A rectangular slice of cake ABED is iced along two sides, the icing being the L-shaped region BCGFDE, so...

geometry/dissections/iced.cake New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

A rectangular slice of cake ABED is iced along two sides, the icing
being the L-shaped region BCGFDE, so that cake plus icing is the
rectangle ACGF:

     F--------------------QG
     |                  .  |
    D+----------------.-E  |
     |              .   |  |
     |            .     |  |
     |          Y       |  |
     |        X         |  |
     |      .           |  |
     |    .             |  |
     |  .               |  |
     AP-----------------B--C

Construct (with a straight knife edge as the only geometrical
instrument) a slicing line PQ which gives equal shares of cake and
icing.

[ Editor: problem statement reconstructed from the postings. ]

Show solution

[ Editor: the puzzle was posted by Barry Harridge
<barry@torps.apana.org.au>; David Karr added the labels X and Y to the
diagram. ]

Using the knife, score the cake along the diagonals AE and BD.  Let
the intersection of these be X.

Next, score the combination of cake+icing along the diagonals AG and
CF.  Let the intersection of these be Y.

Now place the knife across the cake so that the edge lies on points X
and Y.  This gives you your cutting line PXYQ.  Cut straight down.

(This of course assumes either that you *can* be sure of cutting
"straight down", or that the slice is so thin it doesn't matter; the
new wording of the puzzle seems to imply that one of this is the case.
But you could also repeat the construction on the opposite face,
finding points P' and Q', and score "guide lines" in the top and
bottom along lines PP' and QQ'---assuming of course that the slice of
cake doesn't fall apart while you're handling it.)

To see that the quadrilateral APQF contains half the cake and half the
icing, notice that it contains half the cake because the cut goes
through X (the centroid of rectangle ABED), and half the combination
of cake+icing because the cut goes through Y (the centroid of
rectangle ACGF).  By simple algebra, APQF contains half the icing.

Now *my* questions are:

First, given the customary thickness of icing on a wedding cake, how
close are points X and Y, and how susceptible to error is this
construction?

Second, since most wedding cakes I've ever seen were cylindrical in
shape, how did they cut a perfectly rectangular piece of one,
including the icing as shown?

-- David A. Karr <karr@cs.cornell.edu>