# 183 geometry/fence.p

## Description

This article is from the Puzzles FAQ,
by Chris Cole chris@questrel.questrel.com and Matthew Daly
mwdaly@pobox.com with numerous contributions by others.

# 183 geometry/fence.p

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?

geometry/fence.s

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.

geometry/ham.sandwich.p

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?

geometry/ham.sandwich.s

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

fundamental group).

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