Question 69
Joe has a $1 \times 46^2$ dogwood strip consisting of $46^2$ unit squares, which are labelled from 1 to $46^2$ in order. Joe saws the strip into $N$ pieces, each of which consists of a number of consecutive unit squares, and rearranges them (without rotating or flipping) into a $46 \times 46$ grid satisfying the following: if the square in the $i$-th row and $j$-th column is labelled $a_{ij}$, then $a_{ij} - (i + j - 1)$ is divisible by 46. Find the smallest possible value of $N$.