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14.0    Special Magic Squares, Prime Numbers

14.10   Magic Squares (13 x 13), Composed

14.10.1 Introduction

In Section 12.6 a detailed description and analysis has been provided of one of the 13th order Composed Magic Squares for distinct integers, as previously published by William Symes Andrews (1909).

It has been proven and illustrated that comparable squares, composed out of each other overlapping sub squares, can be constructed with a Pan Magic Center Square.

This section will describe how comparable squares can be constructed for prime numbers. While doing so, a number of interesting prime number based sub squares will be found and described as well.

For the sake of the analysis the 13th order Magic Square, composed out of each other overlapping sub squares, will be represented as shown below. The important key variables have been highlighted in blue.

 a(1) a(2) a(3) a(4) a(5) a(6) a(7) a(8) a(9) a(10) a(11) a(12) a(13) a(14) a(15) a(16) a(17) a(18) a(19) a(20) a(21) a(22) a(23) a(24) a(25) a(26) a(27) a(28) a(29) a(30) a(31) a(32) a(33) a(34) a(35) a(36) a(37) a(38) a(39) a(40) a(41) a(42) a(43) a(44) a(45) a(46) a(47) a(48) a(49) a(50) a(51) a(52) a(53) a(54) a(55) a(56) a(57) a(58) a(59) a(60) a(61) a(62) a(63) a(64) a(65) a(66) a(67) a(68) a(69) a(70) a(71) a(72) a(73) a(74) a(75) a(76) a(77) a(78) a(79) a(80) a(81) a(82) a(83) a(84) a(85) a(86) a(87) a(88) a(89) a(90) a(91) a(92) a(93) a(94) a(95) a(96) a(97) a(98) a(99) a(100) a(101) a(102) a(103) a(104) a(105) a(106) a(107) a(108) a(109) a(110) a(111) a(112) a(113) a(114) a(115) a(116) a(117) a(117) a(119) a(120) a(121) a(122) a(123) a(124) a(125) a(126) a(127) a(128) a(129) a(130) a(131) a(132) a(133) a(134) a(135) a(136) a(137) a(138) a(139) a(140) a(141) a(142) a(143) a(144) a(145) a(146) a(147) a(148) a(149) a(150) a(151) a(152) a(153) a(154) a(155) a(156) a(157) a(158) a(159) a(160) a(161) a(162) a(163) a(164) a(165) a(166) a(167) a(168) a(169)

The 13th order Magic Square J, with Magic Sum s1, contains following sub squares:

• One 5th order Pan Magic Center Square C (MC5 = 5 * s1 / 13);
• One 5th order Magic Corner Square G (MC5 = 5 * s1 / 13), the element a(109) = s1 / 13 common with C;
• One 3th order Embedded Semi Magic Square M (MC3 = 3 * s1 / 13), eccentric in G (right top);
• Four 4th order Magic Border Squares (MC4 = 4 * s1 / 13): A and B (left), D and E (bottom);
• Two each other overlapping 7th order Magic Squares (MC7 = 7 * s1 / 13):
- I with C in the left bottom corner and
- L with C in the right top corner;
• Two each other overlapping 9th order Magic Squares (MC9 = 9 * s1 / 13):
- F composed out of B (left top), G (left bottom), D (right bottom) and C (right top)
- H with eccentric embedded I (left bottom)and C (left bottom).
• One 11th order Eccentric Magic Square K (MC11 = 11 * s1 / 13).

14.10.2 Analysis (Sub Squares)

As a consequence of the properties described in Section 14.10.1 above, the 13th order Magic Square J is composed out of:

• a Magic Center Square C with a Magic Sum MC5 = 5 * s1 / 13 and
• 72 pairs, each summing to 2 * s1 / 13, distributed over two layers, surrounding square C.

Magic Corner Square G

If the Magic Corner Square G is represented as:

 a(1) a(2) a(3) a(4) a(5) a(6) a(7) a(8) a(9) a(10) a(11) a(12) a(13) a(14) a(15) a(16) a(17) a(18) a(19) a(20) a(21) a(22) a(23) a(24) a(25)

with Magic Sum s5, the equations defining the Magic Corner Square can be written as:

```a(21) =     s5   - a(22) - a(23) - a(24) - a(25)
a(20) = 2 * s5/5 - a(25)
a(19) = 2 * s5/5 - a(24)
a(18) = 2 * s5/5 - a(23)
a(17) = 2 * s5/5 - a(21)
a(16) = 2 * s5/5 - a(22)
a(13) = 3 * s5/5 - a(14) - a(15)
a(11) = 2 * s5/5 - a(12)
a(10) = 2 * s5/5 - a(15)
a( 9) = 2 * s5/5 - a(13)
a( 8) = 2 * s5/5 - a(14)
a( 7) =(4 * s5/5 - a(12) - a(13) + a(21) - a(22) + a(24) - a(25)) / 2
a( 6) = 2 * s5/5 - a( 7)
a( 5) =     s5/5
a( 4) = 4 * s5/5 - 2 * a(14) - a(15)
a( 3) = 2 * s5/5 - a( 4)
a( 2) = 4 * s5/5 - a( 6) - a(14) - a(15) - a(24) + a(25)
a( 1) = 2 * s5/5 - a( 2)
```

a routine can be written to generate subject Prime Number Magic Corner Squares of order 5 (ref. PriemG5).

Attachment 14.8.5.01 shows for miscellaneous Magic Sums the first occurring Prime Number Magic Corner Square G.

Pan Magic Center Square C

Pan Magic Center Squares with Magic Sum s5 and corner element s5/5 (bottom/left), can be obtained by translation of Ultra Magic Squares.

Based on the equations for Ultra Magic Squares as deducted in Section 14.3.5, a procedure (PriemC5) can be developed:

• to read the previously generated Magic Corner Squares G;
• to generate Ultra Magic Squares, based on the remainder of the available pairs;
• to translate the Ultra Magic Squares into the required Pan Magic Squares C.

Attachment 14.8.5.02 shows for miscellaneous Magic Sums the first occurring Prime Number Pan Magic Center Square C with corner element s5/5.

The obtained Magic Corner Squares G and related Ultra Magic Squares can be used as input for a suitable guessing routine to generate the 9th order Composed Magic Squares F, which will discussed below.

Semi Magic Square M

The 3th order Semi Magic Square M, embedded in the Magic Corner Square G, as mentioned in Section 14.10.1 above, is a consequence of the defining properties of the Magic Corner Square G.

Magic Border Squares A, B, D and E

If the Magic Border Square A is represented as:

 a(1) a(2) a(3) a(4) a(5) a(6) a(7) a(8) a(9) a(10) a(11) a(12) a(13) a(14) a(15) a(16)

with Magic Sum s4, the equations defining the Magic Border Square A can be written as:

```a(15) = s4/2 - a(16)
a(13) = s4/2 - a(14)
a(11) = s4/2 - a(12)
a( 9) = s4/2 - a(10)
a( 8) =(s4   + a(10) - a(12) - a(14) - a(16))/2
a( 7) = s4/2 - a( 8)
a( 6) = s4/2 - a( 7) - a(10) + a(12)
a( 5) = s4/2 - a( 6)
a( 4) = s4/2 - a( 5) - a(12) + a(14)
a( 3) = s4/2 - a( 4)
a( 2) = s4   - a( 4) - a(10) - a(12)
a( 1) = s4/2 - a( 2)
```

Comparable equations can be applied for the Magic Border Squares B, D and E.

It should be noted that the Border Squares D and E require a transposition around the axis a(4) ... a(13).

Subject equations can be incorporated in suitable guessing routines:

• to generate the Composed Magic Squares F, based on the squares B and D (PriemF9);
• to generate Partly Completed Squares J1, based on the squares A and E (PriemJ13);

which will be discussed below.

Magic Square F

The 9th order Magic Square F is composed out of B (left top), G (left bottom), D (right bottom) and C (right top) and includes the Magic Square L

The 4th order Border Magic Squares B and D should be selected in such a way that:

a(55) + a(69) + a(83) + a(97) + a(111) + a(125) + a(139) = MC7

thus ensuring that the Sub Square L will be magic as well.

Comparable with Section 14.7.1 a procedure (PriemF9) can be developed:

• to read the previously generated 5th order Magic Corner Squares G;
• to read the related previously generated 5th order Ultra Magic Squares;
• to translate the Ultra Magic Squares into the required Pan Magic Squares C;
• to generate the additional 4th order Magic Squares B and D, based on the remainder of the available pairs, while ensuring that L is magic;
• to complete the 9th order Composed Magic Squares F.

Attachment 14.8.5.03 shows for miscellaneous Magic Sums the first occurring Prime Number Composed Magic Square F.

Magic Square L

The 7th order Magic Square L is fully defined by B (left top), M (left bottom), D (right bottom) and C (right top) and is included in F (right top) as discussed above.

Magic Squares A and E

The 4th order Border Magic Squares A and E should be selected in such a way that:

a(3) + a(17) + a(129) + a(141) = MC4

thus ensuring that the Sub Square K will be magic as well.

Comparable with procedure PriemF9 as described above for Magic Square F, a procedure (PriemJ13) can be developed:

• to read the previously generated 9th order Magic Squares F;
• to generate the additional 4th order Magic Squares B and D, based on the remainder of the available pairs, while ensuring that K is magic;
• to construct intermediate 13th order Partly Completed Squares J1, composed out of F, A and E.

Attachment 14.8.5.04 shows for miscellaneous Magic Sums the first occurring Intermediate Square J1, composed out of F, A and E.

Magic Square I

The supplementary pairs, which should be added to Square C, to obtain the 7th order Eccentric Magic Square I, should be selected in such a way that:

a(31) + a(45) + a(101) + a(115) = MC7 - a(59) - a(73) - a(87)

thus ensuring that the Square I will be magic.

Based on the above, a procedure (PriemI7) can be developed:

• to read the previously generated 13th order Intermediate Squares J1;
• to select the supplementary pairs, which should be added to Square C, based on the remainder of the available pairs, while ensuring that I is magic;
• to construct intermediate 13th order Partly Completed Squares J2, based on J1 completed with the supplementary pairs of I.

Attachment 14.8.5.05 shows for miscellaneous Magic Sums the first occurring Prime Number Eccentric Magic Square I as a part of the Partly Completed Square J2.

Magic Square H

The supplementary pairs, which should be added to Square I, to obtain the 9th order Eccentric Magic Square H, should be selected in such a way that:

a(5) + a(19) + a(103) + a(117) = MC9 - a(33) - a(47) - a(61) - a(75) - a(89)

thus ensuring that the Square H will be magic.

Further it is convenient to split the supplementary rows and columns into three equal parts each summing to MC9/3.

Based on the above, a fast procedure (PriemH9) can be developed:

• to read the previously generated 13th order Intermediate Squares J2;
• to select the supplementary pairs, which should be added to Square I, based on the remainder of the available pairs, while ensuring that H is magic;
• to complete the 13th order Composed Magic Squares J, based on J2 completed with the supplementary pairs of H.

Attachment 14.8.5.06 shows for miscellaneous Magic Sums the first occurring Prime Number Eccentric Magic Square H as a part of the Completed Magic Square J.

Magic Square K

The 11th order Magic Square K is fully defined by A (left top), L (left bottom), E (right bottom) and H (right top), contains the square I (center) and is included in the overall square J (right top) as discussed above.

Attachment 14.8.5.07 shows for miscellaneous Magic Sums the first occurring Prime Number Composed Magic Square K.

14.10.3 Conclusions (Completed Squares)

In previous section 13th order Prime Number Magic Squares, composed out of each other overlapping sub squares, have been constructed based on:

• The defining equations of the order 4 and 5 sub squares;
• The construction methods for Composed Magic Squares as discussed in Section 14.7.1;
• Application of an intermediate results approach.

It should be possible to find more solutions, within the range of Magic Sums as shown in Attachment 14.8.5.06, by means of:

• Allowing a longer Time Out time for the applicable steps of the construction process (applied time 60 seconds);
• Starting the construction process with other aspects of (or other variables for) the sub squares G and C.

Verification whether the first found solution (MC = 63171) is the smallest has been left open.

14.10.4 Associated Magic Squares (13 x 13) with 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 a transformation of order 13 Composed Magic Squares, as illustrated in Section 14.7.11 for order 9 Magic Squares.

MC = 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 14.10.6 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.

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

An Example of an order 13 Composed Magic Square with Overlapping Sub Squares is provided below:

MC = 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.

14.10.6 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 Magic Corner Square (left bottom) Pan Magic Corner Element s5/5 Composed Out of: B, G, D and C, includes L Intermediate Partly Completed: F, A and D Intermediate Partly Completed: F, A, D and I Composed Completed Square J Completed Square K Composed Overlapping Sub Squares Associated Corner Squares and Rectangles Associated Associated Square Inlays Order 6 and 7 - - - - -
 Comparable routines as listed above, can be used to generate miscellaneous Prime Number Composed Magic Squares, which will be described in following sections.