Prime reciprocal magic square

A prime reciprocal magic square is a magic square using the decimal digits of the reciprocal of a prime number.

Consider 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 72 8 5 7 1 44 2 8 5 7 15 7 1 4 2 87 1 4 2 8 58 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.

Other properties of Prime Reciprocals: Midy's theorem

The repeating pattern of an even number of digits [7-1, 11-1, 13-1, 17-1, 19-1, 29-1, ...] in the quotients when broken in half are the nines-complement of each half:

1/7 = 0.142,857,142,857 ... +0.857,142 --------- 0.999,9991/11 = 0.09090,90909 ... +0.90909,09090 ----- 0.99999,999991/13 = 0.076,923 076,923 ... +0.923,076 --------- 0.999,9991/17 = 0.05882352,94117647 +0.94117647,05882352 ------------------- 0.99999999,999999991/19 = 0.052631578,947368421 ... +0.947368421,052631578 ---------------------- 0.999999999,999999999

Ekidhikena Purvena From: Bharati Krishna Tirtha's Vedic mathematics#By one more than the one before

Concerning the number of decimal places shifted in the quotient per multiple of 1/19:

01/19 = 0.052631578,94736842102/19 = 0.1052631578,9473684204/19 = 0.21052631578,947368408/19 = 0.421052631578,94736816/19 = 0.8421052631578,94736

A factor of 2 in the numerator produces a shift of one decimal place to the right in the quotient.

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...

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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):