document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209737>Question 1209737</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The dots on the opposite faces of a die have a sum of 7. How many different sum of dots on three adjacent faces are there on a die? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850047 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850047\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to figure out the different possible sums of dots on three adjacent faces of a standard die:\r<BR>\n" );
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document.write( "1. **Visualize:** Imagine a die.  Three adjacent faces share a common vertex (corner).\r<BR>\n" );
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document.write( "2. **Consider the possibilities:**  Since opposite faces sum to 7, the possible sets of three faces sharing a vertex are limited. Let's represent the faces by the number of dots on them.\r<BR>\n" );
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document.write( "3. **List the combinations:**  We can't have opposite faces together. Here are the distinct sets of three adjacent faces and their sums:\r<BR>\n" );
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document.write( "   * {1, 2, 3}: Sum = 6<BR>\n" );
document.write( "   * {1, 2, 4}: Sum = 7<BR>\n" );
document.write( "   * {1, 2, 5}: Sum = 8<BR>\n" );
document.write( "   * {1, 2, 6}: Sum = 9<BR>\n" );
document.write( "   * {1, 3, 4}: Sum = 8<BR>\n" );
document.write( "   * {1, 3, 5}: Sum = 9<BR>\n" );
document.write( "   * {1, 3, 6}: Sum = 10<BR>\n" );
document.write( "   * {1, 4, 5}: Sum = 10<BR>\n" );
document.write( "   * {1, 4, 6}: Sum = 11<BR>\n" );
document.write( "   * {1, 5, 6}: Sum = 12<BR>\n" );
document.write( "   * {2, 3, 4}: Sum = 9<BR>\n" );
document.write( "   * {2, 3, 5}: Sum = 10<BR>\n" );
document.write( "   * {2, 3, 6}: Sum = 11<BR>\n" );
document.write( "   * {2, 4, 5}: Sum = 11<BR>\n" );
document.write( "   * {2, 4, 6}: Sum = 12<BR>\n" );
document.write( "   * {2, 5, 6}: Sum = 13<BR>\n" );
document.write( "   * {3, 4, 5}: Sum = 12<BR>\n" );
document.write( "   * {3, 4, 6}: Sum = 13<BR>\n" );
document.write( "   * {3, 5, 6}: Sum = 14<BR>\n" );
document.write( "   * {4, 5, 6}: Sum = 15\r<BR>\n" );
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document.write( "4. **Identify the unique sums:** The unique sums we found are 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15.\r<BR>\n" );
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document.write( "5. **Count:** There are 10 different possible sums.\r<BR>\n" );
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document.write( "Therefore, there are 10 different sums of dots on three adjacent faces of a die.<BR>\n" );
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