Construction of Magic Squares

The construction of perfect panmagic squares of order 4k (k≥2)

Below two new perfect 8×8 panmagic squares are shown, with both franklin and normal panmagic properties. For example, all rows*, columns*, diagonals*, subdiagonals*, and franklin-diagonals* show the sum 260. All half-rows*, half-columns*, half-diagonals*, 2×2 squares* show the sum 130. The quadrants are panmagic as well.
(A subdiagonal is formed by e.g. the numbers 47-40-18-27-61-54-4-9; a franklin-diagonal is e.g. 47-40-18-27-6-15-57-50)

62	4	13	51	46	20	29	35
5	59	54	12	21	43	38	28
52	14	3	61	36	30	19	45
11	53	60	6	27	37	44	22
64	2	15	49	48	18	31	33
7	57	56	10	23	41	40	26
50	16	1	63	34	32	17	47
9	55	58	8	25	39	42	24

60	6	11	53	44	22	27	37
13	51	62	4	29	35	46	20
54	12	5	59	38	28	21	43
3	61	52	14	19	45	36	30
58	8	9	55	42	24	25	39
15	49	64	2	31	33	48	18
56	10	7	57	40	26	23	41
1	63	50	16	17	47	34	32

Both squares have been constructed in April 2007 by Willem Barink, author of the puzzle-game Medjig (strongly magic square related). Inspiration was due to three young master-class students in Nijmegen, the Netherlands, who in March 2007 reached world news because they had constructed a spectacular 12×12 panmagic square (HSA square).

Short explanation of the method of construction

Below you see the first square broken down in its grids, after p = a × 8 + r.

grid for the r-numbers								grid for the a-numbers
6	4	5	3	6	4	5	3	7	0	1	6	5	2	3	4
5	3	6	4	5	3	6	4	0	7	6	1	2	5	4	3
4	6	3	5	4	6	3	5	6	1	0	7	4	3	2	5
3	5	4	6	3	5	4	6	1	6	7	0	3	4	5	2
8	2	7	1	8	2	7	1	7	0	1	6	5	2	3	4
7	1	8	2	7	1	8	2	0	7	6	1	2	5	4	3
2	8	1	7	2	8	1	7	6	1	0	7	4	3	2	5
1	7	2	8	1	7	2	8	1	6	7	0	3	4	5	2

In both grids the rows and columns are filled with 8, and two times 4 different numbers (8/4 system). This is different from all more or less panmagic 8×8 squares I (Willem Barink) found on internet, and which, when broken down, showed grids with 8, and an alternation of 2 different numbers in either the rows, or the columns. The panmagic properties of the existing 8×8 squares could still be improved!

In the above 8/4 system I applied my knowledge about latin squares. There are 12 (and only 12) different structures in latin squares of order 4. I examined the patterns of all of them for their usefulness in constructing the grids, in order to avoid the problem of duplo-numbering when composing the final square. I found three (and only three) latins which were suitable:

Structure A				Structure B				Structure C
A	B	C	D	A	B	C	D	A	B	C	D
B	A	D	C	C	D	A	B	C	D	A	B
D	C	B	A	B	A	D	C	D	C	B	A
C	D	A	B	D	C	B	A	B	A	D	C

Regarding the grids of the first 8×8 square, you find out that they have been constructed by using 2×2 structure A for the a-numbers, and 2×2 structure B for the r-numbers. Decompose the second 8×8 square yourself, and you will find out that its grids have been constructed by using 2×2 structure C and its reflection. Consequently the first square may be called type AB, the second type CC.

I found one difference in magic properties between the two squares AB and CC. In type AB in the columns the sum of pairs of two consecutive numbers is 67, 63, 71, and 59, while in the rows it is an alternation of only 66 and 64. In the type CC square both in the rows and columns there is an alternation of just two sums of pairs (73 / 57 and 66 / 64). I call this feature isotropic (a term borrowed from mineralogy and crystallography). Due to these just two sums of pairs the second square contains more magic than the first: 5 semi-magic 4×4 units versus 2 semi-magic units extra magic (except for the diagonals, e.g. the 4×4 square between the numbers 5, 28, 33 and 64 in the second).

Applying the method to 12×12 and 16×16 squares

The above method can also be applied for constructing a 12×12 magic square (totals of rows, columns, etc. 870). The one shown below has been constructed by using 3×3 structure C for both the a-numbers and the r-numbers (its reflection).

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

The nine 4×4 subsquares, and also the four 8×8 subsquares are all perfectly panmagic. So, the above 12×12 square contains in fact 14 panmagic subsquares!

Note that the square is isotropic in the sense that both in rows and columns the sum of pairs of consecutive numbers is an alternation of six sums (135, 155, 139, 151, 143 and 147 for the columns, 25, 265, 73, 217, 121 and 169 for the rows). Also here this isotropic feature is an easy way to recognize a type CC construction. It is also possible to construct a type CC 12×12 square with horizontally and vertically an alternation of just two sums, see the next square.

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

With this square we have a lot more magic than in foregoing 12×12 square: 16 semi-magic 4×4 units more!

Relative to the well-known Morris 12×12 and the HSA 12×12, this abundance of panmagic and semi-magic subsquares is quite an extra panmagic quality; on the other hand, the square lacks the constant sum of franklin-diagonals*. This lack is due to the position of the diameters, dividing panmagic 4×4 units into non-magic 2×2 parts.

This flaw would not occur when constructing a 16×16 with magic properties as shown above. Consequently, the perfect 16×16 panmagic squares of Benjamin Franklin (1706-1790), recently discovered, and the more recent one of Morris can probably be improved! You see the result in my square below.

For comparison, see the franklin-square, and the panmagic Morris 16×16, and its description on the websites:

http://www.magic-squares.net
formerly: geocities.com/~harveyh/franklin.htm
http://www.bestfranklinsquares.com

In the construction of my 16×16 panmagic square I kept the sums of pairs restricted to two in both rows and columns, as in the second 12×12 square above. The result is a square containing 30 panmagic (sub)squares, and 33 semi-magic 4×4 units! Enjoy the beautiful pattern in the positioning of the numbers, going from 1 to 256.

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

Conclusion

The 8/4, 12/4, 16/4 technique, as described above, for the construction of panmagic squares of order 4k (k≥2) is new, and gives very good results. The method is based on three useful structures in latin squares of order 4, which I called structure A, B and C. In the construction two different combinations of structures are possible: AB and CC. A type CC construction leads to the most impressive magic results.

Willem Barink,
Amsterdam, April 2007.

Use the Magic Square Tester!

62	4	13	51	46	20	29	35
5	59	54	12	21	43	38	28
52	14	3	61	36	30	19	45
11	53	60	6	27	37	44	22
64	2	15	49	48	18	31	33
7	57	56	10	23	41	40	26
50	16	1	63	34	32	17	47
9	55	58	8	25	39	42	24

60	6	11	53	44	22	27	37
13	51	62	4	29	35	46	20
54	12	5	59	38	28	21	43
3	61	52	14	19	45	36	30
58	8	9	55	42	24	25	39
15	49	64	2	31	33	48	18
56	10	7	57	40	26	23	41
1	63	50	16	17	47	34	32

grid for the r-numbers								grid for the a-numbers
6	4	5	3	6	4	5	3	7	0	1	6	5	2	3	4
5	3	6	4	5	3	6	4	0	7	6	1	2	5	4	3
4	6	3	5	4	6	3	5	6	1	0	7	4	3	2	5
3	5	4	6	3	5	4	6	1	6	7	0	3	4	5	2
8	2	7	1	8	2	7	1	7	0	1	6	5	2	3	4
7	1	8	2	7	1	8	2	0	7	6	1	2	5	4	3
2	8	1	7	2	8	1	7	6	1	0	7	4	3	2	5
1	7	2	8	1	7	2	8	1	6	7	0	3	4	5	2

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

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

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

62	4	13	51	46	20	29	35
5	59	54	12	21	43	38	28
52	14	3	61	36	30	19	45
11	53	60	6	27	37	44	22
64	2	15	49	48	18	31	33
7	57	56	10	23	41	40	26
50	16	1	63	34	32	17	47
9	55	58	8	25	39	42	24

60	6	11	53	44	22	27	37
13	51	62	4	29	35	46	20
54	12	5	59	38	28	21	43
3	61	52	14	19	45	36	30
58	8	9	55	42	24	25	39
15	49	64	2	31	33	48	18
56	10	7	57	40	26	23	41
1	63	50	16	17	47	34	32

grid for the r-numbers								grid for the a-numbers
6	4	5	3	6	4	5	3	7	0	1	6	5	2	3	4
5	3	6	4	5	3	6	4	0	7	6	1	2	5	4	3
4	6	3	5	4	6	3	5	6	1	0	7	4	3	2	5
3	5	4	6	3	5	4	6	1	6	7	0	3	4	5	2
8	2	7	1	8	2	7	1	7	0	1	6	5	2	3	4
7	1	8	2	7	1	8	2	0	7	6	1	2	5	4	3
2	8	1	7	2	8	1	7	6	1	0	7	4	3	2	5
1	7	2	8	1	7	2	8	1	6	7	0	3	4	5	2

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

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

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

62	4	13	51	46	20	29	35
5	59	54	12	21	43	38	28
52	14	3	61	36	30	19	45
11	53	60	6	27	37	44	22
64	2	15	49	48	18	31	33
7	57	56	10	23	41	40	26
50	16	1	63	34	32	17	47
9	55	58	8	25	39	42	24

60	6	11	53	44	22	27	37
13	51	62	4	29	35	46	20
54	12	5	59	38	28	21	43
3	61	52	14	19	45	36	30
58	8	9	55	42	24	25	39
15	49	64	2	31	33	48	18
56	10	7	57	40	26	23	41
1	63	50	16	17	47	34	32

grid for the r-numbers								grid for the a-numbers
6	4	5	3	6	4	5	3	7	0	1	6	5	2	3	4
5	3	6	4	5	3	6	4	0	7	6	1	2	5	4	3
4	6	3	5	4	6	3	5	6	1	0	7	4	3	2	5
3	5	4	6	3	5	4	6	1	6	7	0	3	4	5	2
8	2	7	1	8	2	7	1	7	0	1	6	5	2	3	4
7	1	8	2	7	1	8	2	0	7	6	1	2	5	4	3
2	8	1	7	2	8	1	7	6	1	0	7	4	3	2	5
1	7	2	8	1	7	2	8	1	6	7	0	3	4	5	2

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

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

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