document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1031009>Question 1031009</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How to find the number of cubes in an arrangement given the top, front and side views of an arrangement? Are there any strategies or formulas for finding it? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #645894 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=645894\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Assuming that each cube that is not at the bottom is directly on top of another cube,<BR>\n" );
document.write( "the top view will let us know how the cubes on the bottom layer are arranged.<BR>\n" );
document.write( "Each layer above that can be guessed from the front and side views.<BR>\n" );
document.write( "However, with these top, front and side views,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28800%2C300%2C-1%2C15%2C-0.5%2C5.5%2C%0D%0Arectangle%280%2C0%2C1%2C4%29%2Crectangle%283%2C0%2C4%2C4%29%2C%0D%0Arectangle%280%2C0%2C4%2C1%29%2Crectangle%280%2C3%2C4%2C4%29%2C%0D%0Arectangle%280%2C1%2C4%2C2%29%2Crectangle%281%2C0%2C2%2C4%29%2C%0D%0Alocate%281%2C5%2CTOP%29%2Clocate%282%2C5%2CVIEW%29%2C%0D%0Arectangle%285%2C0%2C9%2C1%29%2Crectangle%286%2C0%2C8%2C2%29%2Crectangle%287%2C0%2C9%2C2%29%2C%0D%0Alocate%286%2C3%2CFRONT%29%2Clocate%287%2C3%2CVIEW%29%2C%0D%0Arectangle%2810%2C0%2C14%2C1%29%2Crectangle%2810%2C0%2C12%2C2%29%2Crectangle%2811%2C0%2C13%2C2%29%2C%0D%0Alocate%2810.5%2C3%2CRIGHT%29%2Clocate%2811.5%2C3%2CSIDE%29%2Clocate%2812.5%2C3%2CVIEW%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "how do we know which of the drawing below corresponds to the top view of the top layer?<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28800%2C250%2C-1%2C15%2C-1%2C4%2C%0D%0Arectangle%281%2C0%2C4%2C3%29%2Crectangle%282%2C0%2C3%2C3%29%2Crectangle%281%2C1%2C4%2C2%29%2C%0D%0Arectangle%286%2C0%2C9%2C1%29%2Crectangle%286%2C2%2C9%2C1%29%2C%0D%0Arectangle%287%2C0%2C8%2C3%29%2Crectangle%289%2C0%2C8%2C3%29%2C%0D%0Arectangle%2811%2C0%2C12%2C1%29%2Crectangle%2812%2C0%2C14%2C1%29%2C%0D%0Arectangle%2813%2C0%2C14%2C2%29%2Crectangle%2813%2C2%2C14%2C3%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "FURTHER EXPLANATION ADDED AT A LATER DATE:<BR>\n" );
document.write( "We can count 16 cubes on a bottom layer looking like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1%2C5%2C-0.5%2C5.5%2C%0D%0Arectangle%280%2C0%2C1%2C4%29%2Crectangle%283%2C0%2C4%2C4%29%2C%0D%0Arectangle%280%2C0%2C4%2C1%29%2Crectangle%280%2C3%2C4%2C4%29%2C%0D%0Arectangle%280%2C1%2C4%2C2%29%2Crectangle%281%2C0%2C2%2C4%29%2C%0D%0Alocate%281%2C5%2CTOP%29%2Clocate%282%2C5%2CVIEW%29%2C%0D%0Alocate%280.45%2C0.65%2C1%29%2Clocate%281.45%2C0.65%2C2%29%2Clocate%282.45%2C0.65%2C3%29%2Clocate%283.45%2C0.65%2C4%29%2C%0D%0Alocate%280.45%2C1.65%2C5%29%2Clocate%281.45%2C1.65%2C6%29%2Clocate%282.45%2C1.65%2C7%29%2Clocate%283.45%2C1.65%2C8%29%2C%0D%0Alocate%280.45%2C2.65%2C9%29%2Clocate%281.4%2C2.65%2C10%29%2Clocate%282.4%2C2.65%2C11%29%2Clocate%283.4%2C2.65%2C12%29%2C%0D%0Alocate%280.4%2C3.65%2C13%29%2Clocate%281.4%2C3.65%2C14%29%2Clocate%282.4%2C3.65%2C15%29%2Clocate%283.4%2C3.65%2C16%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > On the top view we see some (7 to 13) of those first layer cubes, AND<BR>\n" );
document.write( "the cubes that are on top of the rest of the first layer.<BR>\n" );
document.write( " <BR>\n" );
document.write( "The front and right side pictures tell us there are some cubes on a second layer.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28575%2C200%2C4%2C15%2C-0.5%2C3.5%2C%0D%0Arectangle%285%2C0%2C9%2C1%29%2Crectangle%286%2C0%2C8%2C2%29%2Crectangle%287%2C0%2C9%2C2%29%2C%0D%0Alocate%286%2C3%2CFRONT%29%2Clocate%287%2C3%2CVIEW%29%2C%0D%0Alocate%285.45%2C0.65%2C1%29%2Clocate%286.45%2C0.65%2C2%29%2Clocate%287.45%2C0.65%2C3%29%2Clocate%288.45%2C0.65%2C4%29%2C%0D%0Alocate%286.45%2C1.65%2CA%29%2Clocate%287.45%2C1.65%2CB%29%2Clocate%288.45%2C1.65%2CC%29%2C%0D%0Arectangle%2810%2C0%2C14%2C1%29%2Crectangle%2810%2C0%2C12%2C2%29%2Crectangle%2811%2C0%2C13%2C2%29%2C%0D%0Alocate%2810.5%2C3%2CRIGHT%29%2Clocate%2811.5%2C3%2CSIDE%29%2Clocate%2812.5%2C3%2CVIEW%29%2C%0D%0Alocate%2810.45%2C0.65%2C4%29%2Clocate%2811.45%2C0.65%2C8%29%2Clocate%2812.4%2C0.65%2C12%29%2Clocate%2813.4%2C0.65%2C16%29%2C%0D%0Alocate%2810.45%2C1.65%2CD%29%2Clocate%2811.45%2C1.65%2CE%29%2Clocate%2812.45%2C1.65%2CF%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .The front and side views show us 3 to 5 second layer cubes,<BR>\n" );
document.write( "each labeled with one or two letters as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C ALIGN=MIDDLE  ALT=\"C\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D ALIGN=MIDDLE  ALT=\"D\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E ALIGN=MIDDLE  ALT=\"E\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F ALIGN=MIDDLE  ALT=\"F\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The problem is we do not know where those labeled cubes are within the second layer.<BR>\n" );
document.write( "We do not know what cubes are hiding \"behind\" them, either.<BR>\n" );
document.write( "All we know is that the second layer has at least <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > cubes,<BR>\n" );
document.write( "and at most <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" > cubes.<BR>\n" );
document.write( "Cube <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > could be on top of cube <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "or it could be on top of cubes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Without any hint of perspective, we cannot tell.<BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > is on top of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "are there other unseen cubes on top of cubes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and/or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" > ?<BR>\n" );
document.write( "We don't know.<BR>\n" );
document.write( "Something like that happens with cubes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C ALIGN=MIDDLE  ALT=\"C\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Regarding the cubes on the right side view,<BR>\n" );
document.write( "Cube <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D ALIGN=MIDDLE  ALT=\"D\"   DISABLED_style= \"max-width:90%; height:auto;\" > could be on top of cube <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "or it could be on top of cubes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Something like that happens with cubes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E ALIGN=MIDDLE  ALT=\"E\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F ALIGN=MIDDLE  ALT=\"F\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The second layer could have 9 cubes:<BR>\n" );
document.write( "cube <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > on top of cube <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "cube <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" > on top of cube <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "a cube on top of cube <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > (labeled as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C ALIGN=MIDDLE  ALT=\"C\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the front view and as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D ALIGN=MIDDLE  ALT=\"D\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the right side view),<BR>\n" );
document.write( "cube <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E ALIGN=MIDDLE  ALT=\"E\"   DISABLED_style= \"max-width:90%; height:auto;\" > on top of cube <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "cube <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F ALIGN=MIDDLE  ALT=\"F\"   DISABLED_style= \"max-width:90%; height:auto;\" > on top of cube <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and another four cubes on top of cubes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7 ALIGN=MIDDLE  ALT=\"7\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11 ALIGN=MIDDLE  ALT=\"11\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "that are not visible in the front or side views.<BR>\n" );
document.write( "The top layer could have as few as 3 cubes,<BR>\n" );
document.write( "strategically arranged so as to appear as cubes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > through <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E ALIGN=MIDDLE  ALT=\"E\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the front and side views.<BR>\n" );
document.write( "One such arrangement of 3 second layer cubes could have<BR>\n" );
document.write( "a cube on top of cube <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (labeled as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D ALIGN=MIDDLE  ALT=\"D\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the front and side views),<BR>\n" );
document.write( "a cube on top of cube <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7 ALIGN=MIDDLE  ALT=\"7\"   DISABLED_style= \"max-width:90%; height:auto;\" > (labeled as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E ALIGN=MIDDLE  ALT=\"E\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the front and side views), and<BR>\n" );
document.write( "a cube on top of cube <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" > (labeled as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C ALIGN=MIDDLE  ALT=\"C\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F ALIGN=MIDDLE  ALT=\"F\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the front and side views).<BR>\n" );
document.write( "Five other possibilities for a 3-cube second layer exist,<BR>\n" );
document.write( "wit cubes on top of cubes<BR>\n" );
document.write( "2, 8, and 1, or<BR>\n" );
document.write( "3, 6, and 12, or<BR>\n" );
document.write( "3, 8, and 10, or<BR>\n" );
document.write( "4, 7, and 10, or<BR>\n" );
document.write( "4, 6, and 11. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );