document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208021>Question 1208021</A>: <I><SPAN STYLE=\"word-break: break-all;\"> On dart board, a player can only score 2, 6 or 9 points for a throw. What is the smallest number of throws needed to get a total score of exactly 82 points? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #846138 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=846138\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "x = number of 2 pointers<BR>\n" );
document.write( "y = number of 6 pointers<BR>\n" );
document.write( "z = number of 9 pointers<BR>\n" );
document.write( "Each variable is some nonnegative whole number {0,1,2,3,...}\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2x+6y+9z = total number of points<BR>\n" );
document.write( "2x+6y+9z = 82\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x+y+z = number of throws<BR>\n" );
document.write( "The goal is to minimize x+y+z \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since z is attached to the largest coefficient, we'll make z as large as possible to find the minimum of x+y+z.<BR>\n" );
document.write( "If we made x as large as possible, then we'd be at the other end of the spectrum. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "82/9 = 9.111 approximately<BR>\n" );
document.write( "This rounds down to 9.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If z = 9, then 9z = 9*9 = 81<BR>\n" );
document.write( "That leaves 82-81 = 1 point left, but of course there's no way to score 1 point with 2 pointers and/or 6 pointers.<BR>\n" );
document.write( "So there's no way to get exactly 82 points with z = 9.<BR>\n" );
document.write( "We'll decrease z by 1.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If z = 8, then 9z = 9*8 = 72<BR>\n" );
document.write( "There would be 82-72 = 10 points left, and now it appears to be possible to achieve a score of 10 with 2 pointers and/or 6 pointers.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The subproblem is now 2x+6y = 10<BR>\n" );
document.write( "This turns into x+3y = 5 after dividing everything by 2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Use a bit of trial-and-error, or you can use the Extended Euclidean Algorithm, to find one solution is (x,y) = (2,1)<BR>\n" );
document.write( "Sure enough 2x+6y = 2*2+6*1 = 4+6 = 10\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "What are some other integer solutions on x+3y = 5?<BR>\n" );
document.write( "Solve for y to get y = (-1/3)x+5/3.<BR>\n" );
document.write( "It has slope -1/3 which means to go from one point to another we move down 1 and right 3.<BR>\n" );
document.write( "Starting at (2,1) follow the \"down 1, right 3\" movement pattern to arrive at (5,0)<BR>\n" );
document.write( "Go back to (2,1). Follow the reverse of that movement pattern to go up 1, left 3, and you'll arrive at (-1,2)<BR>\n" );
document.write( "However, recall that x cannot be negative.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The only practical nonnegative integer solutions to x+3y = 5 are (2,1) and (5,0)<BR>\n" );
document.write( "In the first case we have x+y+z = 2+1+8 = <font color=red>11</font> shots<BR>\n" );
document.write( "In the second case we have x+y+z = 5+0+8 = 13 shots<BR>\n" );
document.write( "The first case is the winner in terms of fewest number of throws.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answer: <font color=red>11</font><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );