Draw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points?

We will analyze the possible configurations of two circles and determine how many points they can have in common.

Two circles may have:

No common points (0 points): If the circles are completely separate from each other, they don't intersect. This happens when the distance between their centers is greater than the sum of their radii or smaller than the difference of their radii (i.e., they are either too far apart or one circle is completely inside the other without touching).

One common point (1 point): The circles can touch each other at exactly one point. This occurs in two cases: a. Externally tangent: The circles touch at one point on the outside. This happens when the distance between their centers is exactly equal to the sum of their radii. b. Internally tangent: One circle is inside the other and they touch at exactly one point. This happens when the distance between their centers is exactly equal to the difference of their radii.

Two common points (2 points): The circles can intersect at exactly two points. This happens when the distance between their centers is greater than the difference of their radii but less than the sum of their radii.

Maximum number of common points: The maximum number of common points occurs when two circles intersect at two points. Therefore, the maximum number of common points between two circles is 2.

Summary:
No common points: 0 points.
One common point: 1 point (either externally or internally tangent).
Two common points: 2 points (if the circles intersect at two points).
Thus, the maximum number of common points between two circles is 2.