document.write( "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 ? \n" ); document.write( "

Algebra.Com's Answer #151049 by vleith(2983) $\"\"$ $\"About$

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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.\r
\n" ); document.write( "\n" ); document.write( "At the top we have a single ball = 1
\n" ); document.write( "Below it are 4 ball, arrange as 2x2 = 4
\n" ); document.write( "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.\r
\n" ); document.write( "\n" ); document.write( "And now you see the pattern.
\n" ); document.write( "Each layer is n^2 balls in.
\n" ); document.write( "First layer is 1^2 =1
\n" ); document.write( "Next 2^2 = 4
\n" ); document.write( "next 3^2 = 9
\n" ); document.write( "And so on
\n" ); document.write( "Just keep adding the next layer as n^2 until you get a total of 91
\n" ); document.write( "answer: 5 layers
\n" ); document.write( "So 25 balls in the bottom layer \n" ); document.write( "

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