Victor's mathematics teacher gave him a $3 \times 3$ grid filled with the numbers $1, 2, 3, 4, 5, 6, 7, 8$ and $9$ in an unspecified order, with each number appearing only once.
Given any row
$\begin{array}{ | l | c | r | } \hline a & b & c \\ \hline \end{array}$
or column
$\begin{array}{ | l | c | r | } \hline d \\ \hline e \\ \hline f \\ \hline \end{array}$
in the grid, Victor is allowed to perform the following operations:
Row:
$a+k$ , $b-k$ , $c-k$
$a-k$ , $b-k$ , $c+k$
Column:
$d+k$ , $e-k$ , $f-k$
$d-k$ , $e-k$ , $f+k$
with $k$ being a non-negative real number and that the numbers in the grid must always be greater than $0$ .
Given that it is possible to perform row and/or column operations such that all the numbers in the grid are equal to a positive real number $N$ , find the maximum value that $N$ can take.
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Since $k$ is nonnegative, the sum of all the numbers would decrease affer every operation. (Consider a row or column with sum $a+b+c$ where $a, b, c$ are the entries in that row/column. The sum after the operation will be $a+k+b-k+c-k=a+b+c-k<a+b+c$ )
Note that the original sum is $1+2+3+...+9=45$ . Clearly the sum must decrease in the end when all numbers are equal from the above, as at leaat one operation is needed.It is obvious that $N$ is an integer, so the largest possible is 4. I will post a possible arrangement in the comments.