Carpet Problem

Date: 10/15/2001 at 08:12:08
From: Nathan Pelletier
Subject: Carpet Problem 10x10 1x8 = 9x12

You have to carpet a 9x12 room, but when you go the store they only 
have a 10x10 carpet and a 1x8 piece of carpet. If you add up the 
number of square units, then 10x10 and 1x8 = 108 and 9x12 = 108, so 
you know that it can fit. However, you can only make one cut in the 
carpet (meaning either entering one side of the carpet and exiting any 
side, or entering one side of the carpet and cutting over yourself).   
Where would this cut be to make the carpet fit in a 9x12 room.?

Date: 10/15/2001 at 14:02:22
From: Doctor Rob
Subject: Re: Carpet Problem 10x10 1x8 = 9x12

Thanks for writing to Ask Dr. Math, Nathan!

Try this:

   o---+---+---+---+---+---+---+---+---+---o
   |                                       |
   o---+---o                               +
   |       |                               |
   +       o---+---o                       +
   |               |                       |
   +               o---+---o               +
   |                       |               |
   +                       o---+---o       +
   |                               |       |
   +       o---+---+---+---+---+---o       +
   |       |                               |
   +       o---+---o                       +
   |               |                       |
   +               o---+---o               +
   |                       |               |
   +                       o---+---o       +
   |                               |       |
   +                               o---+---o
   |                                       |
   o---+---+---+---+---+---+---+---+---+---o

Slide the upper right piece two feet right and then one foot down,
and see what happens.

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

Date: 10/15/2001 at 14:05:31
From: Doctor Greenie
Subject: Re: Carpet Problem 10x10 1x8 = 9x12

Hi, Nathan -

I have seen this problem posed many times, and the instructions are 
virtually never clear. For example, in your presentation of the 
problem, I don't know what "cutting over yourself" means.

Here are a couple of things I think you need in order to have a chance 
of solving this problem:

(1) You can only cut one of the pieces of carpet, giving you a total 
of three pieces. Putting the two original pieces on top of each other 
to make the cut is not allowed, nor is folding the piece you cut so 
that the single cut gives you more than three pieces.

(2) You can't solve the problem by making a single straight cut. It 
has to be some sort of fancy zigzag cut. Note that for the resulting 
pieces to exactly cover a rectangular floor, the cut will have to 
consist of segments all at right angles to each other.

Those are a couple of hints that might help you find the solution; but 
I suspect that even with those hints only a tiny fraction of students 
would be able to do so. (I never figured out the problem for myself - 
I had to be shown how it is done...)

If you don't feel like playing with the problem longer on your own, 
here is a link to a page in the Dr. Math archives where the solution 
to your problem is described:

   Cutting Carpet
   http://mathforum.org/dr.math/problems/andriano.9.9.96.html   

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