# do you know another Magic Square with this property?

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

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 ?

• Your link (or a slightly more helpful version) says the next base-$10$ value that works is $383$ Commented Jan 24, 2014 at 8:12
• 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? Commented Jan 24, 2014 at 10:02
• you are right @JackM ! Commented Jan 24, 2014 at 13:31

The numbers that work are tabulated at the Online Encyclopedia of Integer Sequences, where some references to the literature are given.

• thanks @Grry Myerson this is the definitive answer . Commented Jan 24, 2014 at 13:55

The link died, so here is the array for $$19$$. The numbers would be put into ascending order of the fraction ($$1/19, 2/19, ...$$).

$$\begin{array}& 0&5&2&6&3&1&5&7&8&9&4&7&3&6&8&4&2&1\\ 1&0&5&2&6&3&1&5&7&8&9&4&7&3&6&8&4&2\\ 1&5&7&8&9&4&7&3&6&8&4&2&1&0&5&2&6&3\\ 2&1&0&5&2&6&3&1&5&7&8&9&4&7&3&6&8&4\\ 2&6&3&1&5&7&8&9&4&7&3&6&8&4&2&1&0&5\\ 3&1&5&7&8&9&4&7&3&6&8&4&2&1&0&5&2&6\\ 3&6&8&4&2&1&0&5&2&6&3&1&5&7&8&9&4&7\\ 4&2&1&0&5&2&6&3&1&5&7&8&9&4&7&3&6&8\\ 4&7&3&6&8&4&2&1&0&5&2&6&3&1&5&7&8&9\\ 5&2&6&3&1&5&7&8&9&4&7&3&6&8&4&2&1&0\\ 5&7&8&9&4&7&3&6&8&4&2&1&0&5&2&6&3&1\\ 6&3&1&5&7&8&9&4&7&3&6&8&4&2&1&0&5&2\\ 6&8&4&2&1&0&5&2&6&3&1&5&7&8&9&4&7&3\\ 7&3&6&8&4&2&1&0&5&2&6&3&1&5&7&8&9&4\\ 7&8&9&4&7&3&6&8&4&2&1&0&5&2&6&3&1&5\\ 8&4&2&1&0&5&2&6&3&1&5&7&8&9&4&7&3&6\\ 8&9&4&7&3&6&8&4&2&1&0&5&2&6&3&1&5&7\\ 9&4&7&3&6&8&4&2&1&0&5&2&6&3&1&5&7&8\\ \end{array}$$