Regions of a circle, step by step
Mark points on a circle and join every pair with a chord.
The question
Into how many regions is the pizza cut?
Count the chords and their crossings, do not just double
doubling guess:256
chords:36
crossings:126
36+126=162
162+1=163
- 9 points on a circle, every pair joined. The counts so far run 2, 4, 8, 16, so the guess is to double again to 256. Hold that thought.
- Count it properly instead. Start with 1 whole region; each chord adds one, and there is one chord per pair of points: C(9,2) = 36.
- Each place two chords cross inside adds one more region, and a crossing comes from every set of FOUR points: C(9,4) = 126.
- Add them to the region you started with: 36 + 126 = 162, then 162 + 1 = 163. Not 256 — the doubling breaks.
How it works.
- 01
9 points on a circle, every pair joined. The counts so far run 2, 4, 8, 16, so the guess is to double again to 256. Hold that thought.
- 02
Count it properly instead. Start with 1 whole region; each chord adds one, and there is one chord per pair of points: C(9,2) = 36.
- 03
Each place two chords cross inside adds one more region, and a crossing comes from every set of FOUR points: C(9,4) = 126.
- 04
Add them to the region you started with: 36 + 126 = 162, then 162 + 1 = 163. Not 256 — the doubling breaks.