SOLUTION: A cube is painted red and then cut into 1000 identical smaller cubes. How many of these cubes are painted red on at least two faces?

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 Click here to see ALL problems on Geometry Word Problems Question 236108: A cube is painted red and then cut into 1000 identical smaller cubes. How many of these cubes are painted red on at least two faces?Answer by Edwin McCravy(8879)   (Show Source): You can put this solution on YOUR website!``` We have a 10x10x10 large cube that we are going to cut into 1000 small 1x1x1 cubes. Only the small cubes cut from the edges of the large cube will have at least two faces painted red. All the rest will have either 1 or 0 faces painted red. There are 8 small cubes which have exactly three sides painted. These are the 8 small cubes located at the 8 vertices (corners) of the big cube. So let's say we have counted these 8 with three faces painted red. There are 12 edges of the big cube. Each of these 12 edges of the big cube consists of the edges of 10 small cubes. But 2 of these are at a corner of the big cube, and we have already conted these. So we only need to count the other 8 cubes along each of the 12 edges, which have exactly 2 faces painted red. So that's an additional 12x8 or 96. So the total is 8 small cubes with exactly three faces painted red plus 96 cubes with exactly two faces painted red. That's 8+96 = 104. ------------------- Another way to count them is to start with the 1000 cubes and start subtracting. There are the inner 8x8x8 or 512 small cubes cut from the inside of the big cube which have no faces painted. On each of the 6 faces of the big cube there are 8x8 or 64 faces of small cubes which only have 1 face painted. So that's 64x6 or 384 small cubes which have 1 face painted. So subtracting 1000 small cubes - 512 small cubes which have no faces painted red ----- 488 small cubes which have at least one face painted red. - 384 small cubes which have exactly one face painted red. ----- 104 small cubes with at least two faces painted red. Edwin```