Date: 06/21/2001 at 14:35:06 From: Ste Subject: Painted cube We've been given the painted cube topic in preperation for gcse's next year. I've done loads of diagrams and tables, but just can't see or find a formula for the problem (it must be in algebra). Would you please send me a formula with a brief explenation? Thanks, Dr. Math. STE
Date: 06/21/2001 at 15:42:17 From: Doctor Greenie Subject: Re: Painted cube Hello, Ste - You will learn a lot more about mathematics if you figure out the formulas on your own than if we give them to you. I will try to help you find a way to develop the formulas; if you are able to develop the formulas with the help I give you, then you will be able to provide explanations of the formulas without my help. We have a large cube made up of smaller cubes, with each dimension of the large cube being equivalent to n of the smaller cubes. We paint the large cube, and then we want to know how many of the small cubes have paint on 0, 1, 2, 3, 4, 5, or 6 faces. To do this, you want to visualize the large cube and determine where the small cubes are that have 0 faces painted, where the small cubes are that have 1 face painted, and so on. Let's try to build equations (functions of n) that describe the numbers of small cubes with different numbers of faces painted. In the discussion that follows, I will use f0(n) to denote the function of n defining the number of small cubes with 0 faces painted, f1(n) to denote the function of n defining the number of small cubes with 1 face painted, ..., and f6(n) be the function of n defining the number of small cubes with 6 faces painted. If the "large" cube is a single small cube, then there is one small cube, and it is painted on all 6 faces. That is a special case, since in all the larger cubes there are no small cubes with 6 faces painted. In fact, in all the larger cubes there are no small cubes with more than three painted faces. So we have (for n > 1) f6(n) = 0 f5(n) = 0 f4(n) = 0 Now let's visualize the picture to determine f3(n), f2(n), f1(n), and f0(n). f3(n)... The small cubes that have three faces painted are on the corners of the cube. How many corners are there on an n by n by n cube? The answer will give you f3(n). f2(n)... The small cubes that have two faces painted are on the edges of the cube - except for the small corner cubes, which have three faces painted. How many edges does a cube have? And on an n by n by n cube, how many small cubes are there on each of those edges, not counting the small corner cubes? So how many small edge cubes (not counting corner cubes) does that make on all the edges together? The answer is f2(n). f1(n)... The small cubes that have one face painted are on the faces of the cube - except for the small edge cubes, which are painted on two faces. How many faces does a cube have? And on an n by n by n cube, how many small cubes are there on each of those faces, not counting the edge cubes? So how many small "face" cubes (not counting edge cubes) does that make on all the faces together? The answer is f1(n). f0(n)... The small cubes that have no faces painted are on the interior of the large cube; these small cubes with no painted faces together form a smaller cube inside the large cube. If the large cube is n by n by n, what are the dimensions of the cube inside the large cube formed by all of the small cubes with no painted faces? So how many small cubes are there in that "interior" cube? The answer is f0(n). Now, all together, f0(n), f1(n), ..., and f6(n) must account for all of the small cubes in the large cube. So you can perform a confidence check of your answers for f0(n), f1(n), f2(n), and f3(n) by seeing if together they add to the total number of small cubes in the large cube, which is n^3. If they do, then you probably did the reasoning correctly and have found the correct formulas for f0(n), f1(n), f2(n), and f3(n). If they don't, go back and check your work - at least one of the four formulas you found for f0(n), f1(n), f2(n), and f3(n) is incorrect. See what you can do with this problem on your own now. Write back if you would like more help with this. - Doctor Greenie, The Math Forum http://mathforum.org/dr.math/
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