document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=203845>Question 203845</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a column of hard candies, there is a single red one. Above the red candy, there is one less candy than there is below it. In the entire column, there are twice as many candies as there are below the red one. How many candies are in the column? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #153978 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=153978\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "The total number of candies can be any even number greater than 2.\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> represent the number of candies below the red one.  Then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x - 1\"> must represent the number of candies above the red one, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 2x\"> must be the total number of candies.\r<BR>\n" );
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document.write( "But the total number of candies is also given by adding the number below the red one, the red one, and the number above the red one, so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x + 1 + (x - 1) = 2x\">\r<BR>\n" );
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document.write( "Which, as should be obvious to the most casual observer, is true for all real <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">.\r<BR>\n" );
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document.write( "Since it is reasonable to presume that we are counting whole candies, we can restrict our investigation to the positive integers.  Since we cannot have a negative number of candies above the red one, the smallest value <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> can assume is 1 (making the number of candies above the red one be 0 which is 1 less than 1 and still fits the given conditions), hence the smallest total we can have is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (1) + 1 + ((1) - 1) = 2(1) = 2\">\r<BR>\n" );
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document.write( "There is no upper bound on <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">, hence the total can be as large as we like by taking a sufficiently large <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">.  But we know that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 2x\"> is divisible by 2, therefore the total must be an even integer.\r<BR>\n" );
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document.write( "That's the best I can do unless you left something out of the problem statement.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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