250 If the square in row $i$, column $j$ contains the number $k$, let its 'index' be $i+j-k$. The constraint on adjacent squares now says that if a square has index $r$, the squares to its right and below it each have index $r$ or $r+2$. The upper-left square has index 5, and the lower-right square has index 7, so every square must have index 5 or 7. The boundary separating the two types of squares is a path consisting of upward and rightward steps; it can be extended along the grid's border so as to obtain a path between the lower-left and upper-right corners. Conversely, any such path uniquely determines each square's index and hence the entire array of numbers - except that the two paths lying entirely along the border of the grid fail to separate the upper-left from the lower-right square and thus do not create valid arrays (since these two squares should have different indices). Each path consists of 5 upward and 5 rightward steps, so there are $\binom{10}{5}=252$ paths, but two are impossible, so the answer is 250.