document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=362234>Question 362234</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many triangles can you make if their perimeter can only be 24 units?  The demensions are only in whole numbers.  The triangles can be scalene, equilateral, or isocoles. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #258144 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=258144\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let x,y,z be the measures of the sides of the triangle.  Then x+y+z = 24.  By the symmetries of the triangle inequality, <BR>\n" );
document.write( "x+y>z,<BR>\n" );
document.write( "x+z> y,<BR>\n" );
document.write( "y+z>x.<BR>\n" );
document.write( "Using the the perimeter equation above and the 3 inequalities we arrive at <BR>\n" );
document.write( "12>x, 12>y, 12>z.<BR>\n" );
document.write( "By brute-force method, the following are the possible combinations:<BR>\n" );
document.write( "(8,8,8)    (7,6,11)<BR>\n" );
document.write( "(7,8,9)    (9,6,9)<BR>\n" );
document.write( "(6,10,8)   (9,5,10)<BR>\n" );
document.write( "(5,11,8)   (9,4,11)<BR>\n" );
document.write( "(7,10,7)   (4,10,10)<BR>\n" );
document.write( "(3,10,11)  (2,11,11)<BR>\n" );
document.write( "therefore there are 12 possible traingles with perimeter 24 and integer sides. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );