Humongous Magic Square

Here is one of the biggest magic squares around. It is 15-by-15 squares in all. It was originally published by a German mathematician in 1604. (I got it from Mathematics From the Birth of Numbers. This is not only a nifty magic square, but the explanation that follows it below gives you a general method for creating odd-sided magic squares.

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

Did you find the pattern? Sort of? That's okay, it isn't completely clear until you're given the secret. So how does it work?

You begin by putting a 1 in the square that is just to the right of the center square (which is here the 8th row and 9th column). Then you enter increasing number, moving diagonally upward to the right. In this particular square you go from 1 to 7. Any time you reach the right side of the magic square is continued on the lefthand side of the magic square, one row up. So, in this case, since the 7 hits the last column in the second row, we put the 8 on row one starting in the far left column.

When you come to the top, or first, row, then you continue in on the bottom row of the next column. In this case, 8 happens to be in the top row in the first column, so we have put 9 in the bottom, or fifteenth, row of the next column (which is the second column in this case). Then you continue increasing your numbers diagonally up toward the right like before.

When you come to a square that already has a number in it, then you skip over that number and restart on the same row but two columns over. This happens when 15 runs into 1 in this magic square. Look and see where 16 in comparison to 15 in the magic square.

One special case in this magic square happens when 120 hits the upper right hand corner. Normally it would go in the bottom square of the far left column (the bottom left corner). But it is blocked by 106. So the two-step change gets made by putting 121 in the top square of the next column. Then, because the numbering has now reached the top row, it has to be continued on the bottom row of the next column, which is why 122 is in the third column of the bottom row.

You continue in this way until you get to the last space, and enter 225 just to the left of 113, which is in the center of the whole magic square.

Oh, by the way, did you figure out what every row, column, and diagonal adds up to? You can add one or two of them up if you wish, or you can do the following: Figure out the total number of squares (15 squared, actually, which = 15 X 15 = 225). Then multiply the total number of squares (225) by the total number of squares plus one (226) and divide the answer by 2. That equals (225 X 226) / 2 = 26,425. That's what the total of all of the squares is. Since there are 15 squares in each of the columns or rows and diagonals, you divide 26,425 by 15 and that equals 1,695, which is exactly what each and every row, column, and diagonal adds up to.

Now that's magic!