NationMaster - Encyclopedia: Prime reciprocal magic square

In mathematics, a reciprocal is a number divided into one, like 1/3 or 1/7. In base ten, the remainder, and so the digits, of 1/3 repeats at once: 0�3333... However, the remainders of 1/7 repeat over six, or 7-1, digits: 1/7 = 0�142857142857142857... If you examine the multiples of 1/7, you can see that each is a cyclic permutation of these six digits:

1/7 = 0.1 4 2 8 5 7... 2/7 = 0.2 8 5 7 1 4... 3/7 = 0.4 2 8 5 7 1... 4/7 = 0.5 7 1 4 2 8... 5/7 = 0.7 1 4 2 8 5... 6/7 = 0.8 5 7 1 4 2...

If the digits are laid out as a square, it is obvious that each row will sum to 1+4+2+8+5+7, or 27, and only slightly less obvious that each column will also do so, and consequently we have a magic square:

1 4 2 8 5 7 2 8 5 7 1 4 4 2 8 5 7 1 5 7 1 4 2 8 7 1 4 2 8 5 8 5 7 1 4 2

However, neither diagonal sums to 27, but all other prime reciprocals in base ten with maximum period of p-1 produce squares in which all rows and columns sum to the same total. In the square from 1/19, with maximum period 18 and row-and-column total of 81, both diagonals also sum to 81, and this square is therefore fully magic:

01/19 = 0�0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1... 02/19 = 0�1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2... 03/19 = 0�1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3... 04/19 = 0�2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4... 05/19 = 0�2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5... 06/19 = 0�3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6... 07/19 = 0�3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7... 08/19 = 0�4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8... 09/19 = 0�4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9... 10/19 = 0�5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0... 11/19 = 0�5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1... 12/19 = 0�6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2... 13/19 = 0�6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3... 14/19 = 0�7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4... 15/19 = 0�7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5... 16/19 = 0�8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6... 17/19 = 0�8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7... 18/19 = 0�9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8...

The same phenomenon occurs with other primes in other bases, and the following table lists some of them, giving the prime, base, and magic total (derived from the formula base-1 x prime-1 / 2):

Prime	Base	Total
19	10	81
53	12	286
53	34	858
59	2	29
67	2	33
83	2	41
89	19	792
167	68	5,561
199	41	3,960
199	150	14,751
211	2	105
223	3	222
293	147	21,316
307	5	612
383	10	1,719
389	360	69,646
397	5	792
421	338	70,770
487	6	1,215
503	420	105,169
587	368	107,531
593	3	592
631	87	27,090
677	407	137,228
757	759	286,524
787	13	4,716
811	3	810
977	1,222	595,848
1,033	11	5,160
1,187	135	79,462
1,307	5	2,612
1,499	11	7,490
1,877	19	16,884
1,933	146	140,070
2,011	26	25,125
2,027	2	1,013
2,141	63	66,340
2,539	2	1,269
3,187	97	152,928
3,373	11	16,860
3,659	126	228,625
3,947	35	67,082
4,261	2	2,130
4,813	2	2,406
5,647	75	208,902
6,113	3	6,112
6,277	2	3,138
7,283	2	3,641
8,387	2	4,193

Results from FactBites:

Magic square - Wikipedia, the free encyclopedia (1488 words)
	The earliest magic square of order four was found in an inscription in Khajuraho, India, dating from the eleventh or twelfth century; it is also a panmagic square where, in addition to the rows, columns and main diagonals, the broken diagonals also have the same sum.
	The magic sum of the square is 33, the age of Jesus at the time of the Passion.
	In the magic square of order four, the numbers in the four central squares and one square at each corner are retained in the same place and the others are interchanged with their diametrically opposite numbers.

More results at FactBites »

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