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A prime reciprocal magic square is a magic square using the decimal digits of the reciprocal of a prime number.
Consider a unit fraction, 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:
If the digits are laid out as a square, 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:
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.