It is a centuries-old tradition that when one presents a mathematical recreation or exercise, there must be a humorous or fanciful framing story about a farmer and some animals crossing a river, or a mysterious alien visitor from the fourth dimension, or a lady and a tiger. Maybe there’s just some implicit drunken old tweed-wearing fart fiddling with matchsticks and odd-sized coins in a mythic pub somewhere.

There shall be none of that here today. No Chinamen, no chess boards, no bridges, no salesmen, no pie slicing, no needles, no seating arrangements, no Islands of Liars. No. Not today.

Do this thing: Draw a circle of radius 1, centered at (0,0). Generate a large list of N random pairs of (radius, angle) uniformly from the ranges (0,2) and [0,2π), respectively. For each such pair, draw a circle of radius 1, centered at the point denoted by the pair (radius, angle).

Now look at all the little teeny crescenty shapes into which the N overlapping circles cut the original centered circle. Think of these as "polygons" with curved sides, and vertices wherever two circles intersect. Notice that in this system with curves, there can be nonempty "polygons" with two sides: two intersecting circles always create three "polygons" with two sides each.

Here is an example of what it might look like:

Do this thing: Determine the expected distribution of the number of sides of all the pieces the original circle has been broken into, for N in {1,2,4,8,16,32,44,97,111}. I don't care what happens outside the circumference of that first circle, the one shown in red in the image. I just care about the little pieces inside it. Does the distribution of the number of "polygon" sides approach a Normal distribution as N gets large?

Consider this thing: What if the circle was dissected with straight lines, arranged using the same sort of random points (radius ∈ [0,1), angle ∈ (0,2π]) to denote the points of closest approach of the line to the origin (0,0)? In other words, the point chosen at random is taken to be the position where a line segment extending (radius) distance in direction (angle) from the origin would just touch and be perpendicular to the cutting line you draw. In this case, would the distribution of polygon side counts change significantly?

Do these things: In the first case — the one with the circles — color each “polygon” that has two sides fuligin, three sides chatoyant, four sides ultraviolet, five sides megayellow, six sides octarine, seven sides opaline, and eight or more sides viridian. What proportion of the “polygons” of the same color are adjacent (share at least one edge segment)?

What proportion of all adjacent pairs of “polygons” differ by no more than one step?

What proportion of the area of the original circle will disappear as you remove the two-sided, three-sided and other “polygon” classes, in turn?

Suppose you throw away every piece of the cut circle that is not a 2-sided “polygon”: how much of the area of the circle remains as a function of N? Do it for the 3-sided “polygons”, and the others too. Which class contains the largest area of the original circle? Which the least?

Which class, when sorted out in this way by number of “polygon” sides, contains the largest contiguous chunk of the circle, counting “polygons” as contiguous only if they share edges, but not if they touch only at corners?