Question 200811: basketballs are stacked in a pyramid with a square base. each basketball rests on four basketballs below it except in base . how many basketballs would be in the bottom layer in a pyramid with a total of 91 basketballs ?
Answer by vleith(2983)   (Show Source):
You can put this solution on YOUR website! The 'trick' here is to visualize what each layer looks like. We are told each ball in a layer touches four balls below it. You can imagine that single ball sitting in the 'cavity' made by 4 balls below it.
At the top we have a single ball = 1
Below it are 4 ball, arrange as 2x2 = 4
Here is the tricky layer. It would be easy to imagine the next layer having 16 balls (that is, each ball above has 4 unique balls below it). But that isn't correct. Instead, of that, take 4 balls and put them under the left corner ball above. Then realize that in order to support the right corner ball with 4 balls, you only need 2 more balls. So 6 balls support the 2 above (so far). Now what about the other 2 balls above? In order to have them touch 4 balls each, you only need to add 3 more balls. Thus, the next layer is 3x3 and has 9 balls in it.
And now you see the pattern.
Each layer is n^2 balls in.
First layer is 1^2 =1
Next 2^2 = 4
next 3^2 = 9
And so on
Just keep adding the next layer as n^2 until you get a total of 91
answer: 5 layers
So 25 balls in the bottom layer
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