A geometry problem by Sunaina Jhamb

Let the height of the rectangular grid to have 199 rows and 991 columns. Let the left-bottom corner of the grid be the origin $O(0,0)$ , then the right-top corner is $P(991, 199)$ . It is obvious that the diagonal passes through 199 rows and 991 columns and there is one count of crossing a square when the diagonal crosses a horizontal or vertical grid line. Since both 199 and 991 are primes, the diagonal passes only two corners of the $1\times 1$ square at $O(0,0)$ and $P(991, 199)$ . Therefore, the diagonal crosses 199 horizontal and 991 vertical grid lines with the last two together at $P(991,199)$ , the number of squares crossed is $199+991-1 = \boxed{1189}$ .