It is possible to arrange 3 lines in the plane to create 7 regions. You can arrange 3 circles to create 8 regions. In each case, this is the best you can do. What if you had more lines or circles?
The Challenge:
1. What is the maximum number of regions you can create with n lines?
2. What is the maximum number of regions you can create with n circles?
3. What is the maximum number of regions you can create using a combination of n lines and circles?
The first two questions are fairly easy to answer. Some doodling gives
a sequence of numbers in each case that suggests a simple equation.
A little followup research verifies them:
Partitioning the plane with n lines: 1 + n(n+1)/2 (triangular #s + 1)
Partitioning the plane with n circles: 2 + n(n-1) (double the (n-1)th
line partition value)
My sense of how these originate is: adding a line, you can cross all
previous n lines, thus adding n more partitions. For circles, you can
cross the new circle in two places, doubling the number of new
partitions.
Adding lines to circles makes for me a nightmarish landscape I hesitate
to enter :-). But the question arises as to why, when adding
a line to a pattern of 3 circles, can it cross all but two of the eight
partitions? How do these two curvy shapes avoid being crossed?
I am reminded that topology is arguably considered the most difficult
math...
Edited on March 8, 2025, 1:43 pm
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