How do you cut a rectangular cake in half that already has a rectangular piece removed?

geometry/dissections/slice.cake Added material

Problem

How do you cut a rectangular cake in half that already has a rectangular
piece removed?

Show solution

Cut perpendicular to the plane of the top of the cake along the line that
connects the center of the cake with the center of the removed piece.
Any line through the center of a rectangle cuts the recangle in half,
so the cut through the removed rectangle will cut it in half too.

Additional material

Tanuj Narain wrote:
>Given a rectangular cake with a rectangular piece removed (any size or
>orientation), how would you cut the remainder of the cake into two
>equal halves with one straight cut of a knife?

Neil Patterson wrote:
>Horizontally

No, a horizontal cut does not work.

The cake can have a rectangular piece removed of ANY size/orientation.

Thus, the cake might have a little rectangular piece taken from a top
corner, for instance.  If you cut the remaining cake halfway up
horizontally, the top piece will have less cake than the bottom piece.

Here is a correct answer:

Put a thin flat piece of paper on top of the cake and trace outlines of
the cake and the hole.  Note that the cake will have a rectangular
outline, but the hole might have a hexagonal outline (because the hole
can be at any orientation).

Draw diagonals of the cake outline to determine its center.

Draw diagonals of the (possibly hexagonal) hole outline to determine
its center.

Now cut vertically along the line passing through both of those
centers.

This plane will cut the original cake into two equal volumes, and it
will also cut the hole into two equal volumes.  Therefore, it will cut
the cake minus the hole into two equal volumes.

>>To make it tougher, consider a cake with two rectangular pieces
>>removed (any size or orientation, and both pieces can be of different
>>orientation and size).
>>(Submitted by Ravishankar G Shiroor, shiroor@wipsys.soft.net)

This is significantly tougher.  The basic strategy is to determine the
centers of the cake and both holes.  Then you cut along the plane
determined by these 3 points (if those 3 points are colinear, you have
a choice).

The thing which makes this problem much tougher is that it's hard to
mark these centers 3 dimensionally.

-- Isaac Kuo <kuo@bit.csc.lsu.edu>