Problem:

How would you cut a rectangular cake into two equal pieces when a rectangular piece has already been cut out of it? The cut piece can be of any size and orientation. You are only allowed to make one straight cut.

Solution:

  • Simple question right? There are two possible solutions to this problem. People often overlook the easier solution to this problem. Let’s start with the easiest solution. If you make one straight horizontal cut along the height of the cake, the resulting slices are of equal sizes. But this solution may not work so well on a cake with icing. So let’s rethink.
  • In general, when a straight cut is made at any angle through the center of a  rectangle, the resulting pieces are always of equal area. So let’s consider our situation. What if we make a straight cut such that it passes through the center of both the rectangles? Since the cut halves both the rectangles, the resulting two pieces are guaranteed to have equal area. Each piece has an area equal to half the original cake minus half the area of the missing rectangular piece. This results in two pieces of equal size, assuming the cake’s height is same at all points

 

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