document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197011>Question 1197011</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A rectangular cube with sides measuring 4cm, is made with 1cm³ blocks. Two straight tunnels of 4 cubes each are taken out, as illustrated. All exposed surfaces are painted including inside the tunnels. How many small cubes have been painted on exactly 3 faces?\r<BR>\n" );
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document.write( "Please view the diagram/illustration as here on my google drive link:\r<BR>\n" );
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document.write( "https://drive.google.com/file/d/1YkGkoCOHEX0ZksF9r8UWdviXre3j3RDd/view?usp=sharing </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #830099 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52810)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=830099\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "A rectangular cube with sides measuring 4cm, is made with 1cm³ blocks. <BR>\n" );
document.write( "Two straight tunnels of 4 cubes each are taken out, as illustrated. <BR>\n" );
document.write( "All exposed surfaces are painted including inside the tunnels. <BR>\n" );
document.write( "How many small cubes have been painted on exactly 3 faces?<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;My count is 16 blocks, having three faces painted.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;My reasoning is as follows.\r<BR>\n" );
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document.write( "First, lets consider the 4x4x4 cube with no tunnels.<BR>\n" );
document.write( "I want to explain you ---how to count, analyze and classify the small cubics of this whole cube---.\r<BR>\n" );
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document.write( "There are 8  corner blocks of the 4x4x4-cube. They all have 3 faces painted.\r<BR>\n" );
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document.write( "Next, there are 12 edges of the cube, and along each of these edges, <BR>\n" );
document.write( "we have 2 = 4-2 small cubics, that have 2 (two) painted faces each.\r<BR>\n" );
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document.write( "The rest of small cubics that ARE at the surface of the large cube, have 1 (one)<BR>\n" );
document.write( "painted face.\r<BR>\n" );
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document.write( "The rest of the small cubics, that are INSIDE the large cube and are NOT <BR>\n" );
document.write( "at the surface, do not have painted faces, at all.\r<BR>\n" );
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document.write( "- - - - Now let's introduce our tunnels.<BR>\n" );
document.write( "- - - - From now I consider large cube with the tunnels.\r<BR>\n" );
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document.write( "8 small corner cubics still have 3 painted faces each.<BR>\n" );
document.write( "It gives the number 8 to my counter.\r<BR>\n" );
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document.write( "Which additional cubics have 3 faces painted ?\r<BR>\n" );
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document.write( "Look at your picture, to which you refer to.\r<BR>\n" );
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document.write( "You see large 4x4-face of the cube.<BR>\n" );
document.write( "Focus on the small face in the upper row, 3rd from the left.<BR>\n" );
document.write( "It is easy to understand that this unique cubic, to which this face does belong, has 3 painted faces:<BR>\n" );
document.write( "    &nbsp;&nbsp;&nbsp;&nbsp;one in the tunnel and 2 on the surface. Thus we identified one such cubic.\r<BR>\n" );
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document.write( "Next, focus on the small face in the 2nd row from the top, which (face) is last in this row, counting from the left.<BR>\n" );
document.write( "It is also easy to understand that this unique cubic, to which this face does belong, has 3 painted faces:<BR>\n" );
document.write( "    &nbsp;&nbsp;&nbsp;&nbsp;one in the tunnel and 2 on the surface. Thus we identified second such cubic.\r<BR>\n" );
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document.write( "Thus, we identified 2 small cubics at the ENTRANCE of the tunnel.<BR>\n" );
document.write( "It is easy to understand that TWO OTHER similar cubics are at the EXIT of the tunnel.\r<BR>\n" );
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document.write( "So, with this tunnel, we identified 4 (four) small cubics having 3 faces painted.\r<BR>\n" );
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document.write( "Absolutely in the same way and the same logic tells us, that there are other 4 small cubics <BR>\n" );
document.write( "associated with the entrance and the exit of the other tunnel.\r<BR>\n" );
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document.write( "Thus, we have &nbsp;&nbsp;8 + 4 + 4 = 16 &nbsp;&nbsp;3-face-painted small cubics.\r<BR>\n" );
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document.write( "Doing this way, we completed analyzing 3-face-painted cubics.\r<BR>\n" );
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document.write( "There is no other 3-face-painted cubes.\r<BR>\n" );
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document.write( "<U>ANSWER</U>.  There are 16 3-face-painted cubics in this problem.\r<BR>\n" );
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document.write( "Solved.\r<BR>\n" );
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document.write( "It is how my logic works.\r<BR>\n" );
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