1, 2, 3, 4, 5, 6, 7, 8, 9

Discrete Mathematics Level pending

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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