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Brainteasers

Halving a cake that already has a piece missing

geometrysymmetryclassiccake-cutting

A rectangular cake sits on a table. Someone has already removed a rectangular piece from it, anywhere inside, of any size, and not necessarily aligned with the sides of the cake.

Using one straight cut from edge to edge, divide the remaining cake into two parts of equal area. You may not measure areas; you need a rule that works for every position of the missing piece.

(The cake is uniform in height, so "equal area" means equal cake.)

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What single line divides a rectangle into two equal halves, no matter what its angle?

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The theory behind this question.

Choosing Coordinates That Kill the ProblemAnts on a Pole and Collision ArgumentsBurnside's Lemma and Polya CountingSymmetry Shortcuts for Draws Without ReplacementGeometric Probability Puzzles

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