Splitting a Cake with a Missing Piece

I wanted to offer something a little light today, as I’m in the midst of figuring out the next articles in a couple of my ongoing threads and getting ready for a guest posting. So here, from Mathematics Lounge, please enjoy this nice little puzzle about how to cut, into two even pieces, a cake that’s already had a piece cut out of it. It’s got a lovely answer and it’s worth pondering it and why that answer’s true before reading the solution. And there’s another, grin-worthy, solution offered in the comments.

Mathematics Lounge

Jeremy and Jane would like to divide a rectangular cake in half, but their friend Bob (who can be a jerk sometimes) has already cut out a piece for himself. Bob’s slice is a rectangle of some arbitrary size and rotation. How can Jeremy and Jane divide the remaining cake into two equal portions, using a single cut with a sufficiently long knife?


This is an interesting problem with a fairly elegant solution. It is the type of problem that can be posed as a math puzzle/riddle, and figured out on the spot with some ingenuity.

For this problem, we define a single cut as a separation of the area made by a straight line, viewed from above. For example, a cut that crosses a gap (like below) may intersect the cake in two separate places, but still counts as one cut. (This example, of course, clearly does…

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Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there. He/him.

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