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
