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# Cutting yet another cake

## Question

You are to cut rectangular cake into two pieces of equal size, but a rectangular piece of the cake has already been removed. Your cut must be a single straight cut. (Because the cake is frosted, a horizontal cut won't work. One piece won't have any frosting.) How would you do it? Assume the cake has constant thickness, so the problem is the same as cutting a rectangular piece of paper with a rectangular portion already removed.

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18 hours ago, bonanova said:

When you say equal size, do you mean identical shapes, congruent shapes, or equivalent area?

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Good question. Not similar. Not congruent. Just equal area (viewed from top) any shape.

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Spoiler

A cut parallel to one edge of the cake is always possible...

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Spoiler

Make a cut that is along the line created by the center of the cake (if it were whole) and the center of the rectangular piece missing from the cake.

Since and straight line cut through the center of a rectangle is equal areas when you make a cut that goes through the center of both, you are taking equal area of both the cake and the missing piece so the remaining has to be equal.

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Spoiler

How many slices are needed to remove the rectangle piece? If just one..Cut diagonally.

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On 6/28/2016 at 8:04 PM, Logophobic said:
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You are correct. But where shall the cut be positioned?

12 hours ago, TimeSpaceLightForce said:
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How many slices are needed to remove the rectangle piece? If just one..Cut diagonally.

Probably four. One for each side

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Spoiler

Probably only the parallel cut by Logophobic solves the 3-slice rectangular piece removal.

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On 6/28/2016 at 11:46 AM, tojo928 said:
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Make a cut that is along the line created by the center of the cake (if it were whole) and the center of the rectangular piece missing from the cake.

Since and straight line cut through the center of a rectangle is equal areas when you make a cut that goes through the center of both, you are taking equal area of both the cake and the missing piece so the remaining has to be equal.

Wouldn't this depend on the orientation of the rectangle and the circle.  I can think of some cases when the centers meet on a line where the line intersects the rectangle at an awkward angle and not cut the circle in half. Or am I missing something?

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Wouldn't this depend on the orientation of the rectangle and the circle.  I can think of some cases when the centers meet on a line where the line intersects the rectangle at an awkward angle and not cut the circle in half. Or am I missing something?

Spoiler

No circle is involved here. But any shape that is bisected by a line though its center can be used, both for the entire cake and the missing piece. That includes all shapes that have 180o rotational symmetry: circles, squares, rectangles, parallelograms, any regular polygon with an even number of sides ...

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