Cutting Carpet

Date: 9/9/96 at 21:53:25
From: Thomas Andriano
Subject: One Cut in Two Pieces of Carpet

Two pieces of carpet are to be used to cover a floor.  The two 
rectangular pieces are 1 ft by 8 ft, and 10 ft by 10 ft.  You are 
allowed to make just one cut in one of the two pieces.  The three 
pieces of carpet must then be arranged to fit a 9 ft by 12 ft floor 
exactly.  Make a sketch that shows how this can be done.

Thank you!

Date: 9/16/96 at 15:12:21
From: Doctor Ceeks
Subject: Re: One Cut in Two Pieces of Carpet

Hi,

Because there is no way to fit the 10 by 10 piece into a 9 by 12
rectangle, it is the 10 by 10 piece which must be cut somehow.

Here's how:

I'll start by describing a walk starting at the lower left corner
of the 10 by 10 foot square piece.

up 1 foot. (no cut needed for this part)
right 2 feet.
up 1 foot.
right 2 feet.
up 1 foot.
right 2 feet.
up 1 foot.
right 2 feet.
up 1 foot.
left 6 feet.
up 1 foot
right 2 feet.
up 1 foot.
right 2 feet.
up 1 foot.
right 2 feet.
up 1 foot.
right 2 feet.

Here, you should have cut the square into two pieces, a left piece
and a right piece. (you come out 1 foot below the upper right corner.)

Now, slide the right piece over 2 feet and slide it up 1 foot.
You will see a 9 by 12 foot rectangle with a 1 by 8 hole in the exact
center!

(I arrived at this solution by hoping for a symmetric solution...so
I put the 1 by 8 strip in the exact middle of the 9 X 12 rectangle.
Since I was hoping to deal only with integral length cuts, putting
the 1 x 8 strip in as in the solution was the way to go.)

-Doctor Ceeks,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/