Cube Slicing

Date: 9/1/95 at 12:24:58
From: Anonymous
Subject: math

You are given a 3-inch cube that is painted red. This cube is in 27 one-inch
cubes. What is the number of cubes with red paint on them for 3, 4, 
5, 6,...n inches? 

For 0 faces respectively we get 8, 27, 64, 150. These are not proportional
for each 1-inch increase in the 3-inch cube. How do you figure the number 
of cubes with red paint on the cube of n-inches for 0 faces?  

Please answer right away if you can picture the question.
Thanks.

Date: 9/1/95 at 17:1:11
From: Doctor Ken
Subject: Re: math

Hello!

The easiest way to do this problem is to ask "how many of the smaller
cubes will have paint on none of their faces?"  Since the unpainted 
cubes will form a medium-sized cube themselves inside the big cube, 
there will be a nice formula for the number of unpainted small cubes. 
Then you subtract that from the total number of cubes, and you'll have
your answer.

For starters, the number of unpainted cubes inside a 3-inch cube is 1.

-Doctor Ken,  The Geometry Forum

Date: Sat, 2 Sep 1995 10:27:05 -0600 (MDT)
From: Anonymous
Subject: Re: math

Hello again. Recounting,  I get for a 3, 4, 5, 6,  inch cube for 0 faces: 
1, 8, 27, 64 respectively. 

I still don't understand how you get the formula for n cube inches. 

So for 3,4,5.6,....,n 
0 faces for n is ?

Thanks.

Date: 9/4/95 at 11:5:45
From: Doctor Ken
Subject: Re: math

Hello!

Well, the thing to keep in mind here is that the unpainted faces form a 
CUBE.  So the number of cubes (i.e. the number of cubic inches) that are
unpainted is going to be the length of one of the sides cubed.  So now
you just need to find out what the length of one of the sides is: you've
gone in one unit from each side of the large cube, so the side length of 
the cube we're interested in is (n-2).  Now do you know how to find the
volume of the cube with side length n-2?

-Doctor Ken,  The Geometry Forum

Date: Tue, 5 Sep 1995 10:53:29 -0600 (MDT)
From: Anonymous
Subject: Re: math

Hi, and thanks!

Volume for a cube with side n-2 would be (n-2)^3.
 
What would the number of cubes with red paint on 1 face, 2 
faces, 3 faces, for 3,4,5,6,...n cubic inches? For 1 face respectively 
being 6, 24, 54, 96; 2 faces, 12, 24, 36, 48, and for 3 faces 8 for each. 

What is the number of cubes with red paint on cube of n inches for 1, 2, 
and 3 faces? I know 0 faces is n-2.   

Thanks.

Date: 9/6/95 at 11:16:51
From: Doctor Ken
Subject: Re: math

Looking at the big painted cube, where do the faces that are painted 
on 3 sides show up?  They only appear at the corners, right?  So how
many corners are there on a cube of n cubic inches?  

Similarly, find out where the small cubes with only 2 painted sides will
be; on the edges.  Then find out how many edges the big cube has, and
how many small cubes per edge.  

Do a similar thing for 1 painted face.

- Doctor Ken,  The Geometry Forum