  Consider a line segment placed on a two-dimensional grid of rectangular
tiles. This paper addresses the relationship between the length of the segment
and the number of tiles it visits (i.e. has intersection with). The square grid
is also considered explicitly, as some of the specific problems studied are
more tractable in that particular case. The segment position and orientation
can be modelled as either deterministic or random. In the deterministic
setting, the maximum possible number of visited tiles is characterized for a
given length, and conversely, the infimum segment length needed to visit a
desired number of tiles is analyzed. In the random setting, the average number
of visited tiles and the probability of visiting the maximum number of tiles on
a square grid are studied as a function of segment length. These questions are
related to Buffon's needle problem and its extension by Laplace.
