Cube SlicingDate: 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 |
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