document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203324>Question 1203324</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? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #838788 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13206)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=838788\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "If 4 points on the circle determine a rectangle, then there are 2 of those points on each half of the circle.<br><BR>\n" );
document.write( "So the worst case, if you are trying to get a rectangle with the points you choose, is to first pick all 9 points on one half of the circle.  Then any 2 points on the other half of the circle will determine a rectangle.<br><BR>\n" );
document.write( "ANSWER: 9+2 = 11<br><BR>\n" );
document.write( "Note this solution is easier to see by drawing sketches to consider cases with smaller even numbers of point on the circle.<br><BR>\n" );
document.write( "6 points: choose the 3 on one side of the circle; any one 4th point still does not determine a rectangle, but then any 5th point does. 3+2 = 5<br><BR>\n" );
document.write( "8 points: choose the 4 on one side of the circle; any one 5th point still does not determine a rectangle, but then any 6th point does. 4+2 = 6<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );