document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=890619>Question 890619</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A tiled floor of a room has a dimension m*m sq-m dimensions of tile used are n*n sq-m. All tiles used are green tiles except diagonal tiles are red. After some years some green tiles are replaced by red tiles to form alternate red and green tiles pattern how many green tiles are removed? (m not equal to n) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #539122 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=539122\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If the floor is completely covered with the tiles, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%2Fn ALIGN=MIDDLE  ALT=\"m%2Fn\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a positive integer.<BR>\n" );
document.write( "That is the number of tiles along each side of the square room,<BR>\n" );
document.write( "and it is also the number of tiles in the diagonal.<BR>\n" );
document.write( "The total number of tiles is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28m%2Fn%29%5E2 ALIGN=MIDDLE  ALT=\"%28m%2Fn%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%2Fn ALIGN=MIDDLE  ALT=\"m%2Fn\"   DISABLED_style= \"max-width:90%; height:auto;\" > is even, in an alternating pattern, half of the tiles would be red.<BR>\n" );
document.write( "The final number of red tiles would be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28m%2Fn%29%5E2%2F2 ALIGN=MIDDLE  ALT=\"%28m%2Fn%29%5E2%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The number of green tiles replaced (the number of red tiles added) would be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28m%2Fn%29%5E2%2F2-m%2Fn=%28m%2Fn%29%28m%2F2n-1%29 ALIGN=MIDDLE  ALT=\"%28m%2Fn%29%5E2%2F2-m%2Fn=%28m%2Fn%29%28m%2F2n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%2Fn ALIGN=MIDDLE  ALT=\"m%2Fn\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an odd number, an alternating pattern with a red diagonal would have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28m%2Fn%29%5E2%2B1%29%2F2 ALIGN=MIDDLE  ALT=\"%28%28m%2Fn%29%5E2%2B1%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > red tiles and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28m%2Fn%29%5E2%2B1%29%2F2 ALIGN=MIDDLE  ALT=\"%28%28m%2Fn%29%5E2%2B1%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > green tiles.<BR>\n" );
document.write( "The number of green tiles to be replaced (the number of red tiles to be added) would be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28m%2Fn%29%5E2%2B1%29%2F2-m%2Fn=%281%2F2%29%28%28m%2Fn%29%5E2%2B1-2%28m%2Fn%29%29=%281%2F2%29%28%28m%2Fn%29-1%29%5E2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "UNFORTUNATELY, there is no simple formula that covers both cases (when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%2Fn ALIGN=MIDDLE  ALT=\"m%2Fn\"   DISABLED_style= \"max-width:90%; height:auto;\" > is even, and when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%2Fn ALIGN=MIDDLE  ALT=\"m%2Fn\"   DISABLED_style= \"max-width:90%; height:auto;\" > is odd).<BR>\n" );
document.write( "If you want a single formula, I would have to write something complicated, like<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2%2F2n%5E2-m%2Fn-%281%2F4%29%28-1%2B%28-1%29%5E%28m%2Fn%29%29 ALIGN=MIDDLE  ALT=\"m%5E2%2F2n%5E2-m%2Fn-%281%2F4%29%28-1%2B%28-1%29%5E%28m%2Fn%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%28m%2Fn-1%29%5E2-%281%2F4%29%281%2B%28-1%29%5E%28m%2Fn%29%29 ALIGN=MIDDLE  ALT=\"%281%2F2%29%28m%2Fn-1%29%5E2-%281%2F4%29%281%2B%28-1%29%5E%28m%2Fn%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%2Fn=9 ALIGN=MIDDLE  ALT=\"m%2Fn=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which is odd, we are replacing<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%28%28m%2Fn%29-1%29%5E2=%281%2F2%29%289-1%29%5E2=%281%2F2%29%2A8%5E2=64%2F2=32 ALIGN=MIDDLE  ALT=\"%281%2F2%29%28%28m%2Fn%29-1%29%5E2=%281%2F2%29%289-1%29%5E2=%281%2F2%29%2A8%5E2=64%2F2=32\"   DISABLED_style= \"max-width:90%; height:auto;\" > tiles, or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2%2F2n%5E2-m%2Fn-%281%2F4%29%28-1%2B%28-1%29%5E%28m%2Fn%29%29=9%5E2%2F2-9-%281%2F4%29%28-1%2B%28-1%29%5E9%29=81%2F2-9-%281%2F4%29%28-1%2B%28-1%29%29=81%2F2-9-%281%2F4%29%28-2%29=81%2F2-9%2B1%2F2=41-9=32 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > tiles, or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%28m%2Fn-1%29%5E2-%281%2F4%29%281%2B%28-1%29%5E%28m%2Fn%29%29=%281%2F2%29%289-1%29%5E2-%281%2F4%29%281%2B%28-1%29%5E9%29=%281%2F2%29%2A8%5E2-%281%2F4%29%281%2B%28-1%29%29=%281%2F2%29%2A64-%281%2F4%29%2A0=64%2F2=32 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > tiles.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+400%2C+400%2C+3.13%2C31.5%2C+3.13%2C31.5%2C%0D%0Agreen%28triangle%282pi%2Cpi%2C10pi%2Cpi%2C10pi%2C9pi%29%29%2C%0D%0Agreen%28triangle%28pi%2C2pi%2Cpi%2C10pi%2C9pi%2C10pi%29%29%2C%0D%0Agraph%28+400%2C+400%2C++pi%2C10pi%2C+pi%2C10pi%2C+sin%28x%29%2Asin%28y%29+%3E+0.1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > The new (replacement) red tiles are the ones in the green triangles.<BR>\n" );
document.write( "There are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2B5%2B3%2B1=16 ALIGN=MIDDLE  ALT=\"7%2B5%2B3%2B1=16\"   DISABLED_style= \"max-width:90%; height:auto;\" > in each triangle. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );