A beautiful problem from a Hungarian competition

We need to fill each cell in a \(6\times6\) grid with a distinct integer from 1 to 36. There are two rules:

  • Every pair of consecutive numbers are in adjacent cells that share an edge.
  • Any two cells containing a multiple of 4 cannot share an edge nor a vertex.

Over all valid configurations, how many of the 36 cells could contain the number \(36?\)

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Note: Below is a \(3\times 3\) grid adhering to the rules above.

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