Count the Number of Magic Squares - Solution

If you display it on the surface of a donut, you get 216 magic squares, but on a piece of paper only 35.

Now, compare that with Figure 2.13 shown in John R. Hendricks, Inlaid Magic Squares and Cubes, Second Edition. Mr. Hendricks was the first person in the world to notice that he had miscounted (thank goodness) so he decided to make it a puzzle for the website.

Solution

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

The entire square is a pandiagonal magic square of order 12. One feature of pandiagonal squares is that any number can serve as the top left-hand corner. So there are two answers depending upon whether, or not you allow wrap-around.

Order	No Wrap	Total Wrap
4^th	25	36
8^th	9	36
12^th	1	144
sums	35	216

Notice that every second square is a magic one in both the x and the y directions for both 4^th order & 8^thorder squares.

Home

Last updated Thursday March 22, 2007

Webmaster: Harvey Heinz harveyheinz@shaw.ca