Eleven Sections of Fence

Date: 05/29/2003 at 19:43:23
From: Terry
Subject: Geometry

I have a grid containing 16 squares. Each square represents 1 acre. I 
need to draw 11 sections of fence along the dotted lines of the 
squares so that 4 fields are formed, each containing 4 acres of land.

I can't figure it out!

Date: 05/29/2003 at 23:08:53
From: Doctor Peterson
Subject: Re: Geometry

Hi, Terry.

Interesting question! (That means I don't already know how to solve 
it, so I get to use my brain a bit.)

Let's start with a dot at each corner, and a fence around the outside, 
which I think is assumed:
    ._._._._.
    | . . . |
    | . . . |
    | . . . |
    |_._._._|

Now there are two obvious ways to divide this into four equal areas:
    ._._._._.    ._._._._.
    |_._._._|    | . | . |
    |_._._._|    |_._|_._|
    |_._._._|    | . | . |
    |_._._._|    |_._|_._|

The one on the left, making four strips, takes 12 fences; the one on 
the right, making four squares, takes only 8. What makes the 
difference? A "fatter" shape, like a square, has less "outside" 
(edges) for the same amount of "inside" (area). What if only one or 
two of the sections were squares? Would the amount of fence required 
be between these two cases? It turns out that this idea made it easy 
for me to solve. See if you can do the same.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/