document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=206586>Question 206586</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A cube composed of 64 1 x 1 x 1 cubes is painted and then broken apart again into the smaller cubes. How many of the 1 x 1 x 1 cubes will have exactly 2 painted faces?\r<BR>\n" );
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document.write( "(A) 8 (B) 16 (C) 24 (D) 36 (E) 48 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #156128 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=156128\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you have 64 cubes.<BR>\n" );
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document.write( "cube root of 64 is 4 which means your cubic structure has a length of 4 cubes and a width of 4 cubes and a depth of 4 cubes.<BR>\n" );
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document.write( "out of these 64 cubes there will be 8 cubes that will have 0 faces painted.<BR>\n" );
document.write( "these cubes are in the interior of the cube.<BR>\n" );
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document.write( "that leaves 64 - 8 = 56 cubes that have at least one face painted.<BR>\n" );
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document.write( "if you look at each side of the 6 sides of the cube, you will see that there are 4 cubes on each side that will only have 1 face painted.  Those are the interior cubes of each side.<BR>\n" );
document.write( "since 6 * 4 = 24, you have 24 more cubes that will not have 2 sides painted because they only have 1 side painted.<BR>\n" );
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document.write( "56 - 24 = 32 cubes are left that will have at least 2 sides painted.<BR>\n" );
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document.write( "if you look at the corners of your cube you will see that each corner cube will have 3 sides painted.<BR>\n" );
document.write( "since there are 8 corners to the cube, that makes 8 cubes that will have more than 2 sides painted.<BR>\n" );
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document.write( "32 - 8 = 24 cubes are left that will have exactly 2 sides painted.<BR>\n" );
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document.write( "that makes your answer = 24.<BR>\n" );
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document.write( "another way to look at this is to look at each side of your cube.<BR>\n" );
document.write( "there are 16 faces of cubes on each side.<BR>\n" );
document.write( "out of those 16 only the on the edge of each side of the cube and not on the corners are the candidates for having 2 faces painted.<BR>\n" );
document.write( "there are 8 of these faces on each side of the cube that will have a companion face on the other side of the cube only.<BR>\n" );
document.write( "6 sides of the cube times 8 = 48 faces thaat are painted that are members of the set of 2 faces to a cube.<BR>\n" );
document.write( "48 / 2 = 24 again.<BR>\n" );
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document.write( "i would say 24 cubes is your answer.<BR>\n" );
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document.write( "a picture of your cubes can be seen on this website.<BR>\n" );
document.write( "http://theo.x10hosting.com/<BR>\n" );
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document.write( "go there and click on 206586.<BR>\n" );
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document.write( "if it's not there, give it 1/2 hour and try again.\r<BR>\n" );
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