Office Applications and Entertainment, Magic Squares

14.0    Special Magic Squares, Prime Numbers

14.33   Magic Squares (13 x 13), Composed

14.33.1 Introduction

In Section 12.6 a detailed description and analysis has been provided of one of the 13^th 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 13^th 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.

a1

a2

a3

a4

a5

a6

a7

a8

a9

a10

a11

a12

a13

a14

a15

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a169

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

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

14.33.2 Analysis (Sub Squares)

As a consequence of the properties described in Section 14.33.1 above, the 13^th 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 9^th order Composed Magic Squares F, which will discussed below.

Semi Magic Square M

The 3^th order Semi Magic Square M, embedded in the Magic Corner Square G, as mentioned in Section 14.33.1 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 9^th 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 4^th 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 5^th order Magic Corner Squares G;
• to read the related previously generated 5^th order Ultra Magic Squares;
• to translate the Ultra Magic Squares into the required Pan Magic Squares C;
• to generate the additional 4^th order Magic Squares B and D, based on the remainder of the available pairs, while ensuring that L is magic;
• to complete the 9^th 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 7^th 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 4^th 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 9^th order Magic Squares F;
• to generate the additional 4^th order Magic Squares B and D, based on the remainder of the available pairs, while ensuring that K is magic;
• to construct intermediate 13^th 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 7^th 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 13^th 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 13^th 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 9^th 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 13^th 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 13^th 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 11^th 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.

14.33.3 Conclusions (2017)

In previous section 13^th 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 solutions with smaller Magic Sums, as shown in Attachment 14.8.5.06, exist has been left open.

14.33.4 Addendum (2023)

While using a less restrictive procedure for PriemH9 it appeared to be possible to construct Magic Squares - composed of each other overlapping sub squares - with smaller Magic Sums, as illustrated below:

MC = 66131

9293	881	5237	4937	9473	797	8807	8681	1787	8537	5387	1931	383
4877	5297	10061	113	701	9377	1367	1493	8387	1637	4787	9791	8243
5501	4673	317	9857	41	10103	1511	8693	7583	1877	5801	2297	7877
677	9497	4733	5441	10133	71	8663	1481	2591	4373	8297	1811	8363
8093	2081	107	10067	3593	6173	7817	3371	4481	9011	1163	1307	8867
2447	7727	8741	1433	6101	7211	233	7433	4457	5903	4271	9161	1013
9221	953	2927	7247	4073	5717	2741	9941	2963	6473	3701	761	9413
587	9587	8573	1601	6581	5693	6803	2357	4001	7541	2633	9203	971
743	9431	251	9923	5087	641	7841	2333	9533	431	9743	9521	653
5657	4517	5003	167	10091	4871	7823	6971	683	3851	9227	6047	1223
8747	1427	10007	5171	83	5303	2351	3203	9491	6323	947	4127	8951
137	23	4943	10163	10169	3023	9311	5483	2531	2693	7517	7757	2381
10151	10037	5231	11	5	7151	863	4691	7643	7481	2657	2417	7793

Prime Number Magic Squares of order 13 - composed of each other overlapping sub squares - based on Consecutive Primes will be discussed in Section 14.13.49.

14.33.7 Summary

The obtained results regarding the 13^th order Prime Number Magic Squares and related sub 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 Composed Magic Squares, which will be described in following sections.