Question 266109: hello, can you guys please help me? my question is a about a big pyramid made out of cubes starting with 1 cube at the top, then 8, then 21, and so on. Our question is how many sides will be painted. How many will have 0 sides painted, 1 side, 2 sides, 3 sides, 4 sides, 5 sides, and 6 sides, and all sides. I already know that there's gonna be a certain amount that some sides are not gonna be painted because some sides are going to be covered, like the bottom. We're trying to find out how many sides will be painted in a 10 story building. I also noticed that as it goes down by each floor the number of cubes are a perfect square but the number increases by two. like for the third floor there were 25 cubes, which is 5*5. then the fourth floor was 49 cubes in total. 7*7, so as you can see from 5 it increased by 2. So im trying to find a formula to find the number of shaded ones for each number of painted sides. And a formula for the whole thing. I would greatly appreciate your help, because as you can see I'm confused, and this project is hard!
-Laura! thank you!
Answer by drk(1908)   (Show Source):
You can put this solution on YOUR website! It may be easier to understand the solution, by setting up a table as follows:
levels . 0 face paint//1 face paint//2 face paint//3 face paint//4 face paint//5 face paint//6 face paint//ttl
1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. . . . . . . . . . . . . . . . .5
2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . . . . . . . . . . 4. . . . . . . . . . . . . . . . . . 1. . . . . . . . . . . . . . . . 25
3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 . . . . . . . . . 8. . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . .61
4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 . . . . . . . . . 12 . . . . . . . . . . . . . . . . . 1. . . . . . . . . . . . . . . .113
notice that we have a patterns here:
2 faces = 4*(level - 1)^2
3 faces = 4*(level - 1)
5 faces = 1
The number of total cubes is
(level)*(level + 1)*(2*level + 1)
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If we go to 10 stories, then we get
2 faces = 4*9^2 = 4 * 81 = 324
3 faces = 4*9 = 36
5 faces = 1 = 5
added, we get 365 total faces painted
there are a total of 338,350 cubes
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