On the graph below, all lines with slopes $-1$ or $1$ and integer $y-$intercepts are drawn. The lines form many intersection points, in which some lie in the shaded area. How many of these intersections lie inside the shaded area?
My first instinct would be to list out a general form for the lines we will graph. In a form of $y=mx+b$ we would have a graph of $\displaystyle=(-1 \text{ or } 1)x+(b \in \mathbb{Z}).$ However, I'm not sure how to progress with this idea, and it would be a messy idea to bash and draw all lines satisfying these conditions. Thanks in advance for the help.
I think this will be easier if you first figure out where in the rectangle intersections can possibly take place. Which is going to be either at the corner of a square, or in the center of a square (both intersecting lines are going to have to have equal X and Y distance to an integer at their point of intersection)
To see that note that a line of gradient 1/-1 will hit an integer at 0 if and only if it hits an integer at every other integer
That gives 6+2=8 if you don't include the edges or 8+10=18 if you do