Office Applications and Entertainment, Magic Squares

12.7   Magic Squares (13 x 13)

12.7.1 Composed Magic Squares, Overlapping Sub Squares

Previous section 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 = 1105

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

The Magic Square shown above is composed out of:

• Two 6^th order Partly Compact Associated Magic Corner Squares (s6 = 6 * s1 / 13),
• Two 7^th order Overlapping Magic Corner Squares (s7 = 7 * s1 / 13) each composed of:
  - One 4^th order Associated Magic Square (s4 = 4 * s1 / 13)
  - One 3^th 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 miscellaneous order 13 Composed Magic Squares with Overlapping Sub Squares, which could be found with subject routine.

Alternatively order 13 Magic Squares, with two 7^th order Overlapping Sub Squares with identical Magic Sums can be constructed based on suitable selected Latin Sub Squares as illustrated in Section 25.5.

12.7.2 Composed Magic Squares, 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 17.8.1 for order 7 Magic Squares.

Mc13 = 1105

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

The Magic Square shown above is composed out of:

• One 6^th order Partly Compact Associated Magic Corner Square (s6 = 6 * s1 / 13),
• One 7^th order Semi Magic Corner Square (s7 = 7 * s1 / 13) composed of:
  - One 4^th order Associated Magic Square (s4 = 4 * s1 / 13)
  - One 3^th 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 miscellaneous order 13 Composed Magic Square, which could be found with subject routine.

Attachment 14.10.6 shows the resulting order 13 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.

12.7.3 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 14.7.5 for order 9 Magic Squares.

Mc13 = 1105

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

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.

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

The 13^th order Composed Inlaid Magic Square shown below:

Mc13 = 1105 169 168 167 166 165 164 49 8 9 10 11 12 7 157 24 41 58 127 116 60 150 46 47 118 136 25 156 111 108 77 22 19 89 100 148 50 130 69 26 39 31 34 65 120 123 53 71 96 134 67 129 143 29 113 106 79 20 21 87 140 61 128 35 133 153 28 32 33 66 119 124 52 36 146 138 147 30 154 15 115 104 81 18 23 85 48 125 94 135 107 155 16 117 45 43 73 75 112 151 72 105 62 92 142 17 42 98 70 59 84 40 93 80 139 90 152 141 27 57 44 63 88 101 126 137 86 91 76 78 131 144 83 102 74 38 56 82 51 122 97 132 110 14 145 54 64 103 95 37 149 114 109 68 99 55 13 163 158 159 160 161 162 121 6 5 4 3 2 1

s 426 497 353 594

contains following inlays:

• two each 6^th order Compact Pan Magic Squares - Magic Sums s(1) = 426 and s(4) = 594 - with the center element in common,
• two each 5^th order Simple Magic Squares with Magic Sums s(2) = 497 and s(3) = 353.

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 = 1105 the Magic Sum of the 13^th 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)

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 6.09.3, resulting in the two each other overlapping 6^th order Compact Pan Magic Sub Squares (ref. Priem6a).
• A guessing routine, based on the defining linear equations as deducted in Section 3.2.2, resulting in the two 5^th order Simple Magic Sub Squares (ref. Priem5b).

Attachment 14.9.9 shows a few 13^th order Inlaid Magic Squares for the Magic Sums s(1) = 426 , s(2) = 497, s(3) = 353 and s(4) = 594.

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

12.7.5 Associated Magic Squares, Diamond Inlays Order 6 and 7

The 13^th order Associated Inlaid Magic Square shown below:

Mc13 = 1105

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

contains following Diamond Inlays:

• one each 6^th order Associated Diamond Inlay with Magic Sum s6 = 510,
• one each 7^th order Associated Diamond Inlay with Magic Sum s7 = 595.

As the order 6 and 7 Diamond Inlays contain only odd numbers, the Associated Inlaid Magic Square is a Lozenge Square.

The method to generate order 13 Associated Lozenge Squares with order 6 and 7 Diamond Inlays has been discussed in Section 18.7.4.

12.7.6 Summary

The obtained results regarding the miscellaneous types of order 13 Magic Squares as deducted and discussed in previous sections are summarised in following table:

Comparable routines as listed above, can be used to generate alternative types of order 13 Magic Squares.