The perfect magic square

The perfect magic square must be a 16x16 magic square. I put an almost perfect 16x16 magic square on my website and asked for a challenge.

Ot Ottenheim from the Netherlands won the challenge and produced the real perfect magic square below.

The perfect magic square

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

This magic 16x16 square is the real perfect magic square, because of the following magic features:

(1) The magic square consists of four by four proportional panmagic 4x4 sub-squares and because of that structure each 1/4 row/column/diagonal gives 1/4 of the magic sum. The 4x4 sub-squares are perfectly connected, so the 16x16 magic square is panmagic and fully 2x2 compact (which means that each random chosen 2x2 sub-square gives 1/4 of the magic sum = 514). The conclusion is that the 16x16 magic square is (Franklin panmagic) most perfect.

(2) The magic square has got the tight 'Willem Barink' structure. Horizontally the sum of 2 numbers (of cell 1+2, 3+4, 5+6, 7+8. 9+10, 11+12, 13+14 and 15+16) gives 241 respectively 273. Vertically the sum of 2 numbers (of cell 1+2, 3+4, 5+6, 7+8. 9+10, 11+12, 13+14 and 15+16) gives 225 and 289 respectively.

241	273	241	273	241	273	241	273
273	241	273	241	273	241	273	241
241	273	241	273	241	273	241	273
273	241	273	241	273	241	273	241
241	273	241	273	241	273	241	273
273	241	273	241	273	241	273	241
241	273	241	273	241	273	241	273
273	241	273	241	273	241	273	241
241	273	241	273	241	273	241	273
273	241	273	241	273	241	273	241
241	273	241	273	241	273	241	273
273	241	273	241	273	241	273	241
241	273	241	273	241	273	241	273
273	241	273	241	273	241	273	241
241	273	241	273	241	273	241	273
273	241	273	241	273	241	273	241

225	289	225	289	225	289	225	289	225	289	225	289	225	289	225	289

289	225	289	225	289	225	289	225	289	225	289	225	289	225	289	225

225	289	225	289	225	289	225	289	225	289	225	289	225	289	225	289

289	225	289	225	289	225	289	225	289	225	289	225	289	225	289	225

225	289	225	289	225	289	225	289	225	289	225	289	225	289	225	289

289	225	289	225	289	225	289	225	289	225	289	225	289	225	289	225

225	289	225	289	225	289	225	289	225	289	225	289	225	289	225	289

289	225	289	225	289	225	289	225	289	225	289	225	289	225	289	225

(3) In each 4x4 sub-square you can find a number from each of the 16 sequences, 1 up to 16, 17 up to 32, 33 up to 48, 49 up to 64, 65 up to 80, 81 up to 96, 97 up to 112, 113 up to 128, 129 up to 144, 145 up to 160, 161 up to 176, 177 up to 192, 193 up to 208, 209 up to 224, 225 up to 240 and 241 up to 256 respectively. The numbers of each sequence are in order from low to high, starting from one of the four corners. And as a finishing touch, the first four sequences start from the top left corner, the second four sequences start from the top right corner, the third four sequences start from the do bottom left corner and the fourth four sequences start from the bottom right corner.

A more perfect square does not exist.

For analysis of the perfect magic square, see the download below.

16x16, the perfect magic square.xls

Microsoft Excel werkblad 621.5 KB

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To produce an Ot Ottenheim perfect magic square for each order is multiple of 4, from 8x8 (to infinite), see download (developed from 8x8 up to 32x32) below.

Ot Ottenheim perfect magic squares, orde

Microsoft Excel werkblad 180.3 KB

Download

16x16 magic square