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with the repeating digits of $\frac{1}{19} = 0.052631578947368421$ we can construct an exceptional magic square :

The number 19 is a cyclic number with a period of 18 before the digits start to repeat.

The full term decimal expansion of the prime number 19 when multiplied by the values 1 to 18, may be arranged in a simple magic square of order-18, if the decimal point is ignored. All 18 rows, columns and the two main diagonals sum to the same value. S = 81. Of course this is not a pure magic square because a consecutive series of numbers from 1 to n is not used

example of order 18

Are there other numbers as well 19 and 383 that can be used for making magic squares in such way ?

is it proved that in base 10 only 19 and 383 have this property ? , what is for higher orders ?

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    $\begingroup$ Your link (or a slightly more helpful version) says the next base-$10$ value that works is $383$ $\endgroup$ – Henry Jan 24 '14 at 8:12
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    $\begingroup$ What you're saying is that (1) the periods of the decimal expansions of the numbers $n/19$, $1\leq n\leq18$ are all of length $18$, (2) if we therefore place these expansions each on their own row, we get a square of side length $18$, and (3) this is a magic square. Is that right? $\endgroup$ – Jack M Jan 24 '14 at 10:02
  • $\begingroup$ you are right @JackM ! $\endgroup$ – Fereydoon Shekofte Jan 24 '14 at 13:31
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The numbers that work are tabulated at the Online Encyclopedia of Integer Sequences, where some references to the literature are given.

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  • $\begingroup$ thanks @Grry Myerson this is the definitive answer . $\endgroup$ – Fereydoon Shekofte Jan 24 '14 at 13:55

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