TITLE: Find the largest number of pairs of adjacent cells such that the number in one cell of the pair divides into the other number in the pair.
QUESTION [0 upvotes]: The numbers from 1 to 9 are written in a 3 × 3 grid of cells. Every cell has one number and no number is written more than once. Find the largest number of pairs of adjacent (sharing a side) cells such that the number in one cell of the pair divides into the other number in the pair.
I think 1 is meant to be in the centre because it divides every number but im not sure
Suggestions and solutions would be appreciated.

REPLY [2 votes]: Possible way to solve the question
The max amount of possible adjacent cells is $12$, because a $3$x$3$ square has 12 possible side where cells could be adjacent.
Both $5,7$ could only be adjacent to $1$. The number $9$ could only be adjacent to $3$ or $1$.
Now you could construct an inelegent proof by contradiction, that a grid with $10$ or more adjacent side does not exist. And show this next example with $9$ adjacent sides:
$$ \left[
\begin{array}{}
  5&8&4\\
  7&1&2\\
  9&3&6
\end{array}
\right] $$
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