My (supposed) Problem of the Year

\[ \begin{array}{|c|c|c|} \hline \ \ 5\ \ &\ \ 6\ \ &\ \ 7\ \ \\ \hline \ \ 4\ \ &\ \ 1\ \ &\ \ 8\ \ \\ \hline \ \ 3\ \ &\ \ 2\ \ &\ \ 9\ \ \\ \hline \end{array}\]

How many ways can we arrange the integers 1 through 9 in a \(3\times3\) grid such that any two adjacent entries are coprime? One such instance is shown above.

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Details and Assumptions:

  • Two entries are adjacent if they share a common edge.
  • Two numbers \(a\) and \(b\) are coprime if \(\text{gcd}(a,b) = 1\).
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