Office Applications and Entertainment, Magic Squares

12.6   Overlapping Sub Squares (13 x 13)

12.6.1 Introduction

The 13^th order Composed Magic Square shown below was previously published by William Symes Andrews (ref. Magic Squares and Cubes (1909), Fig. 394).

157	13	23	147	109	31	111	138	36	66	102	100	72
145	25	17	153	61	139	59	32	134	104	68	98	70
16	154	144	26	57	56	30	112	136	99	105	60	110
22	148	156	14	113	114	140	58	34	65	71	133	37
97	73	94	76	151	18	21	89	146	135	35	29	141
79	91	78	92	27	82	150	155	11	63	107	33	137
74	96	75	95	143	159	15	20	88	115	55	101	69
90	80	93	77	19	24	81	149	152	54	116	103	67
164	6	3	167	85	142	158	12	28	64	106	108	62
7	163	168	86	1	132	44	39	125	50	48	118	124
162	8	84	2	169	38	126	131	45	120	122	52	46
5	83	161	10	166	129	43	40	128	123	117	49	51
87	165	9	160	4	41	127	130	42	47	53	121	119

This 13^th order Magic Square J contains following Sub Squares:

• A Magic Center Square C of order 5;
• A Magic Corner Square G of order 5, one element overlapping with C;
• An embedded Semi Magic Square M of order 3, eccentric in G (right top);
• Four Magic Border Squares of order 4: A and B (left), D and E (bottom);
• Two each other overlapping Magic Squares of order 7:
  - I with C in the left bottom corner and
  - L with C in the right top corner;
• Two each other overlapping Magic Squares of order 9:
  - 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).
• An eccentric Magic Square K of order 11.

It can be proven that:

• None of the Magic Squares described above can be Pan Magic, except the Center Square C;
• The Semi Magic Square M can't be Magic;
• The value of the common element of the overlapping squares C and G is 85.

12.6.2 Analysis (Sub Squares)

As a consequence of the properties described in Section 12.6.1 above, the 13^th order Magic Square J is composed out of:

• a Magic Center Square C with a Magic Sum s1 = 425 and
• 72 pairs, each summing to 170, distributed over two layers surrounding square C.

Magic Center Square C

Assuming the outer and inner border variables constant, the number of valid solutions for C can be determined based on the required variable values:

   {11, 12, 15, 18, 19, 20, 21, 24, 27, 28, 81, 82, 85, 88, 89, 142, 143, 146, 149, 150, 151, 152, 158, 159}

which can be written as {c_i} with i = 1 ... 25.

The required Magic Squares of the 5^th order can be generated with:

1. either a routine comparable with MgcSqr5a2 (ref. Section 3.2) with c(25) = 85
2. or a routine comparable with CnstrSqrs5a (ref. Section 3.6) also with c(25) = 85.

The obtained squares have to be mirrored around the vertical axis to meet the required c(21) = 85.

The number of suitable squares is limited by the values of s₂₁ = c(3) + c(9) + c(15), s₂₂ = c(11) + c(17) + c(23) and c(5), as these values determine whether squares L, I, F and H are magic as required.

1. Routine MgcSqr5a4 combines the functionality of routine MgcSqr5a2, the required transformation to {c_i} and reflection around the vertical axis.
   
   For c(22) = 11 the routine MgcSqr5a4 produced 2185 Magic Squares with distinct integers {c_i} and Magic Sum 425.
   
   This number can be quadrupled by means of transposition around the diagonal c(21) - c(5) and complementation of both the generated and the transposed squares.
   
   Note: The complement of a Magic Square {b_i} = {s_p - a_i} for i = 1 ... 25 with s_p = a_min + a_max.
   
   From the collection of 4 * 2185 = 8740 Magic Squares, 36 Magic Center Squares could be filtered based on the requirements s₂₁ = 264, s₂₂ = 325 and c(5) = 146 which are shown in Attachment 12.6.1.
   
   
2. Routine CnstrSqrs5c combines the functionality of routine CnstrSqrs5a, the required transformation to {c_i} and reflection around the vertical axis.
   
   Routine CnstrSqrs5c produced 6912 Magic Squares with distinct integers {c_i} and Magic Sum 425, of which 24 Magic Center Squares could be filtered based on the requirements mentioned above which are shown in Attachment 12.6.2.
   
   

Magic Corner Square G

Assuming the variables outside the Corner Square constant, the number of valid solutions for G can be determined based on the required variable values:

   {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 83, 84, 85, 86, 87, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169}

which can be written as {g_i} with i = 1 ... 25.

The Magic Corner Square G can be 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)

In addition to the defining equations of a 5^th order Magic Square:

a( 1) + a( 2) + a( 3) + a( 4) + a( 5) = s1
a( 6) + a( 7) + a( 8) + a( 9) + a(10) = s1
a(11) + a(12) + a(13) + a(14) + a(15) = s1
a(16) + a(17) + a(18) + a(19) + a(20) = s1
a(21) + a(22) + a(23) + a(24) + a(25) = s1

a( 1) + a( 6) + a(11) + a(16) + a(21) = s1
a( 2) + a( 7) + a(12) + a(17) + a(22) = s1
a( 3) + a( 8) + a(13) + a(18) + a(23) = s1
a( 4) + a( 9) + a(14) + a(19) + a(24) = s1
a( 5) + a(10) + a(15) + a(20) + a(25) = s1

a( 1) + a( 7) + a(13) + a(19) + a(25) = s1
a( 5) + a( 9) + a(13) + a(17) + a(21) = s1

following equations should be added:

The resulting number of equations can be written in matrix representation as:

          →   →
     A_G * a = s

which can be reduced, by means of row and column manipulations, and results in following set of linear equations:

a(21) = 425 - a(22) - a(23) - a(24) - a(25)
a(20) = 170 - a(25)
a(19) = 170 - a(24)
a(18) = 170 - a(23)
a(17) = 170 - a(21)
a(16) = 170 - a(22)
a(13) = 255 - a(14) - a(15)
a(11) = 170 - a(12)
a(10) = 170 - a(15)
a( 9) = 170 - a(13)
a( 8) = 170 - a(14)
a( 7) =(340 - a(12) - a(13) + a(21) - a(22) + a(24) - a(25)) / 2
a( 6) = 170 - a( 7)
a( 5) = 85
a( 4) = 340 - 2 * a(14) - a(15)
a( 3) = 170 - a( 4)
a( 2) = 340 - a( 6) - a(14) - a(15) - a(24) + a(25)
a( 1) = 170 - a( 2)

It should be noted that the common element of the overlapping Corner Square G and Center Square C is fully determined by the defining properties of the Corner Square.

With an optimized guessing routine (MgcSqr5g), based on the equations above, 1216 Magic Corner Squares could be generated within 150 seconds, which are shown in Attachment 12.6.3.

Semi Magic Square M

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

The 1216 Corner Squares shown in Attachment 12.6.3 contain 40 different embedded Semi Magic Squares.

Magic Border Squares A, B, D, E

Assuming the variables outside the applicable Border Square constant, the number of valid solutions for A, B, D and E can be determined based on the applicable variable values:

   {a_i} = { 13, 14, 16, 17, 22, 23, 25, 26, 144, 145, 147, 148, 153, 154, 156, 157 }
   {b_i} = { 73, 74, 75, 76, 77, 78, 79, 80, 90, 91, 92, 93, 94, 95, 96, 97 }
   {d_i} = { 38, 39, 40, 41, 42, 43, 44, 45, 125, 126, 127, 128, 129, 130, 131, 132 }
   {e_i} = { 46, 47, 48, 49, 50, 51, 52, 53, 117, 118, 119, 120, 121, 122, 123, 124 }

The Magic Border Square A can be 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)

In addition to the defining equations of a 4^th order Magic Square:

a( 1) + a( 2) + a( 3) + a( 4) = s1
a( 5) + a( 6) + a( 7) + a( 8) = s1
a( 9) + a(10) + a(11) + a(12) = s1
a(13) + a(14) + a(15) + a(16) = s1


a( 1) + a( 5) + a( 9) + a(13) = s1
a( 2) + a( 6) + a(10) + a(14) = s1
a( 3) + a( 7) + a(11) + a(15) = s1
a( 4) + a( 8) + a(12) + a(16) = s1

a( 1) + a( 6) + a(11) + a(16) = s1
a( 4) + a( 7) + a(10) + a(13) = s1

following equations should be added:

The resulting number of equations can be written in matrix representation as:

          →   →
     A_A * a = s

which can be reduced, by means of row and column manipulations, and results in following set of linear equations:

a(15) = 170 - a(16)
a(13) = 170 - a(14)
a(11) = 170 - a(12)
a( 9) = 170 - a(10)
a( 8) =(340 + a(10) - a(12) - a(14) - a(16))/2
a( 7) = 170 - a( 8)
a( 6) = 170 - a( 7) - a(10) + a(12)
a( 5) = 170 - a( 6)
a( 4) = 170 - a( 5) - a(12) + a(14)
a( 3) = 170 - a( 4)
a( 2) = 340 - a( 4) - a(10) - a(12)
a( 1) = 170 - a( 2)

In addition to this the limiting condition a(3) + a(8) = 176 should be taken into consideration as this value determines whether square K is magic as required.

With an optimized guessing routine (MgcSqr4f), based on the equations above, suitable Magic Border Squares A could be generated.

Comparable results, taken the applicable conditions into account, could be obtained for Border Squares B, D and E.

The applicable conditions and results are summarized below:

Border Square	Variable Values	Limiting Condition (Square)	Results	Attachment 12.6.4
A	{ai}	a(3) + a(8) = 176 (K)	16	Section A
B	{bi}	b(3) + b(8) = 186 (L)	16	Section B
D	{di}	d(3) + d(8) =  84 (L)	16	Section D
E	{ei}	e(3) + e(8) = 164 (K)	16	Section E

It should be noted that for D and E the equations deducted above require a transposition around the a(13) - a(4) axis.

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).

Under the restrictions made in previous sections with regard to the variable values per square and the application of only 60 Center Squares it is already possible to construct 16 * 1216 * 16 * 60 = 18677760 solutions for F.

Magic Square L

The 7^th order Magic Square L is determined by B (left top), M (left bottom), D (right bottom) and C (right top).

Under the same restrictions mentioned above it is possible to construct 16 * 40 * 16 * 60 = 614400 solutions for L, which are all included in the number mentioned above for F.

Magic Square I

The 7^th order Magic Square I is an Eccentric Square as discussed in Section 7.5.2. which can be 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)	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)

Assuming the variables within Magic Square C and outside Magic Square I constant, the number of valid solutions can be determined based on the applicable border variable values:

   {i_m} = {30,34,35,54,55,56,57,58,63,64,65,71,99,105,106,107,112,113,114,115,116,135,136,140}

The supplementary rows and columns (hatched) can be described by following linear equations:

a( 1) + a( 2) + a( 3) + a( 4) + a( 5) + a( 6) + a( 7) = 595
a( 8) + a( 9) + a(10) + a(11) + a(12) + a(13) + a(14) = 595
a( 6) + a(13) + a(20) + a(27) + a(34) + a(41) + a(48) = 585
a( 7) + a(14) + a(21) + a(28) + a(35) + a(42) + a(49) = 595
a( 1) + a( 9) + a(17) + a(25) + a(33) + a(41) + a(49) = 595
a( 1) + a( 8) = 170
a( 2) + a( 9) = 170
a( 3) + a(10) = 170
a( 4) + a(11) = 170
a( 5) + a(12) = 170
a( 6) + a(14) = 170
a( 7) + a(13) = 170
a(20) + a(21) = 170
a(27) + a(28) = 170
a(34) + a(35) = 170
a(41) + a(42) = 170
a(48) + a(49) = 170

Which can be reduced to:

a(48) = 170 -  a(49)
a(41) = 170 -  a(42)
a(34) = 170 -  a(35)
a(27) = 170 -  a(28)
a(20) = 170 -  a(21)
a(13) =-425 +  a(14) + a(21) + a(28) + a(35) + a(42) + a(49)
a( 9) = 510 - (a(10) + a(11) + a(12) + a(13) + a(14) + a(17) + a(25) + a(33) + a(41) + a(49))/2
a( 8) = 595 -  a(9) - a(10) - a(11) - a(12) - a(13) - a(14)
a( 7) = 170 -  a(13)
a( 6) = 170 -  a(14)
a( 5) = 170 -  a(12)
a( 4) = 170 -  a(11)
a( 3) = 170 -  a(10)
a( 2) = 170 -  a( 9)
a( 1) = 170 -  a( 8)

In addition to this the limiting condition s3 = a(3) + a(11) + a(19) + a(27) + a(35) = 352 should be taken into consideration, as this value determines whether square H is magic as required.

With an optimized guessing routine (MgcSqr7h), based on the equations above, 328 suitable Eccentric Magic Squares I could be generated (ref. Attachment 12.6.5).

Magic Square H

The 9^th order Magic Square H is an Eccentric Square as discussed in Section 9.5.2. which can be 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)	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)

Assuming the variables within Magic Square I and outside Magic Square H constant, the number of valid solutions can be determined based on the applicable border variable values {h_m}:

    {29,31,32,33,36,37,59,60,61,62,66,67,68,69,70,72,98,100,101,102,103,104,108,109,110,111,133,134,137,138,139,141}

The supplementary rows and columns can be described by following linear equations:

a( 1) + a( 2) + a( 3) + a( 4) + a( 5) + a( 6) + a( 7) + a( 8) + a( 9) = 765
a(10) + a(11) + a(12) + a(13) + a(14) + a(15) + a(16) + a(17) + a(18) = 765
a( 9) + a(18) + a(27) + a(36) + a(45) + a(54) + a(63) + a(72) + a(81) = 765
a( 8) + a(17) + a(26) + a(35) + a(44) + a(53) + a(62) + a(71) + a(80) = 765
a( 1) + a(11) + a(21) + a(31) + a(41) + a(51) + a(61) + a(71) + a(81) = 765
a( 1) + a(10) = 170
a( 2) + a(11) = 170
a( 3) + a(12) = 170
a( 4) + a(13) = 170
a( 5) + a(14) = 170
a( 6) + a(15) = 170
a( 7) + a(16) = 170
a( 8) + a(18) = 170
a( 9) + a(17) = 170
a(26) + a(27) = 170
a(35) + a(36) = 170
a(44) + a(45) = 170
a(53) + a(54) = 170
a(62) + a(63) = 170
a(71) + a(72) = 170
a(80) + a(81) = 170

Which can be reduced to:

a(80) = 170 -  a(81)
a(71) = 170 -  a(72)
a(62) = 170 -  a(63)
a(53) = 170 -  a(54)
a(44) = 170 -  a(45)
a(35) = 170 -  a(36)
a(26) = 170 -  a(27)
a(17) = 595 +  a(18) - a(26) - a(35) - a(44) - a(53) - a(62) - a(71) - a(80)
a(11) = 680 - (a(12) + a(13) + a(14) + a(15) + a(16) + a(17) + a(18) +
                             + a(21) + a(31) + a(41) + a(51) + a(61) + a(71) + a(81))/2
a(10) = 765 -  a(11) - a(12) - a(13) - a(14) - a(15) - a(16) - a(17) - a(18)
a( 9) = 170 -  a(17)
a( 8) = 170 -  a(18)
a( 7) = 170 -  a(16)
a( 6) = 170 -  a(15)
a( 5) = 170 -  a(14)
a( 4) = 170 -  a(13)
a( 3) = 170 -  a(12)
a( 2) = 170 -  a(11)
a( 1) = 170 -  a(10)

The number of possible solutions for H is determined by the sum of the values of the key variables s3 = a(21) + a(31) + a(41) + a(51) + a(61) = 352 and is quite high.

An optimized guessing routine (MgcSqr9f) produced, based on the equations above, 240 suitable Eccentric Magic Squares H while varying the variables a(12) through a(16), which are shown in Attachment 12.6.6.

While varying 4 more variables - a(18), a(27), a(36) and a(45) - already 168480 Eccentric Magic Squares could be generated.

Magic Square K

The 11^th order Magic Square K is determined by A (left top), B (left center), M (left bottom), D (bottom center), E (bottom right), C (eccentric), border I (top right) and border H (top right).

Under the restrictions made in previous sections with regard to the variable values per square and the application of only 60 Center Squares it is possible to construct 16⁴ * 40 * 60 * 328 * n_h solutions for K, which are all included in the number mentioned in Section 12.6.4 below for J.

12.6.3 Analysis (Complete Square)

For the sake of completeness, the full set of equations - describing the whole 13^th order Magic Square J as defined in Section 12.6.1 above - has been deducted.

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)

Based on the defining equations of:

the Magic Square J (13 x 13);
the Magic Square K (11 x 11);
the Magic Squares F and H (9 x 9);
the Magic Squares I and L (7 x 7);
the Magic Squares C and G (5 x 5);
the Magic Squares A, B, D and E (4 x 4) and
the Semi Magic Square M (3 x 3)

a matrix equation can be composed:

          →   →
     A_J * a = s

which can be reduced, by means of row and column manipulations, and results in following set of linear equations:

a(166) =  340 - a(167) - a(168) - a(169)
a(162) =  340 - a(163) - a(164) - a(165)
a(157) =  425 - a(158) - a(159) - a(160) - a(161)
a(156) =  170 - a(169)
a(155) =  170 - a(168);  a(154) =   170  - a(167); a(153) = 170 - a(166); a(152) = 170 - a(165)  
a(151) =  170 - a(164);  a(150) =   170  - a(163); a(149) = 170 - a(162); a(148) = 170 - a(161)  
a(147) =  170 - a(160);  a(146) =   170  - a(159); a(145) = 170 - a(157); a(144) = 170 - a(158)
a(142) =        a(143) - a(166) + a(167)
a(141) =  340 - a(143) - a(167) - a(169)
a(140) =  340 - a(143) - a(167) - a(168)
a(138) =        a(139) - a(162) + a(163)
a(137) =      - a(138) + a(163) + a(164)
a(136) =  340 - a(137) - a(138) - a(139)
a(133) =  255 - a(134) - a(135)
a(131) =  170 - a(132);  a(130) =   170  - a(143); a(129) = 170 - a(142); a(128) = 170 - a(141)  
a(127) =  170 - a(140);  a(126) =   170  - a(139); a(125) = 170 - a(138); a(124) = 170 - a(137)
a(123) =  170 - a(136);  a(122) =   170  - a(135); a(121) = 170 - a(133)
a(120) =  170 - a(134)
a(119) = (510 - a(132) + a(134) + a(135) - 2 * a(158) - a(159) - 2 * a(161)) / 2
a(118) =  170 - a(119)
a(116) =  170 - a(117)
a(114) =  170 - a(115)
a(110) =  340 - a(111) - a(112) - a(113)
a(108) =  340 - 2 * a(134) - a(135)
a(107) =  170 - a(108)
a(106) =  255 - a(119) - a(132) + a(136) - a(137) + a(157) - a(158) + a(164) - a(165)
a(105) =  170 - a(106)
a(103) =  170 - a(104)
a(101) =  170 - a(102)
a( 96) =  425 - a(97) - a(98) - a(99) - a(100)
a( 94) =  170 - a(95)
a( 92) =  170 - a(93)
a( 90) =  170 - a(91)
a( 88) =  170 - a(89)
a( 83) =  425 - a(84) - a(85) - a(86) - a(87)
a( 81) =  170 - a(82)
a( 80) =  255 - a(83) + a(93) - a(97) - a(111) + a(138) - a(139)
a( 79) =  170 - a(80)
a( 77) =  170 - a(78)
a( 75) =  170 - a(76)
a( 73) =  -85 + a(74) - a(85) + a(87) - a(97) + a(100) + a(113)
a( 71) =(1020 - a(72) - 2 * a(74) + a(83) - a(87) - a(98) - 2 * a(99) - 2 * a(100) - 2 * a(113))/2
a( 70) =  425 - a(71)-a(72)-a(73)-a(74)
a( 69) = (595 - a(82) - a(83) - a(95) - a(97) - a(111) + a(138) - a(139))/2
a( 68) =  170 - a(69)
a( 67) =  340 - a(68) - a(80) - a(81)
a( 66) =  170 - a(67)
a( 64) =  170 - a(65)
a( 62) =  170 - a(63)
a( 61) =  425 - a(73) - a(85) - a(97) - a(109)
a( 60) =  425 - a(73) - a(86) - a(99) - a(112)
a( 59) =  425 - a(72) - a(85) - a(98) - a(111)
a( 58) =  425 - a(71) - a(84) - a(97) - a(110)
a( 57) =  425 - a(58) - a(59) - a(60) - a(61)
a( 56) =  340 - a(68) - a(80) - a(92)
a( 55) =  170 - a(56)
a( 54) =  340 - a(67) - a(80) - a(93)
a( 53) =  170 - a(54)
a( 51) =  170 - a(52)
a( 49) = -425 + a(50) + a(63) + a(76) + a(89) + a(102) + a(115)
a( 45) = (935 - a(46) - a(47) - a(48) - 2 * a(50) - a(63) + a(72) - a(74) - a(76) + 2 * a(85) - 2 * a(87) +
              - a(89) + a(97) + a(98) - a(100) + a(111) - a(113) - 2 * a(115))/ 2
a( 44) =  595 - a(45) - a(46) - a(47) - a(48)  - a(49) - a(50)
a( 42) =  170 - a(43)
a( 40) =  170 - a(41);  a( 38) =   170 - a(39); a(37) =  170 - a(49); a(36) =  170 - a(50)
a( 35) =  170 - a(48);  a( 34) =   170 - a(47); a(33) =  170 - a(46); a(32) =  170 - a(45)
a( 31) =  170 - a(44);  a( 29) =   170 - a(30);
a( 28) =  170 + a(41) - a(129) - a(143)
a( 27) =  170 - a(28)
a( 25) = -595 + a(26) + a(39) + a(52) + a(65) + a(78) + a(91) + a(104) + a(117)
a( 19) = (425 - a(20) - a(21) - a(22) - a(23) - a(24) - a(25) - a(26) + a(46) - a(47) + a(74) + a(76) + a(87) +
              - a(89) + a(100) + a(104) + a(113) - a(117))/2
a( 18) =  765 - a(19) - a(20) - a(21) - a(22) - a(23) - a(24) - a(25) - a(26)
a( 17) = (510 - a(27) - a(30) - a(41) - a(43))/2
a( 16) =  170 - a(17)
a( 15) =  340 - a(16) - a(28) - a(29)
a( 14) =  170 - a(15)
a( 13) =  170 - a(25);  a(12) =  170 - a(26); a(11) =  170 - a(24); a(10) =  170 - a(23)
a(  9) =  170 - a(22);  a( 8) =  170 - a(21); a( 7) =  170 - a(20); a( 6) =  170 - a(19)
a(  5) =  170 - a(18)
a(  4) =        a(15) - a(30) + a(41)
a(  3) =  170 - a( 4)
a(  2) =        a( 4) - a(41) + a(43)
a(  1) =  170 - a( 2)

The equations deducted above have been applied in an Excel Spreadsheet (CnstrSngl13) which illustrates the mutual dependency of the individual sub squares discussed in previous sections.

12.6.4 Summary (1)

Magic Squares of order 13 as defined in Section 12.6.1 above can be constructed based on independent from each other generated sub squares and border sections for corresponding predefined variable values as summarised below:

With n_a = n_b = n_d = n_e = n the total number of squares n_j can be written as:

     n_j = n⁴ * n_c * n_g * n_i * n_h = 16⁴ * 60 * 1216 * 328 * n_h

It should be noted that n_c is limited to those Center Squares (60 ea) which could be found within a reasonable time, which is only a fraction of all possible Center Squares.

For other defined variable ranges {a_m} ... {h_m} comparable amounts of 13^th order Magic Squares can be found.

Attachment 12.6.7 shows for each of the 60 Magic Center Squares C one example of the 13^th order Magic Square J, completed with randomly selected sub squares (A, B, D, E, G) and border sections (I, H).

12.6.5 Pan Magic Center Squares

If to the defining equations of the 13^th order Magic Square J, as discussed in Section 12.6.3 above, the equations of Pan Diagonals for Center Square C are added, the resulting equations suggest that valid solutions are possible.

It requires only minor additional effort to construct possible solutions based on the methods discussed and applied in previous sections.

With routine MgcSqr5a5 - comparable with MgcSqr5a3 (ref. Section 7.7.2) - applied on the applicable variable values {c_i}, 1152 Pan Magic Center Squares with corner element 85 (bottom left) can be generated (ref. Attachment 12.6.8).

As mentioned in Section 12.6.2 above, the number of suitable Center Squares C is limited by the values of s₂₁ = c(3) + c(9) + c(15), s₂₂ = c(11) + c(17) + c(23) and c(5), as these values determine whether squares L, I, F and H are magic as required.

The 1152 Pan Magic Squares shown in Attachment 12.6.8 contain 144 combinations {s₂₁ , s₂₂}, which are summarised in following table:

s22	s21
181	117	123	124	239	245	246	248	254	255
182	117	124	125	239	246	247	248	255	256
188	123	124	131	245	246	253	254	255	262
189	124	125	131	246	247	253	255	256	262
190	117	123	124	248	254	255	257	263	264
191	117	124	125	248	255	256	257	264	265
197	123	124	131	254	255	262	263	264	271
198	124	125	131	255	256	262	264	265	271
312	239	245	246	248	254	255	379	385	386
313	239	246	247	248	255	256	379	386	387
319	245	246	253	254	255	262	385	386	393
320	246	247	253	255	256	262	386	387	393
321	248	254	255	257	263	264	379	385	386
322	248	255	256	257	264	265	379	386	387
328	254	255	262	263	264	271	385	386	393
329	255	256	262	264	265	271	386	387	393

The 16 possible values of s₂₂ require the application of other Border Squares B and D as applied in Section 12.6.2 above.

Without the limiting conditions s_b = b(3) + b(8) = 186 and s_d = d(3) + d(8) = 84 the routine MgcSqr4f will produce 384 Border Squares B and 384 Border Squares D, which are shown in Attachment 12.6.9.

The limiting condition to ensure that the 7^th order Magic Square L is magic is now:

   s₂₂ = 595 - (s_b + s_d)

Attachment 12.6.10 shows the values s₂₂ as a function of s_b and s_d. The possible values of s₂₂, as summarised in the table above, are highlighted (hatched).

The number of combinations {s_b , s_d} which result in correct values of s₂₂ are shown below:

Any of the required values for s_b and s_d corresponds with 16 related Border Squares B and D.

Any of the 16 values of s₂₂ corresponds with 72 Pan Magic Squares C of the 5^th order.

To ensure that I, F and H are magic as required a suitable border for I should be selected for each occurring combination {s₂₁ , c(5)}, which can be achieved with routine MgcSqr7h.

An example of a 13^th order Magic Square J with overlapping Sub Squares and a Pan Magic Center Square C is shown below:

26	144	22	148	139	61	134	32	59	102	66	100	72
14	156	16	154	31	109	36	138	111	68	104	98	70
147	23	145	25	71	56	55	113	106	58	136	60	110
153	17	157	13	99	114	115	57	64	34	112	133	37
91	79	92	78	158	12	24	89	142	105	65	29	141
76	94	73	97	28	81	149	152	15	35	135	33	137
77	93	80	90	143	155	19	20	88	107	63	101	69
96	74	95	75	11	27	82	146	159	116	54	103	67
1	169	8	162	85	150	151	18	21	140	30	108	62
3	167	161	84	10	38	127	45	130	122	120	46	52
166	4	86	9	160	132	43	125	40	48	50	124	118
168	83	163	6	5	126	41	131	42	123	117	49	51
87	2	7	164	165	44	129	39	128	47	53	121	119

The order of magnitude of n_j is comparable with the order of magnitude deducted in Section 12.6.4 above.

12.6.6 Spreadsheet Solutions

In order to ensure that the linear equations deducted in previous sections are correct, they have been applied in following Excel Spread Sheets:

• CnstrSngl4d, Magic Border Square, 4 x 4, MC = 340
  
  
• CnstrSngl5f, Magic Corner Square, 5 x 5, MC = 425
  
  
• CnstrSngl7e, Eccentric Magic Square, 7 x 7, MC = 595
  
  
• CnstrSngl9g, Eccentric Magic Square, 9 x 9, MC = 765
  
  
• CnstrSngl13, Magic Square, 13 x 13, MC = 1105

Only the red figures have to be “guessed” to construct the applicable Magic Squares (wrong solutions are obvious).

12.6.7 Summary (2)

The obtained results regarding the 13^th order Composed 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 Prime Number Magic Squares, which will be described in Section 14 (Special Squares, Prime Numbers).