document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1042421>Question 1042421</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Eighteen points are equally spaced on a circle, from which you will choose a certain number at random. How many do you need to choose to guarantee that you will have the four corners of at least one rectangle?\r<BR>\n" );
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document.write( "Help is much appreciated, do you need to draw a diagram? Thanks. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #657402 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52905)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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/cgi-bin/show-face.mpl?user=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=657402\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "Eighteen points are equally spaced on a circle, from which you will choose a certain number at random. <BR>\n" );
document.write( "How many do you need to choose to guarantee that you will have the four corners of at least one rectangle?<BR>\n" );
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document.write( "You need to have 2 (two) pairs of diametrically opposite points. Two different pairs, I mean.\r<BR>\n" );
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document.write( "Among any of 10 of the given points, there are at least two (one pair) of diametrically opposite points <BR>\n" );
document.write( "(Dirichlet's principle // pigeonhole principle).\r<BR>\n" );
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document.write( "If you randomly choose 10 = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18%2F2+%2B+1 ALIGN=MIDDLE  ALT=\"18%2F2+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > points, you guarantee at least 1 pair of diametrically opposite points.\r<BR>\n" );
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document.write( "If you randomly choose 11 = 10 + 1 points, you guarantee one pair of diametrically opposite points; <BR>\n" );
document.write( "and after removing/withdrawing these two points, still there are at least one pair of diametrically opposite points <BR>\n" );
document.write( "among the remaining 9 = 11-2 points.\r<BR>\n" );
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document.write( "<U>Answer</U>. You need to choose 11 points randomly.\r<BR>\n" );
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