Composite, Proportional (1) b

René Chrétien had noticed the 15x15 composite (4) magic square and showed me it is possible to use the method to construct magic squares of even orders as well.

Construct the 16x16 magic square by using 16 proportional 4x4 panmagic squares. The squares are proportional because all 16 panmagic 4x4 squares have the same magic sum of (1/4 x 2056 = ) 514. We use the basic key method (4x4) to produce the panmagic 4x4 squares. As row coordinates don't use 0 up to 3 but use 0 up to (16x4 -/- 1 = ) 63 instead. Take care that the sum of the row coordinates in each 4x4 square is the same (0+31+32+63 = 1+30+33+62 = ... = 15+16+47+48 = 126) to get proportional squares.

1x row coordinate +64x column coordinate + 1 = panmagic 4x4 square

0	31	32	63	0	3	1	2	1	224	97	192
32	63	0	31	3	0	2	1	225	64	129	96
31	0	63	32	2	1	3	0	160	65	256	33
63	32	31	0	1	2	0	3	128	161	32	193

1	30	33	62	0	3	1	2	2	223	98	191
33	62	1	30	3	0	2	1	226	63	130	95
30	1	62	33	2	1	3	0	159	66	255	34
62	33	30	1	1	2	0	3	127	162	31	194

2	29	34	61	0	3	1	2	3	222	99	190
34	61	2	29	3	0	2	1	227	62	131	94
29	2	61	34	2	1	3	0	158	67	254	35
61	34	29	2	1	2	0	3	126	163	30	195

3	28	35	60	0	3	1	2	4	221	100	189
35	60	3	28	3	0	2	1	228	61	132	93
28	3	60	35	2	1	3	0	157	68	253	36
60	35	28	3	1	2	0	3	125	164	29	196

4	27	36	59	0	3	1	2	5	220	101	188
36	59	4	27	3	0	2	1	229	60	133	92
27	4	59	36	2	1	3	0	156	69	252	37
59	36	27	4	1	2	0	3	124	165	28	197

5	26	37	58	0	3	1	2	6	219	102	187
37	58	5	26	3	0	2	1	230	59	134	91
26	5	58	37	2	1	3	0	155	70	251	38
58	37	26	5	1	2	0	3	123	166	27	198

6	25	38	57	0	3	1	2	7	218	103	186
38	57	6	25	3	0	2	1	231	58	135	90
25	6	57	38	2	1	3	0	154	71	250	39
57	38	25	6	1	2	0	3	122	167	26	199

7	24	39	56	0	3	1	2	8	217	104	185
39	56	7	24	3	0	2	1	232	57	136	89
24	7	56	39	2	1	3	0	153	72	249	40
56	39	24	7	1	2	0	3	121	168	25	200

8	23	40	55	0	3	1	2	9	216	105	184
40	55	8	23	3	0	2	1	233	56	137	88
23	8	55	40	2	1	3	0	152	73	248	41
55	40	23	8	1	2	0	3	120	169	24	201

9	22	41	54	0	3	1	2	10	215	106	183
41	54	9	22	3	0	2	1	234	55	138	87
22	9	54	41	2	1	3	0	151	74	247	42
54	41	22	9	1	2	0	3	119	170	23	202

10	21	42	53	0	3	1	2	11	214	107	182
42	53	10	21	3	0	2	1	235	54	139	86
21	10	53	42	2	1	3	0	150	75	246	43
53	42	21	10	1	2	0	3	118	171	22	203

11	20	43	52	0	3	1	2	12	213	108	181
43	52	11	20	3	0	2	1	236	53	140	85
20	11	52	43	2	1	3	0	149	76	245	44
52	43	20	11	1	2	0	3	117	172	21	204

12	19	44	51	0	3	1	2	13	212	109	180
44	51	12	19	3	0	2	1	237	52	141	84
19	12	51	44	2	1	3	0	148	77	244	45
51	44	19	12	1	2	0	3	116	173	20	205

13	18	45	50	0	3	1	2	14	211	110	179
45	50	13	18	3	0	2	1	238	51	142	83
18	13	50	45	2	1	3	0	147	78	243	46
50	45	18	13	1	2	0	3	115	174	19	206

14	17	46	49	0	3	1	2	15	210	111	178
46	49	14	17	3	0	2	1	239	50	143	82
17	14	49	46	2	1	3	0	146	79	242	47
49	46	17	14	1	2	0	3	114	175	18	207

15	16	47	48	0	3	1	2	16	209	112	177
47	48	15	16	3	0	2	1	240	49	144	81
16	15	48	47	2	1	3	0	145	80	241	48
48	47	16	15	1	2	0	3	113	176	17	208

Put the 16 panmagic 4x4 squares in sequence together.

16x16 magic square

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

The 16x16 magic square is not fully 2x2 compact. Use the Khajuraho method to swap numbers.

Franklin panmagic 16x16 square

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

This 16x16 magic square is panmagic, 2x2 compact and each 1/4 row/column/diagonal gives 1/4 of the magic sum.

I have used composite method, proportional (1) to construct

8x8, 9x9, 12x12a, 12x12b, 15x15a, 15x15b, 16x16a, 16x16b, 18x18, 20x20a, 20x20b, 21x21a, 21x21b, 24x24a, 24x24b, 24x24c, 27x27a, 27x27b, 28x28a, 28x28b, 30x30a, 30x30b,32x32a, 32x32b and 32x32c

16x16, Composite, Prop. (1) b.xls

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16x16 magic square