Question:

How do you cut a rectangular cake into two equal pieces when someone has already removed a rectangular piece of any size or orientation from it? You are allowed just one straight cut.

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Solution: For most people, either they know how to conceptualize the geometry of this puzzle or they don’t. The insight is to understand how to bisect nesting rectangles (see Figure below).

This question definitely has a right answer. The key insight is knowing or realizing that any line passing through the center point of a rectangle bisects the rectangle. Before you remove the rectangular piece from the cake, there are infinitely many lines that bi-sect the cake. After you remove the rectangular piece, there is only one—the line that passes through both the center of the cake and the center of the removed rectangular piece. This line necessarily divides the removed piece in half, and hence the same amount of cake was removed from each half of the remaining portion.

The value in this question is not only seeing if candidates can compute the answer, but watching them eliminate non solutions. The fact that there is no constraint on the location of the removed rectangular piece is key. Perhaps they will ask for constraints (“Can I assume the removed piece is along an edge?”). I wouldn’t say “no.” Rather, I’d say “In what way is that helpful?” They would probably realize after a little trial-and-error that such a constraint is not helpful, and that might guide them toward the solution.

Figure  The line connects the centers of the exterior and interior rectangles.

Now for the alternate solution. Most people will think only of vertical cuts. Some bright thinkers believe they are operating “outside the box” when they propose the solution of simply slicing the cake horizontally to create two equal layers (alternate solution). But every grade school child would criticize this solution because everyone who prefers icing knows that the tops and bottoms of cakes are not and never will be equal. More critically, if the interior rectangular cut is oriented on an angle, the solution is incorrect. Nevertheless, most interviewers give candidates credit for suggesting the horizontal cut.