document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=96371>Question 96371</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the rule for finding the number of boxes in a pyramid display that has 100 boxes in the base?  The boxes are stacked somewhat like this:  <BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #70166 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=70166\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> For a display that is arranged like a regular pyramid, here's an approach:<BR>\n" );
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document.write( "For the first layer of 100 boxes, the boxes are arranged in a 10 by 10 square.<BR>\n" );
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document.write( "The next layer up will be indented 1 row on all 4 sides. That means that the second layer<BR>\n" );
document.write( "will be an 8 by 8 square of boxes.<BR>\n" );
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document.write( "The third layer up will also be indented 1 row from the preceding layer, making it a 6 by 6<BR>\n" );
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document.write( "The fourth layer is indented 1 row on all sides from the third layer, making it a 4 by 4 layer.<BR>\n" );
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document.write( "The fifth layer is indented 1 row on all sides from the fourth layer, making it a 2 by 2 layer.<BR>\n" );
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document.write( "And the 6th layer is a single box that sits on top of the common intersection of the 4 boxes in<BR>\n" );
document.write( "the fifth layer.<BR>\n" );
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document.write( "Adding up all the boxes in the reverse order of what we found to be the size of each layer <BR>\n" );
document.write( "results in 1 + 4 + 16 + 36 + 64 + 100 = 221 boxes<BR>\n" );
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document.write( "Notice that this is of the form: 1 + 2^2 + 4^2 + 6^2 + 8^2 + 10^2<BR>\n" );
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document.write( "This tells you that the box contents of a regular pyramid is 1 plus the sum of the squares<BR>\n" );
document.write( "of consecutive even numbers until you reach the number of layers you desire. In this problem<BR>\n" );
document.write( "the pyramid contained 6 layers. If you wanted to add a seventh layer, what size should the<BR>\n" );
document.write( "base be? The answer is that it should be a 12 by 12 base ... requiring 144 boxes. And<BR>\n" );
document.write( "an eighth layer would be 14^2 which would be 196 boxes.<BR>\n" );
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document.write( "Hope this makes sense to you.  It might help if you sketched a 10 by 10 grid and then indented<BR>\n" );
document.write( "1 row on each side for the next layer. That will show you why the next layer is 8 by 8. Then<BR>\n" );
document.write( "you can extend on that by indenting 1 row on all sides of the 8 by 8 layer. This will make <BR>\n" );
document.write( "it apparent that the next layer has to be 6 by 6. And so on. The final top box is just the<BR>\n" );
document.write( "finishing cap to the pyramid, so it is the odd loner in the arrangement.<BR>\n" );
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