document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1061895>Question 1061895</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The void cube is a 3 𠖧 cube with a hole through the centre of each face. If the side length of each small cube is 1 cm, what is the surface area in square cmon the void cube. Thanks  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #676720 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=676720\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The total of surface area (in square cm) on the void cube is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2872%29 ALIGN=MIDDLE  ALT=\"highlight%2872%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . <BR>\n" );
document.write( " <BR>\n" );
document.write( "ONE APPROACH:<BR>\n" );
document.write( "Let us start with a large cube made of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" > smaller cubes of side length <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1cm ALIGN=MIDDLE  ALT=\"1cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "After making the holes,<BR>\n" );
document.write( "we end up with only the small cubes attached to the edges of the original large cube.<BR>\n" );
document.write( "The large cube originally had (and ends up still having) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2B4%2B4=12 ALIGN=MIDDLE  ALT=\"4%2B4%2B4=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > edges;<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > horizontal edges on the bottom base,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > horizontal edges on the top base,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > vertical connecting edges.<BR>\n" );
document.write( "There were (and are) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2B4=8 ALIGN=MIDDLE  ALT=\"4%2B4=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > vertices:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the bottom base,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >on the top base.<BR>\n" );
document.write( "On the void cube, each edge has<BR>\n" );
document.write( "one central small cube, attached to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > other cubes, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6-2=4 ALIGN=MIDDLE  ALT=\"6-2=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > faces exposed.<BR>\n" );
document.write( "of the remaining small cubes on the edges, each one is on one of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" > attached to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > other small cubes, leaving <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6-3=3 ALIGN=MIDDLE  ALT=\"6-3=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > of its faces exposed.<BR>\n" );
document.write( "There is a total of<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2A4%2B8%2A3=48%2B24=highlight%2872%29 ALIGN=MIDDLE  ALT=\"12%2A4%2B8%2A3=48%2B24=highlight%2872%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > small cube face surfaces.<BR>\n" );
document.write( " <BR>\n" );
document.write( "ANOTHER APPROACH:<BR>\n" );
document.write( "Before making the holes, we had a large solid cube with edge length = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3cm ALIGN=MIDDLE  ALT=\"3cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Volume=%283cm%29%5E3=9cm%5E3 ALIGN=MIDDLE  ALT=\"Volume=%283cm%29%5E3=9cm%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=surface ALIGN=MIDDLE  ALT=\"surface\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=6%283cm%29%5E2=6%289cm%5E2%29=54cm%5E2 ALIGN=MIDDLE  ALT=\"area=6%283cm%29%5E2=6%289cm%5E2%29=54cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the solid cube.<BR>\n" );
document.write( "We could start with a large solid cube made of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" > smaller <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1cm%5E3 ALIGN=MIDDLE  ALT=\"1cm%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > cubes,<BR>\n" );
document.write( "looking like a Rubik cube.<BR>\n" );
document.write( "From the large solid cube,<BR>\n" );
document.write( "we first tunnel top to bottom through the middle of the top face,<BR>\n" );
document.write( "removing 3 small cubes,<BR>\n" );
document.write( "including the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%281%29 ALIGN=MIDDLE  ALT=\"red%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > central core cube,<BR>\n" );
document.write( "which in a Rubik cube would have no colored faces.<BR>\n" );
document.write( "Then, back on our large holey cube,<BR>\n" );
document.write( "we tunnel front to back from the front face,<BR>\n" );
document.write( "this time removing only <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3-red%281%29=2 ALIGN=MIDDLE  ALT=\"3-red%281%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > more small cubes,<BR>\n" );
document.write( "because the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%281%29 ALIGN=MIDDLE  ALT=\"red%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > central core cube has been removed before.<BR>\n" );
document.write( "Finally, we tunnel left to right from the left side through the large holey cube,<BR>\n" );
document.write( "removing another <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > more small cubes.<BR>\n" );
document.write( "We have removed <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2B2%2B2=7 ALIGN=MIDDLE  ALT=\"3%2B2%2B2=7\"   DISABLED_style= \"max-width:90%; height:auto;\" > small cubes from the original large solid un-holey cube.<BR>\n" );
document.write( "If we put back together the small cubes removed,<BR>\n" );
document.write( "the same way they were attached to each other before, <BR>\n" );
document.write( "we have something familiar anyone who ever took apart a Rubik cube.<BR>\n" );
document.write( "There is a central core cube,<BR>\n" );
document.write( "completely covered on all <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > faces by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > other small cubes.<BR>\n" );
document.write( "That is a total of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%286%29 ALIGN=MIDDLE  ALT=\"red%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > small cube face surfaces removed from the original large cube.<BR>\n" );
document.write( "Only <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6-1=5 ALIGN=MIDDLE  ALT=\"6-1=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > faces of those <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > small cubes are exposed<BR>\n" );
document.write( "(the other one is attached to the central core cube).<BR>\n" );
document.write( "Of those <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > exposed faces,<BR>\n" );
document.write( "the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > outermost face was the center of a face of the larger solid cube.<BR>\n" );
document.write( "(If we started with a Rubik cube,<BR>\n" );
document.write( "those <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > outermost faces,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > face on each of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > small cubes, are colored).<BR>\n" );
document.write( "The other <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5-1=4 ALIGN=MIDDLE  ALT=\"5-1=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > faces on each of those still exposed <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > removed small cubes,<BR>\n" );
document.write( "is a newly exposed surface.<BR>\n" );
document.write( "That is a total of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2A4=green%2824%29 ALIGN=MIDDLE  ALT=\"6%2A4=green%2824%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > newly exposed small cube face surfaces.<BR>\n" );
document.write( "Meanwhile, on the original cube, while converting it into the holey void cube,<BR>\n" );
document.write( "we have removed <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%286%29 ALIGN=MIDDLE  ALT=\"red%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > small cube face surfaces,<BR>\n" );
document.write( "and we have newly exposed <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%2824%29 ALIGN=MIDDLE  ALT=\"green%2824%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > small cube face surfaces,<BR>\n" );
document.write( "for a total of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=54-red%286%29%2Bgreen%2824%29=highlight%2872%29 ALIGN=MIDDLE  ALT=\"54-red%286%29%2Bgreen%2824%29=highlight%2872%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > small cube face surfaces. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );