document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208226>Question 1208226</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi<BR>\n" );
document.write( "A sheet of paper 42cm by 66cm has circles of 7cm radius cut out from it . What is the maximum number of circles that can be cut out from the paper. <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 #846515 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=846515\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "To get the densest packing possible, you'll need to arrange your circles into a hexagonal formation.<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://mathworld.wolfram.com/CirclePacking.html\">https://mathworld.wolfram.com/CirclePacking.html</a>\r<BR>\n" );
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document.write( "Here is one possible arrangement.<BR>\n" );
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document.write( "Each row has 4 circles, and there are 3 rows, giving 4*3 = 12 circles.\r<BR>\n" );
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document.write( "Can we do better than 12?<BR>\n" );
document.write( "Turns out we can.<BR>\n" );
document.write( "13 is also possible as shown here:<BR>\n" );
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document.write( "I imagine each column of circles as snowmen. The bigger snowmen are 3 circles tall. There are 3 of these snowmen, so that's 3*3 = 9 circles so far. The shorter snowmen add in 2*2 = 4 extra circles. That's 9+4 = 13 circles total. An alternative would simply be to count the circles in any fashion you want.\r<BR>\n" );
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document.write( "I don't think anything higher than 13 is possible. <BR>\n" );
document.write( "However, I am blanking on a proof at the moment. \r<BR>\n" );
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document.write( "I used <a  rel=nofollow HREF=\"https://www.geogebra.org/\">GeoGebra</a> to ensure each diagram is to scale. That helped me determine if I could pack in any extra circles or not.<BR>\n" );
document.write( "Doing this with pieces of paper is a recommended old-school method if you aren't familiar with technology. Be sure to carefully measure everything a few times over.\r<BR>\n" );
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document.write( "Side note:<BR>\n" );
document.write( "Unfortunately tutor MathLover1 is incorrect.<BR>\n" );
document.write( "18 is not the answer.<BR>\n" );
document.write( "It's not possible to fit 18 circles into this rectangle. <BR>\n" );
document.write( "She did not take into account the empty spaces between the circles. If somehow you could melt the circles (or chop them up into very small pieces), then fitting 18 \"circles\" would be possible. I put that in quotes because after the circles fall to pieces like this, they aren't whole circles anymore.<BR>\n" );
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