1
$\begingroup$

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 (Moved to an answer. The link died.)

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 ?

$\endgroup$
3
  • 1
    $\begingroup$ Your link (or a slightly more helpful version) says the next base-$10$ value that works is $383$ $\endgroup$
    – Henry
    Jan 24, 2014 at 8:12
  • 1
    $\begingroup$ 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, 2014 at 10:02
  • $\begingroup$ you are right @JackM ! $\endgroup$
    – Fereydoon Shekofte
    Jan 24, 2014 at 13:31

2 Answers 2

Reset to default
3
$\begingroup$

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

$\endgroup$
1
  • $\begingroup$ thanks @Grry Myerson this is the definitive answer . $\endgroup$
    – Fereydoon Shekofte
    Jan 24, 2014 at 13:55
1
$\begingroup$

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

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

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .