Office Applications and Entertainment, Magic Squares

12.9   Quadrant Magic Squares (17 x 17)

The concept of Quadrant Magic Squares, as discussed in following sections for order 17 Magic Squares, was introduced by Harvey Heinz (2001/2002).

12.9.1 Definition and Terminology

An order 17 magic square can be divided into four overlapping quadrants of 9 x 9 cells.

Each quadrant might contain symmetric patterns of 17 cells (16 cells + the centre cell) which sum to the Magic Sum s1, as illustrated in following example:

p001 (Plusmagic)

o	o	o	o	o	o	o	o	o
o	o	o	o	o	o	o	o	o
o	o	o	o	o	o	o	o	o
o	o	o	o	o	o	o	o	o
o	o	o	o	o	o	o	o	o
o	o	o	o	o	o	o	o	o
o	o	o	o	o	o	o	o	o
o	o	o	o	o	o	o	o	o
o	o	o	o	o	o	o	o	o

For order 17 magic squares 253 patterns can be recognised (ref.Patterns17), further referred to as Magic Pattern P01 thru P253, which are shown in Attachment 12.8.1.

An order 17 Magic Square is called Quadrant Magic if the same pattern(s) occur in all four quadrants.

The number of patterns which occur in all four quadrants will be referred to as nQ4.

Note:
As only 15 of the possible patterns (ref. Section 12.8.2 below) are archived, the tag numbers of the patterns shown in Attachment 12.8.1 will differ from tag numbers used in other (possible) publications.

12.9.2 Historical Results

The properties of the historical order 17 (Multi) Quadrant Magic Squares, as published by Harvey Heinz, are based on the 15 Quadrant Magic Patterns shown in Attachment 12.8.2 and summarised below:

Note: The squares listed above contain also other patterns, however not in all four quadrants.

Subject squares were filtered from the order 17 (Pan) Magic Squares, which can be constructed based on Latin Diagonal Squares, as discussed in Section 17.2.1.

Following sections will describe and illustrate how comparable squares, not necessarily listed in the table shown above, can be constructed or generated.

12.9.3 Equations Quadrant Magic Patterns

The quadrant properties defined in Section 12.8.1 above can be described by the linear equations as listed in Attachment 12.8.3.

Subject equations can be combined with the equations describing miscellaneous types of order 17 Magic Squares.

Subject equations might be incorporated in procedures to determine all occurring patterns of (previously) generated Quadrant Magic Squares.

12.9.4 Pan Magic, Example 6

The occurring patterns for the above mentioned Pan Magic Square 'Example 6' can be summarised as follows (ref. ChkPtrn17):

• The number of patterns which occur in all 4 Quadrants is nQ4 = 61,
• All 253 patterns occur in the first Quadrant (left/top).

Attachment 12.8.41 shows for 'Example 6'(separately) the 61 Quadrant Magic Patterns.

Attachment 12.8.42 shows for 'Example 6'(separately) the 253 Patterns occurring in the first Quadrant (left/top).

12.9.5 Ultra Magic

In Section 17.2.2 is described how order 17 Ultra Magic Squares can be constructed based on pairs of Orthogonal Latin Diagonal Squares A and T(A).

Note: Square A is of type R2, L2, L3, R3, L4, R4, L5, R5, L6, R6, L7, R7, L8 and R8. T(A) is the transposed of A.

With the type R2 based routine UltraLat17 10.321.920 Quadrant Ultra Magic Squares could be generated as summarised in Attachment 12.8.52 and following graph:

An example of a Quadrant Ultra Magic Square with nQ4 = 2 (P73/P229) is shown below:

P73

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

P229

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

Attachment 12.8.51 shows (separately) all Quadrant Magic Patterns for the first occurring Quadrant Ultra Magic Square with nQ4 = 98 (maximum).

Notes:

1. The squares described above contain also numerous other patterns, however not in all four quadrants.
2. It can be noticed that all Ultra Magic Squares constructed as described in Section 17.2.2 are (Multi) Quadrant Ultra Magic.

12.9.7 Bordered, Centre Square Order 13

Order 17 Bordered Quadrant Magic Squares with order 13 Ultra Magic Centre Squares can be constructed based on:

• Latin Square based Ultra Magic Centre Squares as discussed in Section 13.2.2
• Semi Latin Square based borders as discussed in Section 17.2.4

The construction method is dependent from the extend in which the Quadrant Magic Property is effected by variations in the border(s).

Case 1 P61, P140, P143, P147, P148, P154

The Quadrant Magic Property based on the patterns listed above is invariant for variations of the borders as illustrated by following example:

P154

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	253	88	116	148	180	198	39	60	132	211	76	159	225	254	255
52	135	40	55	134	207	82	165	231	249	90	111	149	177	195	155	238
69	67	228	246	91	106	151	173	201	46	61	130	209	77	166	223	221
102	169	196	47	58	127	210	72	168	224	252	97	112	147	175	121	188
119	70	164	226	247	98	109	144	176	191	49	54	133	216	78	220	171
136	118	182	197	45	56	128	217	75	161	227	242	100	105	150	172	154
170	186	71	167	233	248	96	107	145	183	194	42	57	123	219	104	120
187	87	140	185	190	48	63	129	215	73	162	234	245	93	108	203	103
204	189	212	74	157	236	241	99	114	146	181	192	43	64	126	101	86
205	240	115	143	178	193	38	66	122	218	80	163	232	243	94	50	85
222	206	124	213	81	160	229	244	89	117	139	184	199	44	62	84	68
239	257	95	113	141	179	200	41	59	125	208	83	156	235	250	33	51
256	237	65	131	214	79	158	230	251	92	110	142	174	202	37	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

Suitable Centre Squares can be selected from the 5760 unique Ultra Magic Squares with routine ChkPtrn17a.

Attachment 12.8.71 shows the first occurring Bordered Quadrant Magic Square for each of the patterns listed above.

Case 2 P38, P39, P45, P51, P134, P135, P138, P139

The Quadrant Magic Property based on the patterns listed above is invariant for variations of the outer border.

Exhibit P38 describes how P38 Quadrant Bordered Magic Squares, with Ultra Magic Centre Squares, can be constructed based on Semi Latin Borders as discussed in Section 17.2.4.

Centre Squares as described for Case 1 above (ref. Attachment 12.8.71) will result in two way Quadrant Bordered Magic Squares e.g. (P38/P61) as illustrated below:

P38

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

P61

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

Attachment 12.8.73 shows for each of the patterns listed above the first occurring Bordered Quadrant Magic Square for miscellaneous centre squares.

Case 3 P112, P115, P116, P119, P120, P126, P244

Also the patterns listed above will result in Quadrant Magic Properties which are invariant for variations of the outer border.

Quadrant Bordered Magic Squares, with Ultra Magic Centre Squares, can be constructed based on Semi Latin Borders as discussed in Section 17.2.4 and a method comparable with the one described in Exhibit P38.

Centre Squares as described for Case 1 above (ref. Attachment 12.8.71) will result in two way Quadrant Bordered Magic Squares e.g. (P112/P61).

Attachment 12.8.75 shows for each of the patterns listed above the first occurring Bordered Quadrant Magic Square for miscellaneous centre squares.

12.9.8 Summary

The obtained results regarding the miscellaneous types of order 17 Quadrant 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 17 Magic Squares.