 3527 241 181 1663 3511
179 3433 3457 71 79 101 2039 3347 3359 163 3313 199 2081 3169
2017 3067 271 227 3049 2269 37 1987 2797 571 3343 2801 2357 197
3181 631 1543 239 2293 1303 1759 3259 859 1237 3331 2447 1289 1619
157 1657 3541 2311 13 1783 3559 109 1699 3547 1259 107 1709 3539
769 1213 3373 2927 1583 773 2999 521 1301 3533 643 1553 503 3299
1123 2281 1951 3221 311 2711 2333 1277 2267 1811 349 389 2939 2027
3463 1861 31 313 3461 1871 23 3557 1787 11 3257 3413 1913 29
3391 137 113 257 3491 3469 1531 223 211 3407 3499 3371 1489 401
1283 1049 3023 2089 1567 727 3061 317 2141 2897 1481 251 2243 2861
683 2399 2273 3079 487 2617 2251 1571 1367 2417 491 1901 1013 2441
3389 1907 59 3187 3301 2011 43 3467 1847 41 383 3203 2099 53

is an example of a Prime Number Inlaid Magic Square with Magic Sum s14, composed of:

  • Three each 6th order (Pan) Magic Square Inlays (s6 = 6 * s14 / 14),
  • One 6th order (Pan) Magic Center Square (s6 = 6 * s14 / 14),
  • Twentysix supplementary pairs, each summing to 2 * s14 / 14

Subject Inlaid Magic Squares can be obtained by transformation of Composed Magic Squares as discussed in Section 14.32.4 above.

Attachment 14.34.37 shows for a few Magic Sums the first occurring 14th order Prime Number Inlaid Magic Square.

14.34.7 Associated Magic Squares (14 x 14)
        Composed of Semi Magic Squares (7 x 7)

Associated Magic Squares, composed of four each Semi Magic Squares, contain two sets of Complementary Anti Symmetric Semi Magic Squares, which can be arranged as illustrated below:

Mc14 = 85414
719 683 10883 12119 11939 5711 653 3911 4919 9473 11633 7829 3449 1493
131 941 11321 12011 11813 2789 3701 4253 5099 8951 10331 10253 2711 1109
11393 10631 2111 269 419 8423 9461 5939 8273 4091 2039 4409 7853 10103
12161 12149 89 5 233 7481 10589 10301 6113 1889 401 1913 10391 11699
3191 1229 9851 10133 3659 3371 11273 8669 6173 3461 9803 5081 5981 3539
3203 5903 7229 8069 2963 8369 6971 8123 4679 5501 6899 10223 2579 4703
11909 11171 1223 101 11681 6563 59 1511 7451 9341 1601 2999 9743 10061
2141 2459 9203 10601 2861 4751 10691 12143 5639 521 12101 10979 1031 293
7499 9623 1979 5303 6701 7523 4079 5231 3833 9239 4133 4973 6299 8999
8663 6221 7121 2399 8741 6029 3533 929 8831 8543 2069 2351 10973 9011
503 1811 10289 11801 10313 6089 1901 1613 4721 11969 12197 12113 53 41
2099 4349 7793 10163 8111 3929 6263 2741 3779 11783 11933 10091 1571 809
11093 9491 1949 1871 3251 7103 7949 8501 9413 389 191 881 11261 12071
10709 8753 4373 569 2729 7283 8291 11549 6491 263 83 1319 11519 11483

Due to the chosen arrangement the Composed Associated Magic Square shown above contains an order 6 Associated Magic Centre Square and an order 8 Associated Square Inlay.

  • Attachment 14.34.41 shows of few more examples of suitable 7th order Anti Symmetric Semi Magic Squares,
    containing order 3 and 4 Semi Magic Sub Squares (ref. PriemSqrs7).

  • Attachment 14.34.42 shows for miscellaneous Magic Sums the related 14th order Associated Magic Squares;

  • Attachment 14.34.43 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, Mc14 = 85414:

A1 (Associated)
719 683 10883 12119 11939 5711 653 3911 4919 9473 11633 7829 3449 1493
131 941 11321 12011 11813 2789 3701 4253 5099 8951 10331 10253 2711 1109
11393 10631 2111 269 419 8423 9461 5939 8273 4091 2039 4409 7853 10103
12161 12149 89 5 233 7481 10589 10301 6113 1889 401 1913 10391 11699
3191 1229 9851 10133 3659 3371 11273 8669 6173 3461 9803 5081 5981 3539
3203 5903 7229 8069 2963 8369 6971 8123 4679 5501 6899 10223 2579 4703
11909 11171 1223 101 11681 6563 59 1511 7451 9341 1601 2999 9743 10061
2141 2459 9203 10601 2861 4751 10691 12143 5639 521 12101 10979 1031 293
7499 9623 1979 5303 6701 7523 4079 5231 3833 9239 4133 4973 6299 8999
8663 6221 7121 2399 8741 6029 3533 929 8831 8543 2069 2351 10973 9011
503 1811 10289 11801 10313 6089 1901 1613 4721 11969 12197 12113 53 41
2099 4349 7793 10163 8111 3929 6263 2741 3779 11783 11933 10091 1571 809
11093 9491 1949 1871 3251 7103 7949 8501 9413 389 191 881 11261 12071
10709 8753 4373 569 2729 7283 8291 11549 6491 263 83 1319 11519 11483
B1 (Associated)
719 3911 683 4919 10883 9473 12119 11633 11939 7829 5711 3449 653 1493
2141 12143 2459 5639 9203 521 10601 12101 2861 10979 4751 1031 10691 293
131 4253 941 5099 11321 8951 12011 10331 11813 10253 2789 2711 3701 1109
7499 5231 9623 3833 1979 9239 5303 4133 6701 4973 7523 6299 4079 8999
11393 5939 10631 8273 2111 4091 269 2039 419 4409 8423 7853 9461 10103
8663 929 6221 8831 7121 8543 2399 2069 8741 2351 6029 10973 3533 9011
12161 10301 12149 6113 89 1889 5 401 233 1913 7481 10391 10589 11699
503 1613 1811 4721 10289 11969 11801 12197 10313 12113 6089 53 1901 41
3191 8669 1229 6173 9851 3461 10133 9803 3659 5081 3371 5981 11273 3539
2099 2741 4349 3779 7793 11783 10163 11933 8111 10091 3929 1571 6263 809
3203 8123 5903 4679 7229 5501 8069 6899 2963 10223 8369 2579 6971 4703
11093 8501 9491 9413 1949 389 1871 191 3251 881 7103 11261 7949 12071
11909 1511 11171 7451 1223 9341 101 1601 11681 2999 6563 9743 59 10061
10709 11549 8753 6491 4373 263 569 83 2729 1319 7283 11519 8291 11483

Inlaid Four Way V type ZigZag Croswise Symmetric Magic Square B2, Mc14 = 85414:

A2 (Complete)
719 683 10883 12119 11939 5711 653 1493 3449 7829 11633 9473 4919 3911
131 941 11321 12011 11813 2789 3701 1109 2711 10253 10331 8951 5099 4253
11393 10631 2111 269 419 8423 9461 10103 7853 4409 2039 4091 8273 5939
12161 12149 89 5 233 7481 10589 11699 10391 1913 401 1889 6113 10301
3191 1229 9851 10133 3659 3371 11273 3539 5981 5081 9803 3461 6173 8669
3203 5903 7229 8069 2963 8369 6971 4703 2579 10223 6899 5501 4679 8123
11909 11171 1223 101 11681 6563 59 10061 9743 2999 1601 9341 7451 1511
10709 8753 4373 569 2729 7283 8291 11483 11519 1319 83 263 6491 11549
11093 9491 1949 1871 3251 7103 7949 12071 11261 881 191 389 9413 8501
2099 4349 7793 10163 8111 3929 6263 809 1571 10091 11933 11783 3779 2741
503 1811 10289 11801 10313 6089 1901 41 53 12113 12197 11969 4721 1613
8663 6221 7121 2399 8741 6029 3533 9011 10973 2351 2069 8543 8831 929
7499 9623 1979 5303 6701 7523 4079 8999 6299 4973 4133 9239 3833 5231
2141 2459 9203 10601 2861 4751 10691 293 1031 10979 12101 521 5639 12143
B2 (Crosswise Symmetric)
719 1493 683 3449 10883 7829 12119 11633 11939 9473 5711 4919 653 3911
10709 11483 8753 11519 4373 1319 569 83 2729 263 7283 6491 8291 11549
131 1109 941 2711 11321 10253 12011 10331 11813 8951 2789 5099 3701 4253
11093 12071 9491 11261 1949 881 1871 191 3251 389 7103 9413 7949 8501
11393 10103 10631 7853 2111 4409 269 2039 419 4091 8423 8273 9461 5939
2099 809 4349 1571 7793 10091 10163 11933 8111 11783 3929 3779 6263 2741
12161 11699 12149 10391 89 1913 5 401 233 1889 7481 6113 10589 10301
503 41 1811 53 10289 12113 11801 12197 10313 11969 6089 4721 1901 1613
3191 3539 1229 5981 9851 5081 10133 9803 3659 3461 3371 6173 11273 8669
8663 9011 6221 10973 7121 2351 2399 2069 8741 8543 6029 8831 3533 929
3203 4703 5903 2579 7229 10223 8069 6899 2963 5501 8369 4679 6971 8123
7499 8999 9623 6299 1979 4973 5303 4133 6701 9239 7523 3833 4079 5231
11909 10061 11171 9743 1223 2999 101 1601 11681 9341 6563 7451 59 1511
2141 293 2459 1031 9203 10979 10601 12101 2861 521 4751 5639 10691 12143

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.

14.34.8 Magic Squares, Composed (14 x 14)

The Composed Associated Magic Square shown in Section 14.34.6 above. can be transformed - by means of row and column permutations - to the Composed Associated Magic Square shown below:

Mc14 = 85414
3659 3371 11273 3191 1229 9851 10133 9803 5081 5981 3539 8669 6173 3461
2963 8369 6971 3203 5903 7229 8069 6899 10223 2579 4703 8123 4679 5501
11681 6563 59 11909 11171 1223 101 1601 2999 9743 10061 1511 7451 9341
11939 5711 653 719 683 10883 12119 11633 7829 3449 1493 3911 4919 9473
11813 2789 3701 131 941 11321 12011 10331 10253 2711 1109 4253 5099 8951
419 8423 9461 11393 10631 2111 269 2039 4409 7853 10103 5939 8273 4091
233 7481 10589 12161 12149 89 5 401 1913 10391 11699 10301 6113 1889
10313 6089 1901 503 1811 10289 11801 12197 12113 53 41 1613 4721 11969
8111 3929 6263 2099 4349 7793 10163 11933 10091 1571 809 2741 3779 11783
3251 7103 7949 11093 9491 1949 1871 191 881 11261 12071 8501 9413 389
2729 7283 8291 10709 8753 4373 569 83 1319 11519 11483 11549 6491 263
2861 4751 10691 2141 2459 9203 10601 12101 10979 1031 293 12143 5639 521
6701 7523 4079 7499 9623 1979 5303 4133 4973 6299 8999 5231 3833 9239
8741 6029 3533 8663 6221 7121 2399 2069 2351 10973 9011 929 8831 8543

The Composed Associated Magic Square shown above contains an order 8 Associated Magic Centre Square and an order 6 Associated Square Inlay.

Attachment 14.34.38 shows for a few Magic Sums the first occurring 14th order Prime Number Composed Magic Square.

14.34.9 Magic Squares, Order 7 Magic Cube Based
        Composed (14 x 14)

Order 14 Prime Number Magic Squares composed of Order 7 (Semi-) Magic Sub Squares can be constructed based on Prime Number Concentric Magic Cubes, as deducted in Section 7.5 of Chapter 'Prime Number Magic Cubes'.

A typical examples of an order 14 Prime Number Composed Magic Square (Mc14 = 151186), based on planes of a Prime Number Magic Cube of half the Magic Sum, is shown below.

Mc14 = 151186
20921 6719 9587 197 16319 19979 1871 19961 5927 7517 9791 17579 4391 10427
7757 251 13781 20939 3539 19559 9767 4937 13691 14369 41 4787 21107 16661
1229 19469 7547 21089 17921 6311 2027 2729 3467 16067 12539 6131 15791 18869
2309 20771 14411 8807 17837 2297 9161 12227 11 18341 5171 17471 13001 9371
20789 20759 4079 419 2141 11447 15959 14627 19157 3917 15737 15077 107 6971
1709 2477 19697 21341 449 14519 15401 9941 17669 1301 20507 10529 3989 11657
20879 5147 6491 2801 17387 1481 21407 11171 15671 14081 11807 4019 17207 1637
3557 20147 2417 15497 6977 6449 20549 191 16451 15107 18797 4211 20117 719
18461 17609 20297 1091 11069 3929 3137 11831 21347 7817 659 18059 2039 13841
16127 5807 5531 9059 15467 18131 5471 19571 2129 14051 509 3677 15287 20369
15017 8597 3257 16427 4127 21587 6581 12437 827 7187 12791 3761 19301 19289
8231 21491 17681 5861 6521 2441 13367 5639 839 17519 21179 19457 10151 809
13151 491 7229 21557 16811 7907 8447 6197 19121 1901 257 21149 7079 19889
1049 1451 19181 6101 14621 15149 18041 19727 14879 12011 21401 5279 1619 677

It can be noticed that, for the example shown above (Center Cube with Magic Top Plane), also the Semi Diagonals and the Main Bent Diagonals sum to the Magic Sum Mc14.

Attachment 14.34.6 shows for miscellaneous Magic Sums a few of the Composed Magic Squares described above.

14.34.10 Summary

The obtained results regarding the 14th order Prime Number Magic Squares and related sub squares, as deducted and discussed in previous sections, are summarized in following table:

Type

Characteristics

Subroutine

Results

Concentric

-

Priem14a

Attachment 14.34.1

Bordered

Miscellaneous Types

Priem14a

Ref. Sect. 14.34.2

Eccentric

-

Priem14b

Attachment 14.34.2

Composed

Magic Cube Based

-

Attachment 14.34.6

Center Cross, Pan Magic Sub Squares

-

Attachment 14.34.36

Inlaid

Pan Magic Center Square and Square Inlays

-

Attachment 14.34.37

Ultra Magic Square Inlays

-

Attachment 14.34.3

Composed

Associated

-

Attachment 14.34,42

Pan Magic, Complete

Euler

Attachment 14.34.43

-

-

-

-

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

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