Question 1
A $3 \times 3$ grid made up of $9$ $1 \times 1$ squares is given. Suppose you want to distribute $9$ distinct positive integers chosen from the integers greater than or equal to $1$ and less than or equal to $9$ into $9$ square boxes of the grid. How many distinct ways of distributing the $9$ numbers are there if for any pair of boxes sharing a side the difference of the numbers inserted must be $3$ or less? Even when the two configurations of the result of distribution coincide under a rotation or flipping over, regard the configurations distinct.