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.
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:
$$ \left[ \begin{array}{} 5&8&4\\ 7&1&2\\ 9&3&6 \end{array} \right] $$