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/
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