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 #830093 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=830093\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: 18\r<BR>\n" );
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document.write( "Reason:\r<BR>\n" );
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document.write( "For now let's ignore the tunnels and consider a fully intact 4x4x4 Rubik's Cube.\r<BR>\n" );
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document.write( "The cube has four levels which I'll call A,B,C,D.<BR>\n" );
document.write( "Think of these as floors of a four story building.<BR>\n" );
document.write( "I'll have level A at the top floor and D at the bottom. <BR>\n" );
document.write( "The order could easily be reversed without affecting the answer. This is because of symmetry.\r<BR>\n" );
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document.write( "For each level, we have 4*4 = 16 cubes.\r<BR>\n" );
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document.write( "On level A...<BR>\n" );
document.write( "the 1st row goes from block A1 to A4<BR>\n" );
document.write( "the 2nd row goes from block A5 to A8<BR>\n" );
document.write( "the 3rd row goes from block A9 to A12<BR>\n" );
document.write( "the 4th row goes from block A13 to A16<BR>\n" );
document.write( "Similar labeling applies for levels B,C and D.\r<BR>\n" );
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document.write( "It might help to make a table like this<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >2</TD><TD >3</TD><TD >4</TD></tr><tr><TD >5</TD><TD >6</TD><TD >7</TD><TD >8</TD></tr><tr><TD >9</TD><TD >10</TD><TD >11</TD><TD >12</TD></tr><tr><TD >13</TD><TD >14</TD><TD >15</TD><TD >16</TD></tr>\n" );
document.write( "</TABLE>then stick letters A,B,C, or D before the numbers (either on paper or just mentally) so you can remember where all the blocks are positioned.\r<BR>\n" );
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document.write( "In this configuration where we haven't formed any tunnels, I hope you agree that the four corners on the top floor (A1,A4,A13,A16) and the four corners on the bottom floor (D1,D4,D13,D16) are the only blocks that have 3 faces painted.\r<BR>\n" );
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document.write( "That means we have 4+4 = 8 blocks with 3 faces painted so far.<BR>\n" );
document.write( "Check out this page<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"http://jwilson.coe.uga.edu/EMT668/EMT668.Student.Folders/SeitzBrian/EMT669/painted.cube/painted.html\">http://jwilson.coe.uga.edu/EMT668/EMT668.Student.Folders/SeitzBrian/EMT669/painted.cube/painted.html</a><BR>\n" );
document.write( "to see a diagram where the professor has painted 3 faces at each corner. The diagram is around the middle of the page.\r<BR>\n" );
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document.write( "After we dig out the tunnels, we will add to this count. \r<BR>\n" );
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document.write( "This is where things get a bit tricky (in my opinion).<BR>\n" );
document.write( "The 3D nature of the problem is a big reason why. \r<BR>\n" );
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document.write( "It might help to draw out the four levels of blocks as shown in the diagram below.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/u2oLiwF.png\"><BR>\n" );
document.write( "Levels A,B,C, and D are shown side by side. <BR>\n" );
document.write( "I've unstacked the blocks to \"flatten\" out the drawing a bit.<BR>\n" );
document.write( "Imagine this is a sort of birds-eye-view of the blocks.<BR>\n" );
document.write( "The empty spaces represent where the tunnels are carved out. \r<BR>\n" );
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document.write( "The blue faces are the original exterior faces. These are the exterior painted faces before the tunnels were carved out. The pink faces are the result of forming the tunnels. These are the tunnel floor, ceiling and wall faces.<BR>\n" );
document.write( "The white faces are not painted as they are glued to another white face, and it's hidden from exterior view (no matter how much you rotate the 3D figure).\r<BR>\n" );
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document.write( "Unfortunately I can't think of a way to have this shown in 3D, to rotate the figure around and such. So you'll have to use your imagination.\r<BR>\n" );
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document.write( "The blocks marked with an X represent blocks that have exactly 3 faces painted. Note the four corner blocks mentioned on level A, and the four on level D.<BR>\n" );
document.write( "The other 5 X's on level A are due to the tunnels. <BR>\n" );
document.write( "Block A2 for instance has a hidden pink face in the back because of the tunnel along blocks B2,B6,B10,B14\r<BR>\n" );
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document.write( "Counting the X's in the diagram we have:<BR>\n" );
document.write( "9 X's on level A <BR>\n" );
document.write( "3 X's on level B <BR>\n" );
document.write( "0 X's on level C <BR>\n" );
document.write( "6 X's on level D\r<BR>\n" );
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document.write( "In total there are 9+3+0+6 = 18 blocks with exactly 3 painted faces after the two tunnels form.\r<BR>\n" );
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document.write( "Let me know if I missed any blocks that have 3 painted faces. Also, feel free to ask further questions. \r<BR>\n" );
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document.write( "There might be some clever formula for this type of problem, but I'm blanking on it. Another tutor may know.<BR>\n" );
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