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Cake numbers, step by step

Cut a cake with straight flat cuts, moving nothing between cuts, and count the most pieces you can make.

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The question

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
  1. 6 flat cuts. The counts so far run 2, 4, 8, so the guess is that it keeps doubling to 64. Hold it.
  2. 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.
  3. 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.

  1. 01

    6 flat cuts. The counts so far run 2, 4, 8, so the guess is that it keeps doubling to 64. Hold it.

  2. 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.

  3. 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.

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