Cute Numbers

Date: 06/05/2001 at 05:05:16
From: Emily
Subject: Cute numbers

If a square can be cut into n squares of at the most two different 
sizes, then n is called a cute number. For example, 4 and 10 are cute 
numbers.
 _______
|   |   |
|___|___|
|   |   |
|___|___|

 _______
|_|_|   |
|_|_|___|
|_|_|   |
|_|_|___|

a. Show that 6 is a cute number.
   I have already succeeded in doing so by having 1 square of 2 units 
   squared and 5 squares of 1 unit squared.

b. Show that 2001 is a cute number.

c. Show that every integer greater than 5 is a cute number.

Date: 06/05/2001 at 12:44:17
From: Doctor Peterson
Subject: Re: Cute numbers

Hi, Emily.

This is a "cute" problem to play with on your own, so I don't want to 
spoil it by giving any direct help.

The next step after solving part a is to try extending the solution 
you discovered there to larger numbers. See what kind of numbers you 
can show to be "cute" by using the same way of dividing a square.

Then you will have to keep playing, trying other numbers and other 
patterns, and looking for a way to make the numbers you couldn't make 
yet. (It's very helpful to have the last question, which lets you know 
that all larger numbers CAN be done, so this is worth the effort.)

Once you have found a pattern that works for 2001, you will have the 
third part; and by then I think you will also know how to handle any 
number.

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

Date: 06/05/2001 at 12:52:15
From: Doctor Rob
Subject: Re: Cute numbers

Thanks for writing to Ask Dr. Math, Emily!

Every even number from 4 up is a cute number, because of the
following kind of dissection:

                2*n
     o---o---o---o---o---o---o
     |       2*n-2       |   |2
     o                   o---o
     |                   |   |2
     o                   o---o
     |2*n-2              |   |2
 2*n o                   o---o
     |                   |   |2
     o                   o---o
     |                   |   |2
     o---o---o---o---o---o---o
     |   |   |   |   |   |   |2
     o---o---o---o---o---o---o
       2   2   2   2   2   2

This cuts a square up into 2*n smaller squares, for any whole
number n >= 2.  (Why are there 2*n smaller squares in the above
picture?)

Now if 2*n is a cute number, so is 2*n + 3. You get this by taking the 
above dissection of a square into 2*n smaller squares, and cutting the 
large square into quarters. (Why does this work?)

This is enough to show that all whole numbers greater than 5 are
cute. (How can you show this?)

I leave the rest to you.

By the way, another way to show that 2001 is a cute number is to take 
a square of side 45, and cut it up into 1 square of side 5 and 2000 
squares of side 1. There are LOTS of ways to do a dissection for large 
numbers.

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