Maggie graphs the six possible lines of the form $y=m x+b$ where $m$ is either 1 or -2 , and $b$ is either 0,1 or 2 . For example, one of the lines is $y=x+2$. The lines are all graphed on the same axes. There are exactly $n$ distinct points, each of which lies on two or more of these lines. What is the value of $n$ ?
If two lines are parallel, then they will not intersect unless they are the same line. This means two lines that have the same slope but different $y$-intercepts cannot intersect.

On the other hand, two lines that have different slopes always intersect at exactly one point.

The six lines that Maggie graphs can be broken into two sets of three distinct parallel lines. As discussed above, each of the three lines with slope 1 intersects each of the three lines with slope -2 exactly once.

This gives a total of $3 \times 3=9$ intersections.

We must check that none of the points have been counted more than once. For this to have happened, we would need to have at least three of the six lines intersecting at the same point. There are only two possible slopes, so among any three of the lines, at least two must have the same slope.

Therefore, if three of the lines intersect at the same point, then two distinct lines of equal slope intersect.

This would mean that two distinct parallel lines intersect, which cannot happen.

Therefore, the nine points are distinct, so $n=9$.

ANSWER: 9