You may have heard of the Borromean rings (a famous pattern of 3 equal circles) or the Olympic rings (a famous pattern of 5 equal circles). The unchanging rings (which aren't famous because we just made up the name) are a pattern that can be made from different numbers of circles, with some special points on them marked.
A system of unchanging rings must satisfy three unchanging rules:

I. If two circles meet at a marked point, they share exactly two marked points.
II. If two marked points lie on a common circle, they must share exactly two common circles.
III. We can get from any circle to any other by some number of hops along marked points.
One possible system of unchanging rings is shown below:

a. Draw a system of unchanging rings with more than 4 circles, all equal in size.
Maybe you thought this problem was about geometry? Just kidding! You see, there is a special kind of circle called a math circle: it's not a geometric object, but a group of people doing math together. Let's say that a "marked point" in such a case is a person participating in one or more of the math circles. We can form a system of unchanging rings out of math circles as well, by satisfying the unchanging rules.
b. Prove that every system of unchanging rings (whether made up of ordinary circles or math circles) has an unchanging constant n, such that every circle contains exactly n marked points, and every marked point is contained in exactly n circles. (In the example drawn above, n=3.)
c. In a system of unchanging rings with unchanging constant n, what is the maximum number of circles?
d. For some n, find a nonempty system of unchanging rings with unchanging constant n such that it has fewer circles than the maximum you found in part c.