Regions of a circle
The question
9 people sit evenly around a round pizza and every pair of them is joined by a straight cut, arranged so that no three cuts meet at one inside point.
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. Let's check.
- Count it properly. 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. Let's check.
- 02
Count it properly. 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.