Magic-Square preserving functional equation(Self-made)

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Fm Jakaria
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Magic-Square preserving functional equation(Self-made)

Unread post by Fm Jakaria » Fri Sep 19, 2014 5:43 am

Find all functions f:$\mathbb{N} \to \mathbb{N}$ that preserves 'general magic square's. That is, if a n*n square's entries are serially in standard written as ($a_1, a_2,........, a_{n^2}$); then ($a_1, a_2,........, a_{n^2}$)(positive integer entries) is a 'general magic square' if and only if so is ($f(a_1), f(a_2),........, f(a_{n^2})$).
Here we defined 'general magic square' as an arbitrary square with the sum of entries in each column, each row, each main diagonal- all equal.
You cannot say if I fail to recite-
the umpteenth digit of PI,
Whether I'll live - or
whether I may, drown in tub and die.

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Samiun Fateeha Ira
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Re: Magic-Square preserving functional equation(Self-made)

Unread post by Samiun Fateeha Ira » Fri Sep 19, 2014 11:12 am

If we have a $n*n$ magic square with entries $(a_1, a_2,..., a_{n^2})$ , we can simply make another magic square with entries $(a_1\pm r, a_2\pm r,..., a_{n^2}\pm r)$ , where $r$ is a positive integer.
because then the sum of each row or column will be increased or decreased by $nr$ .

And the same goes for multiplication & division by $r$.

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Samiun Fateeha Ira
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Re: Magic-Square preserving functional equation(Self-made)

Unread post by Samiun Fateeha Ira » Fri Sep 19, 2014 12:06 pm

Anyway, you can only divide by r, if this is a common factor of all the entries!

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Fm Jakaria
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Re: Magic-Square preserving functional equation(Self-made)

Unread post by Fm Jakaria » Sun Sep 21, 2014 8:58 pm

I have noted that f should be injective. This is easy to see: if f(a)=f(b); we construct the 'general magic square'
(f(a),f(a),......($n^2-1$ times),f(b)); we get so is (a,a,a,.....($n^2-1$ times),b). So a=b.

I have another idea, which may be interesting to try: that is, using a latin square is a general magic square. Then if we have ($f(a_1), f(a_2), ......., f(a_{n^2})$) is a latin square, ($a_1, a_2,.............,a_n$) is a general magic square.(Though we already proved injectivity....)

Another idea is: if ($a_1, a_2,.............,a_n$) is a general magic square, we have the total sum of entries of first and last column equal to that of total sum in first and last row; always excluding the four corner entries. Can we construct such useful collection of 'broken' first and last columns and rows; which can be extended to general magic squares? Then it may be possible to get useful informations converting that 'broken' sums under f(since f preserves such squares).
You cannot say if I fail to recite-
the umpteenth digit of PI,
Whether I'll live - or
whether I may, drown in tub and die.

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*Mahi*
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Re: Magic-Square preserving functional equation(Self-made)

Unread post by *Mahi* » Mon Sep 22, 2014 7:28 pm

Using this generalized latin square \[\begin{matrix}
{n-1} & {n+1} & n \\
{n+1} & n & {n-1} \\
n & {n-1} & {n+1}
\end{matrix}\]
We get $f(n) = \frac{f(n+1)+f(n-1)}2$ which implies $f$ is linear, i.e $f(x)$ has the form $ax+b$.
It can be easily verified that this $f$ satisfies the conditions.
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