This article is from the Puzzles FAQ, by Chris Cole email@example.com and Matthew Daly firstname.lastname@example.org with numerous contributions by others.
A farmer wishes to enclose the maximum possible area with 100 meters of fence.
The pasture is bordered by a straight cliff, which may be used as part of the
fence. What is the maximum area that can be enclosed?
A circle is the plane figure with highest ratio of area to perimeter.
The cliff can be used to bisect a circle of radius 100/pi meters. By
symmetry, this will form the pen of largest area. The resulting pen
will contain 5000/pi meters squared.
Consider a ham sandwich, consisting of two pieces of bread and one of
ham. Suppose the sandwich was dropped into a machine and spindled,
torn and mutilated. Is it still possible to divide the ham sandwich
with a straight knife cut such that both the ham and each slice of
bread are divided in two parts of equal volume?
Yes. There is a theorem in topology called the Ham Sandwich Theorem,
which says: Given 3 (finite) volumes (each may be of any shape, and in
several pieces), there is a plane that cuts each volume in half. You
would learn about it typically in a first course in algebraic topology,
or maybe in a course on introductory topology (if you studied the