Office Applications and Entertainment, Latin Squares

17.0   Latin Squares (17 x 17)

A Latin Square of order 17 is a 17 x 17 square filled with 17 different symbols, each occurring only once in each row and only once in each column.

17.1   Latin Diagonal Squares (17 x 17)

Latin Diagonal Squares are Latin Squares for which the 17 different symbols occur also only once in each of the main diagonals.

17.2   Magic Squares, Natural Numbers

17.2.1 Pan Magic Squares

Pan Magic Square M of order 17 with the integers 1 ... 289 can be written as M = A + 17 * B + 1 where the squares A and B contain only the integers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ... 16.

Consequently order 17 Pan Magic Squares can be based on pairs of Orthogonal Latin Diagonal Squares (A, B).

The required Orthogonal Latin Diagonal Squares (A, B) for Pan Magic Squares can be constructed as follows:

1. Fill the first row of square A and square B with the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ... 16.
   While starting with 0 there are 16! = 2,09228 10¹³ possible combinations for each square.
   
   
2. Complete square A and B by copying the first row into the following rows of the applicable square,
   according to one of the following 91 schemes:
   
   

   A/B

   L2

   R2

   L3

   R3

   L4

   R4

   L5

   R5

   L6

   R6

   L7

   R7

   L8

   R8

   L2

   -

   y

   y

   y

   y

   y

   y

   y

   y

   y

   y

   y

   y

   y

   L3

   y

   y

   -

   y

   y

   y

   y

   y

   y

   y

   y

   y

   y

   y

   L4

   y

   y

   y

   y

   -

   y

   y

   y

   y

   y

   y

   y

   y

   y

   L5

   y

   y

   y

   y

   y

   y

   -

   y

   y

   y

   y

   y

   y

   y

   L6

   y

   y

   y

   y

   y

   y

   y

   y

   -

   y

   y

   y

   y

   y

   L7

   y

   y

   y

   y

   y

   y

   y

   y

   y

   y

   -

   y

   y

   y

   L8

   y

   y

   y

   y

   y

   y

   y

   y

   y

   y

   y

   y

   -

   y

   
   Ln = shift n columns to the left  (n = 2, 3, 4, 5, 6, 7, 8)
   Rn = shift n columns to the right (n = 2, 3, 4, 5, 6, 7, 8)

Attachment 17.2.1 shows the 14 types Latin Diagonal Squares based on the construction method described above.

An example of such a pair (A, B) and the resulting Pan Magic Square M is shown below:

A(L2)

0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	0	1
4	5	6	7	8	9	10	11	12	13	14	15	16	0	1	2	3
6	7	8	9	10	11	12	13	14	15	16	0	1	2	3	4	5
8	9	10	11	12	13	14	15	16	0	1	2	3	4	5	6	7
10	11	12	13	14	15	16	0	1	2	3	4	5	6	7	8	9
12	13	14	15	16	0	1	2	3	4	5	6	7	8	9	10	11
14	15	16	0	1	2	3	4	5	6	7	8	9	10	11	12	13
16	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	0
3	4	5	6	7	8	9	10	11	12	13	14	15	16	0	1	2
5	6	7	8	9	10	11	12	13	14	15	16	0	1	2	3	4
7	8	9	10	11	12	13	14	15	16	0	1	2	3	4	5	6
9	10	11	12	13	14	15	16	0	1	2	3	4	5	6	7	8
11	12	13	14	15	16	0	1	2	3	4	5	6	7	8	9	10
13	14	15	16	0	1	2	3	4	5	6	7	8	9	10	11	12
15	16	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14

B(R2)

0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
15	16	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14
13	14	15	16	0	1	2	3	4	5	6	7	8	9	10	11	12
11	12	13	14	15	16	0	1	2	3	4	5	6	7	8	9	10
9	10	11	12	13	14	15	16	0	1	2	3	4	5	6	7	8
7	8	9	10	11	12	13	14	15	16	0	1	2	3	4	5	6
5	6	7	8	9	10	11	12	13	14	15	16	0	1	2	3	4
3	4	5	6	7	8	9	10	11	12	13	14	15	16	0	1	2
1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	0
16	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
14	15	16	0	1	2	3	4	5	6	7	8	9	10	11	12	13
12	13	14	15	16	0	1	2	3	4	5	6	7	8	9	10	11
10	11	12	13	14	15	16	0	1	2	3	4	5	6	7	8	9
8	9	10	11	12	13	14	15	16	0	1	2	3	4	5	6	7
6	7	8	9	10	11	12	13	14	15	16	0	1	2	3	4	5
4	5	6	7	8	9	10	11	12	13	14	15	16	0	1	2	3
2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	0	1

M = A + 17 * B + 1

1	19	37	55	73	91	109	127	145	163	181	199	217	235	253	271	289
258	276	5	23	41	59	77	95	113	131	149	167	185	203	221	222	240
226	244	262	280	9	27	45	63	81	99	117	135	153	154	172	190	208
194	212	230	248	266	284	13	31	49	67	85	86	104	122	140	158	176
162	180	198	216	234	252	270	288	17	18	36	54	72	90	108	126	144
130	148	166	184	202	220	238	239	257	275	4	22	40	58	76	94	112
98	116	134	152	170	171	189	207	225	243	261	279	8	26	44	62	80
66	84	102	103	121	139	157	175	193	211	229	247	265	283	12	30	48
34	35	53	71	89	107	125	143	161	179	197	215	233	251	269	287	16
274	3	21	39	57	75	93	111	129	147	165	183	201	219	237	255	256
242	260	278	7	25	43	61	79	97	115	133	151	169	187	188	206	224
210	228	246	264	282	11	29	47	65	83	101	119	120	138	156	174	192
178	196	214	232	250	268	286	15	33	51	52	70	88	106	124	142	160
146	164	182	200	218	236	254	272	273	2	20	38	56	74	92	110	128
114	132	150	168	186	204	205	223	241	259	277	6	24	42	60	78	96
82	100	118	136	137	155	173	191	209	227	245	263	281	10	28	46	64
50	68	69	87	105	123	141	159	177	195	213	231	249	267	285	14	32

Each type Latin Diagonal Square described above, corresponds with 16! = 2,09228 10¹³ Latin Diagonal Squares.

The possible combinations of square A and B described above will result in 91 * (2,0922 10¹³)² / 4 = 9,95911 10²⁷ unique solutions.

Each unique Pan Magic Square results in a Class C_n and finally in 8 * 289 * 9,95911 10²⁷ = 2,30255 10³¹ possible Pan Magic Squares of the 17^th order.

Attachment 17.2.2 shows one Pan Magic Square for each valid type combination (A, B) as defined above.

17.2.2 Ultra Magic Squares

A numerical example of the construction of an order 17 Ultra Magic Square M based on pairs of Orthogonal Latin Diagonal Squares (A, B), for the untegers {a_i, i = 0 ... 16} and {b_j, j = 0 ... 16} is shown below:

A

16	0	15	14	13	12	11	10	9	8	7	6	5	4	3	2	1
2	1	16	0	15	14	13	12	11	10	9	8	7	6	5	4	3
4	3	2	1	16	0	15	14	13	12	11	10	9	8	7	6	5
6	5	4	3	2	1	16	0	15	14	13	12	11	10	9	8	7
8	7	6	5	4	3	2	1	16	0	15	14	13	12	11	10	9
10	9	8	7	6	5	4	3	2	1	16	0	15	14	13	12	11
12	11	10	9	8	7	6	5	4	3	2	1	16	0	15	14	13
14	13	12	11	10	9	8	7	6	5	4	3	2	1	16	0	15
0	15	14	13	12	11	10	9	8	7	6	5	4	3	2	1	16
1	16	0	15	14	13	12	11	10	9	8	7	6	5	4	3	2
3	2	1	16	0	15	14	13	12	11	10	9	8	7	6	5	4
5	4	3	2	1	16	0	15	14	13	12	11	10	9	8	7	6
7	6	5	4	3	2	1	16	0	15	14	13	12	11	10	9	8
9	8	7	6	5	4	3	2	1	16	0	15	14	13	12	11	10
11	10	9	8	7	6	5	4	3	2	1	16	0	15	14	13	12
13	12	11	10	9	8	7	6	5	4	3	2	1	16	0	15	14
15	14	13	12	11	10	9	8	7	6	5	4	3	2	1	16	0

B = T(A)

16	2	4	6	8	10	12	14	0	1	3	5	7	9	11	13	15
0	1	3	5	7	9	11	13	15	16	2	4	6	8	10	12	14
15	16	2	4	6	8	10	12	14	0	1	3	5	7	9	11	13
14	0	1	3	5	7	9	11	13	15	16	2	4	6	8	10	12
13	15	16	2	4	6	8	10	12	14	0	1	3	5	7	9	11
12	14	0	1	3	5	7	9	11	13	15	16	2	4	6	8	10
11	13	15	16	2	4	6	8	10	12	14	0	1	3	5	7	9
10	12	14	0	1	3	5	7	9	11	13	15	16	2	4	6	8
9	11	13	15	16	2	4	6	8	10	12	14	0	1	3	5	7
8	10	12	14	0	1	3	5	7	9	11	13	15	16	2	4	6
7	9	11	13	15	16	2	4	6	8	10	12	14	0	1	3	5
6	8	10	12	14	0	1	3	5	7	9	11	13	15	16	2	4
5	7	9	11	13	15	16	2	4	6	8	10	12	14	0	1	3
4	6	8	10	12	14	0	1	3	5	7	9	11	13	15	16	2
3	5	7	9	11	13	15	16	2	4	6	8	10	12	14	0	1
2	4	6	8	10	12	14	0	1	3	5	7	9	11	13	15	16
1	3	5	7	9	11	13	15	16	2	4	6	8	10	12	14	0

M = A + 17 * B + 1

289	35	84	117	150	183	216	249	10	26	59	92	125	158	191	224	257
3	19	68	86	135	168	201	234	267	283	44	77	110	143	176	209	242
260	276	37	70	119	137	186	219	252	13	29	62	95	128	161	194	227
245	6	22	55	88	121	170	188	237	270	286	47	80	113	146	179	212
230	263	279	40	73	106	139	172	221	239	16	32	65	98	131	164	197
215	248	9	25	58	91	124	157	190	223	272	273	50	83	116	149	182
200	233	266	282	43	76	109	142	175	208	241	2	34	52	101	134	167
185	218	251	12	28	61	94	127	160	193	226	259	275	36	85	103	152
154	203	236	269	285	46	79	112	145	178	211	244	5	21	54	87	136
138	187	205	254	15	31	64	97	130	163	196	229	262	278	39	72	105
123	156	189	238	256	288	49	82	115	148	181	214	247	8	24	57	90
108	141	174	207	240	17	18	67	100	133	166	199	232	265	281	42	75
93	126	159	192	225	258	274	51	69	118	151	184	217	250	11	27	60
78	111	144	177	210	243	4	20	53	102	120	169	202	235	268	284	45
63	96	129	162	195	228	261	277	38	71	104	153	171	220	253	14	30
48	81	114	147	180	213	246	7	23	56	89	122	155	204	222	271	287
33	66	99	132	165	198	231	264	280	41	74	107	140	173	206	255	1

The Latin Diagonal Square B is the transposed square of A (rows and columns exchanged).

The Latin Diagonal Squares A can be determined based on the defining equations of the top and bottom row, as provided below for Latin Diagonal Squares type R2:

a(288) = 2 * s1 / 17 - a(289)
a(280) = 3 * s1 / 17 - a(288) - a(289)
a(279) = 2 * s1 / 17 - a(281)
a(278) = 2 * s1 / 17 - a(282)
a(277) = 2 * s1 / 17 - a(283)
a(276) = 2 * s1 / 17 - a(284)
a(275) = 2 * s1 / 17 - a(285)
a(274) = 2 * s1 / 17 - a(286)
a(273) = 2 * s1 / 17 - a(287)

a(10) = 3 * s1 / 17 - a(288) - a(289)
a( 9) = 2 * s1 / 17 - a(281)
a( 8) = 2 * s1 / 17 - a(282)
a( 7) = 2 * s1 / 17 - a(283)
a( 6) = 2 * s1 / 17 - a(284)
a( 5) = 2 * s1 / 17 - a(285)
a( 4) = 2 * s1 / 17 - a(286)
a( 3) = 2 * s1 / 17 - a(287)

a(17) = a(287)
a(16) = a(286)
a(15) = a(285)
a(14) = a(284)
a(13) = a(283)
a(12) = a(282)
a(11) = a(281)
a( 2) = a(289)
a( 1) = a(288)

The solutions can be obtained by guessing the 8 parameters:

     a(i) for i = 281, 282, 283, 284, 285, 286, 287 and 289

filling out these guesses in the abovementioned equations, and completing the square by copying the first row into the following rows, while shifting 2 columns to the right.

With an appropriate guessing routine (UltraLat17) 16 * 14 * 46080 = 10321920 Ultra Magic Squares can be constructed based on Diagonal Latin Squares type R2.

Comparable results can be obtained for the other types of Diagonal Latin Squares defined above.

17.2.3 Composed Magic Squares

Overlapping Sub Squares

Order 17 Magic Squares, containing order 9 Overlapping Sub Squares with identical Magic Sum, based on Latin Diagonal Sub Squares, have been discussed in Section 25.7 and Section 25.8.

17.2.4 Concentric Magic Squares

A numerical example of the construction of an order 17 Concentric Magic Square M based on pairs of Orthogonal Concentric Semi-Latin Squares (A, B), for the untegers {a_i, i = 0 ... 16} and {b_j, j = 0 ... 16} is shown below:

A

8	1	2	3	4	5	6	7	9	10	11	12	13	14	15	16	0
0	8	2	3	4	5	6	7	9	10	11	12	13	14	15	1	16
0	1	8	3	4	5	6	7	9	10	11	12	13	14	2	15	16
0	15	2	8	4	5	6	7	9	10	11	12	13	3	14	1	16
0	15	2	3	12	5	6	4	7	9	10	11	8	13	14	1	16
16	15	14	13	4	11	5	6	7	9	10	8	12	3	2	1	0
16	1	14	13	4	11	10	6	7	9	8	5	12	3	2	15	0
16	15	2	3	12	5	6	9	7	8	10	11	4	13	14	1	0
16	15	2	3	12	5	10	7	8	9	6	11	4	13	14	1	0
16	1	2	3	12	11	6	8	9	7	10	5	4	13	14	15	0
16	1	14	3	4	5	8	10	9	7	6	11	12	13	2	15	0
0	1	14	13	4	8	11	10	9	7	6	5	12	3	2	15	16
0	1	2	13	8	11	10	12	9	7	6	5	4	3	14	15	16
0	1	14	13	12	11	10	9	7	6	5	4	3	8	2	15	16
0	15	14	13	12	11	10	9	7	6	5	4	3	2	8	1	16
16	15	14	13	12	11	10	9	7	6	5	4	3	2	1	8	0
16	15	14	13	12	11	10	9	7	6	5	4	3	2	1	0	8

B

0	0	0	0	16	16	16	16	16	16	16	16	0	0	0	0	8
1	1	1	1	15	15	15	1	1	15	15	15	1	1	15	8	15
2	2	2	2	2	14	14	14	14	14	14	2	2	2	8	14	14
3	7	3	3	3	3	13	13	13	13	3	3	13	8	13	9	13
4	3	9	13	8	12	12	4	4	4	12	12	4	3	7	13	12
5	9	4	12	11	8	11	5	11	11	5	5	5	4	12	7	11
6	4	10	11	10	10	8	6	10	10	6	6	6	5	6	12	10
7	6	5	10	9	9	7	8	7	9	9	7	7	6	11	10	9
9	10	7	9	7	7	6	9	8	7	10	9	9	7	9	6	7
10	5	11	7	4	6	9	7	9	8	7	10	12	9	5	11	6
11	11	6	6	6	5	10	10	6	6	8	11	10	10	10	5	5
12	14	12	5	5	11	5	11	5	5	11	8	11	11	4	2	4
13	12	13	4	12	4	4	12	12	12	4	4	8	12	3	4	3
14	15	14	8	13	13	3	3	3	3	13	13	3	13	2	1	2
15	13	8	14	14	2	2	2	2	2	2	14	14	14	14	3	1
16	8	15	15	1	1	1	15	15	1	1	1	15	15	1	15	0
8	16	16	16	0	0	0	0	0	0	0	0	16	16	16	16	16

M = A + 17 * B + 1

9	2	3	4	277	278	279	280	282	283	284	285	14	15	16	17	137
18	26	20	21	260	261	262	25	27	266	267	268	31	32	271	138	272
35	36	43	38	39	244	245	246	248	249	250	47	48	49	139	254	255
52	135	54	60	56	57	228	229	231	232	63	64	235	140	236	155	238
69	67	156	225	149	210	211	73	76	78	215	216	77	65	134	223	221
102	169	83	218	192	148	193	92	195	197	96	94	98	72	207	121	188
119	70	185	201	175	182	147	109	178	180	111	108	115	89	105	220	171
136	118	88	174	166	159	126	146	127	162	164	131	124	116	202	172	154
170	186	122	157	132	125	113	161	145	129	177	165	158	133	168	104	120
187	87	190	123	81	114	160	128	163	144	130	176	209	167	100	203	103
204	189	117	106	107	91	179	181	112	110	143	199	183	184	173	101	86
205	240	219	99	90	196	97	198	95	93	194	142	200	191	71	50	85
222	206	224	82	213	80	79	217	214	212	75	74	141	208	66	84	68
239	257	253	150	234	233	62	61	59	58	227	226	55	230	37	33	51
256	237	151	252	251	46	45	44	42	41	40	243	242	241	247	53	34
289	152	270	269	30	29	28	265	263	24	23	22	259	258	19	264	1
153	288	287	286	13	12	11	10	8	7	6	5	276	275	274	273	281

A pair of order 17 Orthogonal Semi-Latin Borders can be constructed for each pair of order 15 Orthogonal Concentric Semi-Latin Squares (A15, B15).

Each pair of order 17 Orthogonal Semi-Latin Borders corresponds with numerous pairs, which can be obtained by permutation of the border pairs.

17.3   Summary

The obtained results regarding the order 17 (Semi) Latin - and related Magic Squares, as deducted and discussed in previous sections, are summarized in following table:

Attachment	Subject	Subroutine
-	-	-
Attachment 17.2.1	Latin Diagonal Squares	-
Attachment 17.2.2	Pan Magic Squares	CnstrSqrs17a
-	-	-

Comparable methods as described above, can be used to construct higher order (Semi) Latin - and related (Pan) Magic Squares.