Office Applications and Entertainment, Magic Squares

Vorige Pagina Volgende Pagina Index About the Author

14.0    Special Magic Squares, Prime Numbers

14.33   Magic Squares (13 x 13)

14.33.1 Magic Squares, Concentric (13 x 13)

A 13th order Prime Number Concentric Magic Square consists of a Prime Number Concentric Magic Square of the 11th order with a border around it.

For Prime Number Concentric Magic Squares of order 13 with Magic Sum s13, it is convenient to split the supplementary rows and columns into parts summing to s4 = 4 * s13 / 13 and s5 = 5 * s13 / 13:

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

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(13) = s4 - a(12) - a(11) - a(10)

a(169) = Pr4 - a(1)
a(158) = Pr4 - a(2)
a(159) = Pr4 - a(3)
a(160) = Pr4 - a(4)
a(161) = Pr4 - a(5)
a(162) = Pr4 - a(6)
a(163) = Pr4 - a(7)
a(164) = Pr4 - a(8)
a(165) = Pr4 - a(9)
a(166) = Pr4 - a(10)
a(167) = Pr4 - a(11)
a(168) = Pr4 - a(12)
a(157) = Pr4 - a(13)

a( 40) = s4 - a( 27) - a( 14) - a(  1)
a(105) = s5 - a( 92) - a( 79) - a( 66) - a(53)
a(144) = s4 - a(131) - a(118) - a(157)

a( 26) = Pr4 - a(14)
a( 39) = Pr4 - a(27)
a( 52) = Pr4 - a(40)
a( 65) = Pr4 - a(53)
a( 78) = Pr4 - a(66)
a( 91) = Pr4 - a(79)
a(104) = Pr4 - a(92)
a(117) = Pr4 - a(105)
a(130) = Pr4 - a(118)
a(143) = Pr4 - a(131)
a(156) = Pr4 - a(144)

which enable the development of a fast procedure to generate Prime Number Concentric Magic Squares of order 13 (ref. Priem13a).

Miscellaneous Prime Number Concentric Magic Squares of order 13, based on 11th order Concentric Magic Squares as discussed in Section 14.9.1, are shown in Attachment 14.33.11.

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

A method to generate order 13 Concentric Mgic Squares with order 7 Diamond Inlays will be discussed in Section 20.1.5.

14.33.2 Magic Squares, Bordered (13 x 13)

Based on the collections of 11th order Composed and miscellaneous Bordered Magic Squares, as discussed in Section 14.9.2 through 14.9.10, also following 13th order Bordered Magic Squares can be generated with routine Priem13a:

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.

14.33.3 Magic Squares, Eccentric (13 x 13)

Also for Prime Number Eccentric Magic Squares of order 13 it is convenient to split the supplementary rows and columns into: parts summing to s4 = 4 * s13 / 13 and s5 = 5 * s13 / 13:

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

This enables, based on the same principles, the development of a fast procedure (ref. Priem13b):

  • to read the previously generated Eccentric Magic Squares of order 11;
  • 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 13 with a 2 x 4 and a 2 x 5 Magic Rectangle.

Attachment 14.33.2 shows, based on the 11th order Eccentric Magic Squares as discussed in Section 14.9.4, 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.

14.33.4 Composed Magic Squares (13 x 13)
        Overlapping Sub Squares Order 3 and 7

Previous page described a classical Magic Square of order 13, with miscellaneous each other – asymmetrically - Overlapping Sub Squares.

An alternative example of an order 13 Composed Magic Square with each other Overlapping Sub Squares is provided below:

Mc13 = 87893
13499 13421 12503 521 563 59 4241 12239 3803 12821 9521 2549 2153
71 53 5981 8039 13163 13259 6563 6719 7001 1481 3221 10193 12149
12161 2351 809 10733 2099 12413 6521 6803 6959 1373 3329 10301 12041
1109 11423 2789 12713 11171 1361 9719 1283 9281 11369 10973 4001 701
263 359 5483 7541 13469 13451 4673 10331 5279 8999 6863 6971 4211
13463 12959 13001 1019 101 23 8849 8243 3191 1973 6869 6653 11549
9743 2909 6311 8081 12659 863 6761 1709 11813 9311 6551 6659 4523
5099 10163 3359 8423 1451 7349 11483 13331 12953 12539 293 1031 419
5441 7211 10613 3779 6173 12071 2039 479 281 6329 7883 12473 13121
12923 10499 1721 1901 12329 3251 4703 7949 5669 1319 12011 5399 8219
1619 2003 10781 12641 4283 4091 11909 5303 8123 1511 12203 7853 5573
881 2741 11519 11903 1613 9431 9239 401 1049 5639 7193 13241 13043
11621 11801 3023 599 8819 10271 1193 13103 12491 13229 983 569 191

The Magic Square shown above is composed out of:

  • Two 6th order Partly Compact Associated Magic Corner Squares (s6 = 6 * s1 / 13),
  • Two 7th order Overlapping Magic Squares (s7 = 7 * s1 / 13) each composed of:
    - One 4th order Associated Magic Square (s4 = 4 * s1 / 13)
    - One 3th order Semi       Magic Square (s3 = 3 * s1 / 13)
    - Two Associated Magic Rectangles order 3 x 4.

Based on this definition a routine can be developed to generate subject Composed Magic Squares (ref. Priem13d).

Attachment 14.10.7 shows for miscellaneous Magic Sums the first occurring Prime Number Composed Magic Square.

14.33.5 Composed Magic Squares (13 x 13)
        Associated Square Inlays Order 6 and 7

Associated Magic Squares of order 13 with Square Inlays of order 6 and 7 can be obtained by means of transformation of order 13 Composed Magic Squares, as illustrated in Section 14.7.13 for order 9 Magic Squares.

Mc13 = 87893
13499 13421 12503 521 563 59 12821 4241 3221 3023 599 11411 12011
71 53 5981 8039 13163 13259 6269 9521 4493 5351 11801 2039 7853
12161 2351 809 10733 2099 12413 1193 6521 12569 11909 7883 6833 419
1109 11423 2789 12713 11171 1361 13103 6689 5639 1613 953 7001 12329
263 359 5483 7541 13469 13451 5669 11483 1721 8171 9029 4001 7253
13463 12959 13001 1019 101 23 1511 2111 12923 10499 10301 9281 701
13331 6659 293 13241 6563 479 11969 863 4523 9689 12611 2663 5009
6551 3461 10271 12203 3989 4091 5153 9059 5399 7433 3371 5939 10973
401 10163 9719 1031 8819 10433 6089 8123 4463 8369 2549 7583 10151
1481 5849 12953 569 7673 12041 3833 8999 12659 1553 8513 10859 911
3089 4703 12491 3803 3359 13121 12641 3671 1259 9473 10781 3329 6173
9431 9533 1319 3251 10061 6971 3593 4349 9173 9929 2153 6761 11369
13043 6959 281 13229 6863 191 4049 12263 9851 881 7349 10193 2741

The Magic Square shown above is composed out of:

  • One 6th order Partly Compact Associated Magic Corner Square (s6 = 6 * s1 / 13),
  • One 7th order Semi Magic Square (s7 = 7 * s1 / 13) composed of:
    - One 4th order Associated Magic Square (s4 = 4 * s1 / 13)
    - One 3th order Simple     Magic Square (s3 = 3 * s1 / 13)
    - Two Associated Magic Rectangles order 3 x 4,
  • Two Associated Magic Rectangles order 6 x 7
    each with two Embedded order 3 Semi Magic Squares (s3 = 3 * s1 / 13).

Based on this definition a routine can be developed to generate subject Composed Magic Squares (ref. Priem13c).

  • Attachment 14.10.5 shows for miscellaneous Magic Sums the first occurring Prime Number Composed Magic Square.
    Attachment 14.10.4 shows the corresponding Magic Squares with Associated Corner Squares and Rectangles.

  • Attachment 14.10.6 shows the corresponding Associated Magic Squares with order 6 and 7 Square Inlays.

It should be noted that the reversed transformation is not necessarily possible because of the bottom-left / top-right Main Diagonal.

14.33.6 Associated Magic Squares
        Associated Center Square Order 7

Associated Magic Squares of order 13 with an Associated Center Square of order 7 can be obtained by means of transformation of order 13 Composed Magic Squares as illustrated in Section 14.7.14 for order 9 Magic Squares.

Mc13 = 87893
13499 13421 12503 12821 4241 3221 3023 599 11411 12011 521 563 59
71 53 5981 6269 9521 4493 5351 11801 2039 7853 8039 13163 13259
12161 2351 809 1193 6521 12569 11909 7883 6833 419 10733 2099 12413
13331 6659 293 11969 12611 863 2663 4523 5009 9689 13241 6563 479
6551 3461 10271 12641 10781 3671 3329 1259 6173 9473 12203 3989 4091
401 10163 9719 5153 3371 9059 5939 5399 10973 7433 1031 8819 10433
1481 5849 12953 3593 2153 4349 6761 9173 11369 9929 569 7673 12041
3089 4703 12491 6089 2549 8123 7583 4463 10151 8369 3803 3359 13121
9431 9533 1319 4049 7349 12263 10193 9851 2741 881 3251 10061 6971
13043 6959 281 3833 8513 8999 10859 12659 911 1553 13229 6863 191
1109 11423 2789 13103 6689 5639 1613 953 7001 12329 12713 11171 1361
263 359 5483 5669 11483 1721 8171 9029 4001 7253 7541 13469 13451
13463 12959 13001 1511 2111 12923 10499 10301 9281 701 1019 101 23

Attachment 14.10.8 shows the Associated Magic Squares with order 7 Associated Center Squares, corresponding with the Composed Magic Squares as shown in Attachment 14.10.4.

It should be noted that the reversed transformation is not necessarily possible because of the bottom-left / top-right Main Diagonal.

14.33.7 Composed Magic Squares, Associated Border
        Square Inlays Order 5 and 6 (overlapping)

The 13th order Composed Inlaid Magic Square shown below:

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

is composed out of an Associated Border and following inlays:

  • two each 6th order (Pan) Magic Squares - Magic Sums s(1) and s(4) with the center element in common,
  • two each 5th order (Associated) Magic Squares with Magic Sums s(2) and s(3).

The relation between the Magic Sums s(1), s(2), s(3) and s(4) is: 

 s(1) = 12 * s1 / 13 - s(4)
 s(2) = 10 * s1 / 13 - s(3)

With s1 the Magic Sum of the 13th order Inlaid Magic Square.

The Associated Border can be described by following linear equations: 

a(162) =  - s1 / 13 + a(164) - s(3) + s(4)
a(161) =  - s1 / 13 + a(165) - s(3) + s(4)
a(160) =  - s1 / 13 + a(166) - s(3) + s(4)
a(159) =  - s1 / 13 + a(167) - s(3) + s(4)
a(158) = -  s1 / 13 + a(168) - s(3) + s(4)
a(157) = 18*s1 / 13 - a(163) - 2*a(164) - 2*a(165) - 2*a(166) - 2*a(167) - 2*a(168) - a(169) + 5*s(3) - 5*s(4)
a(144) =    s1      - a(156) - s(3) - s(4)
a(131) =    s1      - a(143) - s(3) - s(4)
a(118) =    s1      - a(130) - s(3) - s(4)
a(105) =    s1      - a(117) - s(3) - s(4)
a( 92) =    s1      - a(104) - s(3) - s(4)
a( 91) = 66*s1 / 13 - 2*a(104) - 2*a(117) - 2*a(130) - 2*a(143) - 2*a(156) + a(157) - a(169) - 5*s(3) - 5*s(4)

a(79) = 2 * s1 / 13 - a( 91)
a(78) = 2 * s1 / 13 - a( 92)
a(66) = 2 * s1 / 13 - a(104)
a(65) = 2 * s1 / 13 - a(105)
a(53) = 2 * s1 / 13 - a(117)
a(52) = 2 * s1 / 13 - a(118)
a(40) = 2 * s1 / 13 - a(130)
a(39) = 2 * s1 / 13 - a(131)

a(27) = 2 * s1 / 13 - a(143)
a(26) = 2 * s1 / 13 - a(144)
a(14) = 2 * s1 / 13 - a(156)
a(13) = 2 * s1 / 13 - a(157)
a(12) = 2 * s1 / 13 - a(158)
a(11) = 2 * s1 / 13 - a(159)
a(10) = 2 * s1 / 13 - a(160)
a( 9) = 2 * s1 / 13 - a(161)

a(8) = 2 * s1 / 13 - a(162)
a(7) = 2 * s1 / 13 - a(163)
a(6) = 2 * s1 / 13 - a(164)
a(5) = 2 * s1 / 13 - a(165)
a(4) = 2 * s1 / 13 - a(166)
a(3) = 2 * s1 / 13 - a(167)
a(2) = 2 * s1 / 13 - a(168)
a(1) = 2 * s1 / 13 - a(169)

Which can be incorporated in an optimised guessing routine MgcSqr13k1.

The Magic Center Squares can be constructed by means of:

  • A guessing routine, based on the defining linear equations as deducted in Section 14.3.4, resulting in the two 5th order Associated Magic Sub Squares,
  • A guessing routine, based on the defining linear equations as deducted in Section 14.4.12, resulting in the two each other overlapping 6th order Pan Magic Sub Squares.

Attachment 14.10.9 shows for a few Magic Sums the first occurring Prime Number Inlaid Magic Square of order 13.

Each square shown corresponds with numerous solutions, which can be obtained by variation of the four inlays and the border.

14.33.8 Associated Magic Squares, Diamond Inlays Order 6 and 7

The 13th order Associated Inlaid Magic Square shown below:

Mc13 = 102401
257 15287 14867 1097 2267 15107 15077 6581 353 1367 14447 14783 911
15137 653 13841 1877 13313 3947 5 9857 13421 1973 9533 4283 14561
14327 14771 2423 3527 13907 15683 7457 83 521 1721 9437 10313 8231
1787 797 2063 4871 6917 2837 15737 13457 15473 12911 5483 14057 6011
15641 15053 1013 8783 1871 8501 2003 7331 14723 593 1667 9491 15731
4973 14813 15581 3923 7307 7583 1163 8963 8111 13163 15647 863 311
1511 293 14303 15527 14897 491 7877 15263 857 227 1451 15461 14243
15443 14891 107 2591 7643 6791 14591 8171 8447 11831 173 941 10781
23 6263 14087 15161 1031 8423 13751 7253 13883 6971 14741 701 113
9743 1697 10271 2843 281 2297 17 12917 8837 10883 13691 14957 13967
7523 5441 6317 14033 15233 15671 8297 71 1847 12227 13331 983 1427
1193 11471 6221 13781 2333 5897 15749 11807 2441 13877 1913 15101 617
14843 971 1307 14387 15401 9173 677 647 13487 14657 887 467 15497

contains following Diamond Inlays:

  • one each 6th order Associated Diamond Inlay with Magic Sum s6 = 6 * s1 / 13,
  • one each 7th order Associated Diamond Inlay with Magic Sum s7 = 7 * s1 / 13.

The method to generate order 13 Associated Mgic Squares with order 6 and 7 Diamond Inlays will be discussed in Section 20.2.5.

14.33.9 Summary

The obtained results regarding the 13th 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

-

Priem13a

Attachment 14.33.11

Bordered

Miscellaneous Types

Priem13a

Ref. Sect. 14.33.2

Eccentric

-

Priem13b

Attachment 14.33.2

Composed

Overlapping Sub Squares

Priem13d

Attachment 14.10.7

Associated Corner Squares and Rectangles

Priem13c

Attachment 14.10.4

Associated

Associated Square Inlays Order 6 and 7

-

Attachment 14.10.6

Associated

Associated Center Square Order 7

-

Attachment 14.10.8

Inlaid

Square Inlays Order 5 and 6 (overlapping)

MgcSqr13k1

Attachment 14.10.9

-

-

-

-

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

Vorige Pagina Volgende Pagina Index About the Author