A Puzzle by Midnight Math: September 2013

midnightmathMidnight Math is run by Kevin O’Toole ’15.
Have you ever wanted a slice of pizza with no crust? Do you usually feed your crust your dogs?

Find a way of cutting a circular pizza into finitely many congruent pieces such that at least one piece has no crust.

More formally, find a set of simply connected regions (X1, X2…Xn) such that:

  • The intersection (X1 U X2 U… Xn) is the unit disk, D, on ℝ2.
  • For each i, j < n there is a rigid, possibly orientation-reversing transformation of the plane which converts Xi to Xj
  • For some i, λ(Xi ∩ ∂D) = 0, where λ is the Lebesgue measure*, and ∂D is the boundary of the unit disk, D.

* This Lebesgue measure will be zero if no open or closed interval along the boundary is contained in a slice. That is, a piece that touches the disk’s edge only at its corner(s) is considered to have “no crust”.

Send your solutions (with proof) to midnight.math@outlook.com. If you are correct, you will be given the highest of accolades: your name mentioned here, next issue.