$(1)$ Write a program that inputs a "topographical" map and outputs the locations that are in the shade, given a certain predetermined position of the sun.

The input is an $nn$ matrix representing $n^2$ evenly spaced points on a square piece of land. The values in the matrix are integers ranging from $0$ to $100.$ Each value represents the "altitude" of the land at the corresponding point.

The output is an $nn$ matrix of $0s$ and $1s,$ where the output value is $0$ if the land at that point is in the sunlight and $1$ if the land at the point is in the shade.

Let us assume that the sun is shining from the west $($W$).$ A location is in the shade if there is a hill to its west high enough to block the sun. To compute which locations are in the shade: First, the locations in the leftmost column are by definition in the sunlight. For each remaining location $($i$,$ j$),$ it is in the shade if there is another location $($i$,j-k)$ whose altitude is at least $4k$ greater than the altitude at $($i$,$ j$).$ This is an "exhaustive" search for different locations with different $i,j$ and $k.$