document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199136>Question 1199136</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If a polygonal n sides has (n/2)(n - 3) diagonals, how many sides will a polygon with 65 diagonals have? Is there a polygon with 80 diagonals?\r<BR>\n" );
document.write( "\n" );
document.write( "Let me see.\r<BR>\n" );
document.write( "\n" );
document.write( "I gotta set (n/2)(n - 3) equal to 65 and solve for n. The same with 80.\r<BR>\n" );
document.write( "\n" );
document.write( "Yes? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #832842 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52798)</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=832842\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "\r\n" );
document.write( "Your first idea is right.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    +-----------------------------------------------------------------+\r\n" );
document.write( "    |    But in such problems, you ALWAYS must make one more (next)   |\r\n" );
document.write( "    |    step forward and to check,  if integer solution does exist.  |\r\n" );
document.write( "    +-----------------------------------------------------------------+\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "In case of  65  diagonals it does exist: the number of sides is n = 13.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "In case of  \" 80  diagonals \"  integer solution  DOES  NOT  exist.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "You will easily detect it, when you apply the quadratic formula.\r\n" );
document.write( "</PRE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Not for every given &nbsp;\" number of diagonals \"  &nbsp;the integer solution does exist.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So, &nbsp;very often such problems have a &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28highlight%28HUGE%29%29 ALIGN=MIDDLE  ALT=\"highlight%28highlight%28HUGE%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > &nbsp;hidden underwater stone &nbsp;(as a trap),\r<BR>\n" );
document.write( "\n" );
document.write( "and the duty of a person who solves such problem is to detect it / (to recognize it),\r<BR>\n" );
document.write( "\n" );
document.write( "if such trap does present in the hidden form in the problem.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "////////////////\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "By the way, &nbsp;the formulation of the problem starts with the words \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "    If a polygonal n sides has  (n/2)(n - 3)  diagonals . . . \r\n" );
document.write( "</PRE>\r<BR>\n" );
document.write( "\n" );
document.write( "The word  &nbsp;\" if \" &nbsp;is excessive in this phrase : &nbsp;the number of diagonals \r<BR>\n" );
document.write( "\n" );
document.write( "of any convex &nbsp;n-gon &nbsp;is  &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%2A%28n-3%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28n%2A%28n-3%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  &nbsp;&nbsp;for any &nbsp;n > 3, &nbsp;and without any  &nbsp;\" if&nbsp;\".\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );