From the Fun Fact files, here is a Random Fun Fact, at the Medium level: Ham Sandwich Theorem
Figure 1

Here is one of my favorite theorems
from topology, called the Ham Sandwich Theorem.
It says: given globs of ham,
bread, and cheese (in any shape), placed any way you
like, there exists one flat slice of a knife (a plane)
that will bisect each of the ham, bread, and cheese.
In other words you can share it with a friend so that
both of you get exactly the same amounts of all three globs!
Moreover, the ham, bread, and cheese don't even need to
be near each other, nor do they have to be connected!
Presentation Suggestions:
Get your class to envision
various scenarios for the location of the ham, bread, and
cheese, and see if they can figure out where
the plane should go. By the way, a common error is
to think that the plane should pass through the
centers of mass of the three objects. But this is not
necessarily the case, as the reader can check by
constructing some simple examples using lopsided volumes.
The Math Behind the Fact:
There is a version that holds in Ndimensional space,
that says any N globs of positive volume can be
simultaneously bisected by a single hyperplane.
Like the Brouwer fixed point theorem and the
BorsukUlam theorem, this has an existence
proof... it doesn't say where the plane is!
Actually, the Ham Sandwich Theorem can be proved using the
BorsukUlam theorem.
How to Cite this Page:
Su, Francis E., et al. "Ham Sandwich Theorem."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
The Link for this Fun Fact:
is directly accessible here.
