Math PicnicEvery sheet is a promise

The lazy caterer, step by step

Cut a pancake with straight lines and count the most pieces you can make.

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The question

What is the greatest number of pieces?

Count cuts and their crossings, do not just double

doubling guess:64
cuts / pairs:615
6+15=21
21+1=22
  1. 6 straight cuts. The counts so far run 2, 4, so the guess is that it keeps doubling to 64. Hold it.
  2. Count it properly. Start with 1 whole pie and add one piece per cut, C(6,1) = 6, and one per pair of cuts that cross, C(6,2) = 15.
  3. Add them: 6 + 15 = 21, and the whole pie makes 21 + 1 = 22. Not 64 — the doubling has broken.

How it works.

  1. 01

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

  2. 02

    Count it properly. Start with 1 whole pie and add one piece per cut, C(6,1) = 6, and one per pair of cuts that cross, C(6,2) = 15.

  3. 03

    Add them: 6 + 15 = 21, and the whole pie makes 21 + 1 = 22. Not 64 — the doubling has broken.

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