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Suppose we have to cut a pie into n pieces, using only straight cuts. What is the fewest number of cuts we can do it in?

When n is even, it is always possible to do it in n/2 cuts: with each cut being a diameter. When n is odd, a similar technique makes it possible to cut it in (n+1)/2 cuts. Can the n/2 bound ever be beaten?

For instance, it is possible to cut a pie into 7 pieces with 3 cuts, but impossible for these pieces to be equal.

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Any thoughts on whether the n/2 bound can be beaten to create equi-area peices? Note: the cuts need not go from one edge of the circle to another.

Thomas Delaney
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  • It seems the answer is clearly no - to be equal all parts need to have part of the circumference. – Moti Feb 19 '18 at 19:33
  • I think you're probably right about the answer being no. However, it is possible to have equal parts, in which some aren't part of circumference. for instance, you could divide the circle into 5 equal-area parts: a square in the middle of side pi/5, and lines going from each corner to the circumference – Thomas Delaney Feb 20 '18 at 14:32
  • I missed the comment that allow various cuts – Moti Feb 21 '18 at 02:28
  • Related: https://math.stackexchange.com/q/440825/510296 https://math.stackexchange.com/a/3453528/510296 – Rob Dec 07 '21 at 00:16

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