Office Applications and Entertainment, Magic Squares

11.0   Magic Squares (11 x 11)

11.1.1 Concentric Magic Squares (11 x 11)

An 11^th order Concentric Magic Square consists of a Concentric Magic Square of the 9^th order, as discussed in Section 9.6.1, with a border around it.

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

Based on the linear equations defining the border of a Concentric Magic Square of order 11:

a(111) = 671 - a(112) - a(113) - a(114) - a(115) - a(116) - a(117) - a(118) - a(119) - a(120) - a(121)
a( 22) = 671 - a( 11) - a( 33) - a( 44) - a( 55) - a( 66) - a( 77) - a( 88) - a( 99) - a(110) - a(121)

a routine can be written to generate the borders for subject Concentric Magic Squares (ref. Priem11a).

Attachment 14.9.1 shows a few suitable borders for Concentric Magic Squares of order 11.

Each border shown corresponds with (9!)² = 1,32 * 10¹¹ borders with the same corner pairs, which can be obtained by permutation of the horizontal/vertical (non corner) pairs.

Note: The Concentric Magic Center Squares should be based on the consecutive integers 20, 21, ... 101.

11.1.2 Bordered Magic Squares (11 x 11), Miscellaneous Inlays

Also other Magic Squares of the 9^th order as described and constructed in Section 9.1 thru Section 9.7, can be used as Center Squares for 11^th order Bordered Magic Squares.

The Embedded Magic Squares will have a Magic Sum s9 = 549 and can be based on the consecutive integers 20, 21, ... 101.

Attachment 14.9.2 contains - based on some of the described Magic Squares of order 9 - examples of Bordered Magic Squares for the same border.

11.1.3 Bordered Magic Squares (11 x 11), Split Border

Alternatively an 11^th order Bordered Magic Square with Magic Sum s11 = 671 can be constructed based on:

• a Symmetric Magic Center Square of order 7 with Magic Sum s7 = 427;
• 36 pairs, each summing to 122, surrounding the (Concentric) Magic Center Square;
• a split of the supplementary rows and columns into three parts:
  two summing to s3 = 183 and one to s5 = 305.

as illustrated 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

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

Typical Corner Section (3 x 3):

a'(1) a'(2) a'(3) a'(4) a'(5) a'(6) a'(7) a'(8) -

a'(3) = 183 - a'(2) - a'(1) a'(4) = 122 - a'(2) a'(5) = 122 - a'(1) a'(6) = 122 - a'(3) a'(7) = 183 - a'(4) - a'(1) a'(8) = 122 - a'(7)

Typical Border Rectangle (2 x 5):

a'(1) a'(2) a'(3) a'(4) a'(5) a'(6) a'(7) a'(8) a'(9) a'(10)

a'( 5) = 305 - a'(4) - a'(3) - a'(2) - a'(1) a'( 6) = 122 - a'(1) a'( 7) = 122 - a'(2) a'( 8) = 122 - a'(3) a'( 9) = 122 - a'(4) a'(10) = 122 - a'(5)

Based on the equations above, procedures can be developed:

• to generate, based on the distinct integers {1 ... 121}, four Corner Squares (3 x 3);
• to complete the exterior border with four Magic Rectangles (2 x 5);
• to construct, based on the remaining 49 distinct integers, the border of the Concentric Center Square (7 x 7);
• to construct, based on the remaining 25 distinct integers, the embedded Center Symmetric Magic Square of order 5.

The first occuring Bordered Magic Square is shown below:

2	62	119	10	11	69	107	108	114	6	63
60	120	3	112	111	53	15	14	8	59	116
121	1	93	49	41	90	89	37	28	118	4
98	24	80	88	87	47	45	38	42	106	16
97	25	40	39	50	51	79	86	82	104	18
52	70	30	58	48	61	74	64	92	55	67
31	91	44	36	43	71	72	83	78	21	101
27	95	46	84	77	75	35	34	76	19	103
117	5	94	73	81	32	33	85	29	115	7
9	65	109	20	22	68	96	99	105	12	66
57	113	13	102	100	54	26	23	17	56	110

The border shown above corresponds with (4! * 8⁴) * (4! * 240⁴) = 7,83 * 10¹⁵ borders as:

1. The Corner Squares can be arranged in 4! ways and belong each to a collection of 8 Corner Squares;
2. The Rectangles can be arranged in 4! ways and belong each to a collection of 2 * 5! = 240 Rectangles.

Based on the distinct integers applied in the border shown above, 470552 suitable sets of Corner Squares can be found.

11.1.4 Eccentric Magic Squares (11 x 11)

An 11^th order Eccentric Magic Square consists of one Magic Corner Square of the 9^th order, as discussed in Section 9.6.4, supplemented with two rows and two columns.

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

Based on the linear equations defining the supplementary rows, columns and main diagonal of an Eccentroc Magic Square of order 11:

a(11) = 671 - a(21) - a(31) - a(41) - a(51) - a(61) - a(71) - a(81) - a(91) - a(101) - a(111)
a(12) = 671 - a(13) - a(14) - a(15) - a(16) - a(17) - a(18) - a(19) - a(20) - a( 21) - a( 22)
a(24) = 671 - a( 2) - a(13) - a(35) - a(46) - a(57) - a(68) - a(79) - a(90) - a(101) - a(112)

a routine can be written to generate subject Eccentric Magic Squares (ref. Priem11c).

Attachment 14.9.4 shows a few 11^th order Eccentric Magic Squares, based on 9^th order Eccentric Magic Squares.

Each Eccentric Magic Square shown corresponds with (7!)² = 2,54 * 10⁷ Eccentric Magic Squares with the same corner square and corner pairs, which can be obtained by permutation of the horizontal/vertical (non corner) pairs.

Note: The Eccentric Magic Corner Squares are based on the consecutive integers 20, 21, ... 101.

11.1.5 Bordered Magic Squares (11 x 11), Composed Border (1)

The 11^th order Composed Magic Square shown below, with Magic Sum s11 = 671, consists of:

• a Border composed out of:
  - 4 Semi Magic Anti Symmetric Corner Squares of order 3 with Magic Sum s3 = 183;
  - 4 Anti Symmetric Magic Rectangles order 3 x 5 with s3 = 183 and s5 = 305.
• a Center Symmetric Magic Center Square of order 5 with Magic Sum s5 = 305;

As a consequence of the defining properties mentioned above the 11^th order Composed Magic Square is Center Symmetric.

70	104	9	24	21	42	102	116	19	95	69
108	7	68	40	44	76	79	66	67	17	99
5	72	106	119	118	65	2	1	97	71	15
11	58	114	109	84	37	28	47	110	39	34
30	41	112	63	31	29	86	96	100	48	35
89	49	45	32	60	61	62	90	77	73	33
87	74	22	26	36	93	91	59	10	81	92
88	83	12	75	94	85	38	13	8	64	111
107	51	25	121	120	57	4	3	16	50	117
23	105	55	56	43	46	78	82	54	115	14
53	27	103	6	20	80	101	98	113	18	52

Based on the principles described in previous sections, a fast procedure (Priem11d) can be developed:

• to read the previously generated Center Symmetric Magic Squares (5 x 5);
• to generate, based on the remainder of the pairs, the four Anti Symmetric Semi Magic Squares of order 3;
• to complete the Composed Border of order 11 with the four 3 x 5 Anti Symmetric Magic Rectangles.

Attachment 14.9.7 shows miscellaneous order 11 Associated Composed Magic Squares.

11.1.6 Bordered Magic Squares (11 x 11), Composed Border (2)

The 11^th order Composed Magic Square shown below, with Magic Sum s11 = 671, consists of:

• a Border composed out of:
  - 4 Associated Magic Squares of order 4 with Magic Sum s4 = 244
  - 4 Associated Magic Rectangles order 3 x 4 with s3 = 183 and s4 = 244
• a Magic Center Square of order 3 with Magic Sum s3 = 183

112	111	20	1	60	30	93	109	106	25	4
9	12	103	120	90	36	57	14	15	96	119
2	19	110	113	65	86	32	3	26	107	108
121	102	11	10	29	92	62	118	97	16	13
83	49	42	70	72	53	58	91	51	35	67
48	54	68	74	47	61	75	37	45	77	85
52	80	73	39	64	69	50	55	87	71	31
99	94	43	8	59	40	84	105	100	33	6
24	27	78	115	81	46	56	18	21	88	117
7	44	95	98	66	76	41	5	34	101	104
114	79	28	23	38	82	63	116	89	22	17

The Composed Square shown above corresponds with n11 = 8 * (4! * 384⁴) * (4! * 16⁴) = 6,57 * 10¹⁸ squares as:

1. The Center Square belongs to a collection of 8 Center Squares;
2. The Corner Squares can be arranged in 4! ways and belong each to a collection of 384 Corner Squares;
3. The Border Rectangles can be arranged in 4! ways and belong each to a collection of 16 Rectangles.

Based on the principles described in previous sections, a fast procedure (Priem11e) can be developed:

• to read the previously generated 3 x 3 Magic Center Square (850 unique);
• to generate, based on the remainder of the pairs, four 4 x 4 Associated Magic Squares;
• to complete the Composed Border of order 11 with four 3 x 4 Associated Magic Rectangles.

Attachment 14.9.6 shows miscellaneous order 11 Composed Border Magic Squares. Each (unique) square shown corresponds with n11 squares.

Note:

If the applied properties are changed to:

• the opposite Semi Magic Corner Squares (4 x 4) are Anti Symmetric and Complementary;
• the opposite Magic Rectangles (3 x 4) are Anti Symmetric and Complementary;
• the Magic Center Square (3 x 3) is Center Symmetric (per definition);

the 11^th order Composed Magic Square will be associated

Attachment 14.9.8 shows miscellanous order 11 Associated Composed Magic Squares (ref. Priem11f).

11.2.1 Composed Magic Squares (11 x 11), Overlapping Sub Squares (1)

This 11^th order Composed Magic Square, with overlapping sub squares, is a variation on a well known 13^th order Composed Magic Square (Andrews, 1909) which will be discussed in Section 12.6.

The 11^th order Magic Square K, with Magic Sum s11 = 671, contains following sub squares:

• One 9^th order Eccentric Magic Square H (right top K):
  - with embedded 7^th order Eccentric Magic I (left bottom H)
  - with embedded 5^th order Magic Square C (left bottom I)
• One 3^th order Semi Magic Square M, element a(91) common with C (left bottom K);
• Four 2 x 4 Magic Rectangles: A and B (left), D and E (bottom);
• Another 7^th order Magic Square L with C in the right top corner (overlapping I);

as illustrated below:

92	30	83	49	51	90	47	46	98	21	64
31	91	39	73	71	32	75	76	24	58	101
36	86	100	23	28	93	95	19	69	104	18
85	37	22	99	94	29	27	53	103	77	45
114	8	4	106	59	15	121	111	11	72	50
52	70	13	119	5	108	60	25	97	34	88
12	110	109	62	14	117	3	79	43	44	78
66	56	118	1	107	63	16	38	84	82	40
81	41	61	17	120	2	105	102	20	57	65
48	68	67	9	112	7	116	33	80	35	96
54	74	55	113	10	115	6	89	42	87	26

Attachment 14.8.5.07 shows, miscellaneous of these 11^th order Composed Magic Squares (ref. MgcSqr11).

11.2.2 Composed Magic Squares (11 x 11), Overlapping Sub Squares (2)

Following 11^th order Composed Magic Square, with overlapping sub squares, is a variation on another higher order Composed Magic Square as discussed by William Symes Andrews (ref. Magic Squares and Cubes, Fig. 352).

The 11^th order Magic Square E, with Magic Sum s11 = 671, contains following sub squares:

• One 3^th order Magic Center Square C;
• Two each other overlapping 5^th order Eccentric Magic Squares A1 and A2;
• Two each other overlapping 7^th order Eccentric Magic Squares B1 and B2;
• Two 4^th order Pan Magic Squares PM1 and PM2;

as illustrated below:

38	35	83	88	15	86	41	102	104	16	63
78	93	33	40	107	36	81	20	18	59	106
39	34	84	87	1	117	4	116	67	108	14
89	82	44	29	121	5	118	55	6	24	98
96	26	113	9	65	58	60	13	109	100	22
27	95	10	112	56	61	66	119	3	103	19
31	91	101	21	62	64	57	2	120	17	105
90	32	12	53	114	11	115	46	47	74	77
23	99	69	110	8	111	7	71	80	43	50
92	54	37	28	70	49	97	48	45	76	75
68	30	85	94	52	73	25	79	72	51	42

Attachment 14.9.8.1 shows, miscellaneous order 11 Composed Magic Squares (ref. PriemE11).

The corresponding Composed Magic Squares of order 15 contain, in addition to the sub squares mentioned above, following Corner Squares:

• Two 6^th order Eccentric Magic Squares F1 and F2 with embedded PM1 and PM2;
• Two 9^th order Eccentric Magic Squares D1 and D2 with embedded B1 and B2;

Attachment 14.9.8.2 shows, for the sake of completeness, miscellanous Composed Magic Squares of order 15 (ref. PriemG15).

11.3.1 Associated Magic Squares (11 x 11)
       Associated Square Inlays Order 5 and 6

Associated Magic Squares of order 11 with Square Inlays of order 5 and 6 can be obtained by means of transformation of order 11 Composed Magic Squares, as illustrated in Section 14.7.4 for order 9 Magic Squares.

102	108	23	42	39	22	91	36	26	97	85
114	121	59	120	10	57	111	4	60	3	12
24	58	95	49	81	84	33	44	93	70	40
53	54	51	32	79	9	21	94	55	116	107
34	17	75	92	76	77	72	103	74	16	35
13	5	66	7	104	61	18	115	56	117	109
87	106	48	19	50	45	46	30	47	105	88
15	6	67	28	101	113	43	90	71	68	69
82	52	29	78	89	38	41	73	27	64	98
110	119	62	118	11	65	112	2	63	1	8
37	25	96	86	31	100	83	80	99	14	20

The Associated Square shown above is composed out of:

• One 5^th order Associated (Pan) Magic Square Inlay with Magic Sum s5 = 305,
• One 6^th order Associated Magic Square Inlay with Magic Sum s6 = 366 and
• Two Associated Magic Rectangle Inlays order 5 x 6 with s5 = 305 and s6 = 366

Based on this definition a routine can be developed to generate the required Composed Magic Squares (ref. Prime11c1).

Attachment 14.8.15 shows miscellaneous Composed Magic Square.

Attachment 14.8.16 shows the corresponding Associated Magic Squares with order 5 and 6 Square Inlays.

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

11.3.2 Associated Magic Squares (11 x 11)
       Associated Center Square Order 5

Associated Magic Squares of order 11 with an Associated Center Square of order 5 can be obtained by means of transformation of order 11 Composed Magic Squares as illustrated in Section 14.7.5 for order 9 Magic Squares.

102	23	39	108	42	22	36	97	91	26	85
24	95	81	58	49	84	44	70	33	93	40
34	75	76	17	92	77	103	16	72	74	35
114	59	10	121	120	57	4	3	111	60	12
53	51	79	54	32	9	94	116	21	55	107
13	66	104	5	7	61	115	117	18	56	109
15	67	101	6	28	113	90	68	43	71	69
110	62	11	119	118	65	2	1	112	63	8
87	48	50	106	19	45	30	105	46	47	88
82	29	89	52	78	38	73	64	41	27	98
37	96	31	25	86	100	80	14	83	99	20

Attachment 14.8.17 shows the Associated Magic Squares with order 5 Associated Center Squares, corresponding with the Composed Magic Squares as shown in Attachment 14.8.15.

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

11.3.3 Associated Magic Squares (11 x 11)
       Square Inlays Order 4 and 5 (Overlapping)

The 11^th order Associated Inlaid Magic Square shown below:

105 54 24 13 10 35 76 79 90 120 65 104 25 60 116 52 88 91 5 74 44 12 100 50 85 33 58 115 77 41 93 3 16 15 66 113 49 83 30 38 69 11 96 101 14 82 28 63 121 47 8 99 36 71 102 19 118 55 80 27 61 95 42 67 4 103 20 51 86 23 114 75 1 59 94 40 108 21 26 111 53 84 92 39 73 9 56 107 106 119 29 81 45 7 64 89 37 72 22 110 78 48 117 31 34 70 6 62 97 18 57 2 32 43 46 87 112 109 98 68 17

341 214 274 269

contains following inlays:

• two each 5^th order Pan Magic Squares - Magic Sums s(1) = 341 and s(4) = 269 - with the center element in common,
• two each 4^th order Simple Magic Squares with Magic Sums s(2) = 214 and s(3) = 274.

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

 s(1) = 10 * s11 / 11 - s(4)
 s(2) =  8 * s11 / 11 - s(3)

With s11 = 671 the Magic Sum of the 11^th order Inlaid Magic Square.

The Associated Border can be described by following linear equations:

a(115) =   -s11/11 + a(117) - s(3) + s(4)
a(114) =   -s11/11 + a(118) - s(3) + s(4)
a(113) =   -s11/11 + a(119) - s(3) + s(4)
a(112) =   -s11/11 + a(120) - s(3) + s(4)
a(111) = 15*s11/11 - a(116) - 2 * a(117) - 2 * a(118) - 2 * a(119) - 2 * a(120) - a(121) + 4 * s(3) - 4 * s(4)
a(100) =    s11    - a(110) - s(3) - s(4)
a( 89) =    s11    - a( 99) - s(3) - s(4)
a( 78) =    s11    - a( 88) - s(3) - s(4)
a( 67) =    s11    - a( 77) - s(3) - s(4)
a( 66) = 60*s11/11 - 2*a(77) - 2*a(88) - 2*a(99) - 2*a(110) - a(116) - 2*a(117) - 2*a(118) +
                                                                     - 2*a(119) - 2*a(120) - 2*a(121) - 8*s(4)

Which can be incorporated in an optimised guessing routine MgcSqr11k2.

The Magic Center Squares can be constructed by means of suitable selected Latin Squares (ref. MgcSqr11k1), based on resp. order 4 and 5 Magic Lines for the integers 0 ... 10 as shown in Attachment 14.9.9a.

Attachment 14.9.9b shows a few 11^th order Inlaid Magic Squares for miscellaneous possible Magic Sums s(3) and s(4).

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

11.3.4 Associated Magic Squares (11 x 11)
       Diamond Inlays Order 5 and 6

The 11^th order Associated Inlaid Magic Square shown below:

116	110	10	8	26	113	120	46	60	38	24
92	64	32	108	59	25	49	18	88	66	70
20	82	44	67	117	23	27	17	100	94	80
16	72	91	45	21	79	81	89	37	86	54
48	35	107	65	83	119	19	109	53	29	4
1	11	115	47	51	61	71	75	7	111	121
118	93	69	13	103	3	39	57	15	87	74
68	36	85	33	41	43	101	77	31	50	106
42	28	22	105	95	99	5	55	78	40	102
52	56	34	104	73	97	63	14	90	58	30
98	84	62	76	2	9	96	114	112	12	6

contains following Diamond Inlays:

• one each 5^th order Associated Diamond Inlay with Magic Sum s5 = 305,
• one each 6^th order Associated Diamond Inlay with Magic Sum s6 = 366.

As the order 5 and 6 Diamond Inlays contain only odd numbers, the Associated Inlaid Magic Square is a Lozenge Square.

The method to generate order 11 Associated Lozenge Squares with order 5 and 6 Diamond Inlays will be discussed in Section 18.5.3.

11.4   Summary

The obtained results regarding the miscellaneous types of order 11 Magic Squares as deducted and discussed in previous sections are summarized in following table:

Comparable routines as listed above, can be used to generate Magic Squares of order 12 and higher, which will be described in following sections.