Cake numbers
The question
A round melon is cut 6 times, every cut a flat plane passing right through it, and the melon is not moved between cuts.
What is the greatest number of pieces?
Count cuts, pairs and triples, do not just double
doubling guess:64
cuts / pairs / triples:61520
6+15=21
21+20=41
41+1=42
- 6 flat cuts. The counts so far run 2, 4, 8, so the guess is that it keeps doubling to 64. Let's check.
- Count it properly. Start with 1 whole melon and add one piece per cut, C(6,1) = 6; one per pair of cuts (each pair meets in a line), C(6,2) = 15; and one per triple (three meet at a point), C(6,3) = 20.
- Add them up: 6 + 15 = 21, then 21 + 20 = 41, and the whole melon makes 41 + 1 = 42. Not 64 — the doubling has broken.
How it works.
- 01
6 flat cuts. The counts so far run 2, 4, 8, so the guess is that it keeps doubling to 64. Let's check.
- 02
Count it properly. Start with 1 whole melon and add one piece per cut, C(6,1) = 6; one per pair of cuts (each pair meets in a line), C(6,2) = 15; and one per triple (three meet at a point), C(6,3) = 20.
- 03
Add them up: 6 + 15 = 21, then 21 + 20 = 41, and the whole melon makes 41 + 1 = 42. Not 64 — the doubling has broken.