How Many Rectangular Solids in a Cube?

Date: 09/13/2001 at 01:38:22
From: RightBrain
Subject: 3D objects, solid rectangles inside cubes

Is there any standard way of finding out how many different possible 
rectangular solids can fit into a cube made up by n^3?

I have tried it with 2d objects and found the following to be true:
A 10-square-foot area will contain a total of (10^3)+1 rectangles. 
This takes into account the following emergence of patterns, shown in 
order of rectangular size (1x1, 1x2, 1x3, 2x2, 2x3) etc., where:

n=2	= 1 + 4 + 4
n=3	= 1 + 9 + 12 + 6
n=4	= 1 + 16 + 24 + 16 + 8
n=5	= 1 + 25 + 40 + 30 + 20 + 10
n=6	= 1 + 36 + 54 + 42 + 36 + 24 + 12 

...and so on.

It forms a nice pattern, too, so how does this work with cubes? I 
thought it might be similar, but I am running into all sorts of 
problems. As soon as I think I have a new pattern, a perfect cube 
shows up in the system (2x2x2, 3x3x3) and throws me off. Just curious.

Date: 09/13/2001 at 10:14:14
From: Doctor Rob
Subject: Re: 3D objects, solid rectangles inside cubes

Thanks for writing to Ask Dr. Math.

The standard way to do this for rectangles of size m-by-n is this.
Draw lines 1 unit apart parallel to the sides, cutting the rectangle 
up into 1-by-1 squares. There will be, including the sides of the 
rectangle, m+1 lines in one direction and n+1 in the other. Now pick 
two lines in each direction to be the opposite sides of the rectangle.  
In one direction, you can do this in (m+1)*m/2 ways, and in the other 
direction, in (n+1)*n/2 ways. Altogether, you have

   m*(m+1)*n*(n+1)/4

rectangles, including the starting rectangle.

For squares, just set m = n, and the answer is n^2*(n+1)^2/4. For
n = 10, you get 10^2*11^2/4 = 25*121 = 3025, including the given
square, not 10^3 + 1 as you stated above.

For a 3-by-3 square, the answer is 36, not the 28 you got above.
Try counting again:

   1-by-1, 9
   1-by-2, 6
   2-by-1, 6
   1-by-3, 3
   3-by-1, 3
   2-by-2, 4
   2-by-3, 2
   3-by-2, 2
   3-by-3, 1
   ----------
   Total, 36

For a rectangular solid that is m-by-n-by-p, cut it up into
1-by-1-by-1 cubes with planes parallel to the faces and 1 unit apart.
Then, including the faces themselves, there are m+1 planes in one
direction, n+1 in another, and p+1 in the last. Now select two planes 
in each direction to be the opposite faces of the rectangular solid 
contained, and count the number of ways this can be done. For a cube, 
set p = m = n.

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