document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165989>Question 1165989</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Starting with a positive integer, apply the following operations any number<BR>\n" );
document.write( "of times and in any order to produce a list of numbers:<BR>\n" );
document.write( "1. double the current term, or<BR>\n" );
document.write( "2. delete the last digit of the current term.<BR>\n" );
document.write( "An example of such a list is<BR>\n" );
document.write( "231, 23, 46, 92, 9, 18.<BR>\n" );
document.write( "(a) Find a list which starts with 51 and ends in 129.<BR>\n" );
document.write( "(b) Show that every starting number can produce a list ending in 1.<BR>\n" );
document.write( "A cycle is a list which eventually returns to the starting number, such as<BR>\n" );
document.write( "24, 48, 96, 9, 18, 36, 3, 6, 12, 24.<BR>\n" );
document.write( "(c) Show that every number from 1 to 41 occurs in a cycle with at most<BR>\n" );
document.write( "13 distinct terms. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852891 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(53332)</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=852891\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "Starting with a positive integer, apply the following operations any number<BR>\n" );
document.write( "of times and in any order to produce a list of numbers:<BR>\n" );
document.write( "1. double the current term, or<BR>\n" );
document.write( "2. delete the last digit of the current term.<BR>\n" );
document.write( "An example of such a list is<BR>\n" );
document.write( "231, 23, 46, 92, 9, 18.<BR>\n" );
document.write( "(a) Find a list which starts with 51 and ends in 129.<BR>\n" );
document.write( "(b) Show that every starting number can produce a list ending in 1.<BR>\n" );
document.write( "A cycle is a list which eventually returns to the starting number, such as<BR>\n" );
document.write( "24, 48, 96, 9, 18, 36, 3, 6, 12, 24.<BR>\n" );
document.write( "(c) Show that every number from 1 to 41 occurs in a cycle with at most<BR>\n" );
document.write( "13 distinct terms.<BR>\n" );
document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\r<BR>\n" );
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document.write( "I don't like very much how @CPhill answered question &nbsp;(a).\r<BR>\n" );
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document.write( "Indeed, &nbsp;this is what he promised at the beginning of his post\r<BR>\n" );
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document.write( "  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Below is a **full worked solution** to all parts (a), (b), and (c).\r<BR>\n" );
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document.write( "  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;I keep each explanation clear and constructive so you can compare with your own work.\r<BR>\n" );
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document.write( "He made several attempts related to this part &nbsp;(a), &nbsp;but no one attempt was &nbsp;**full worked solution**.\r<BR>\n" );
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document.write( "There are several sections in his post, &nbsp;related to part &nbsp;(a), &nbsp;and each section started very promising, &nbsp;like these\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;### A clean correct chain (verified):\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;## **Correct, short solution**\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;## **Final clean correct chain**\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;# **I give you a correct, minimal finished answer for (a):**\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;**Final correct answer — verified:**\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;# **Given the increasing messiness, here is the correct canonical solution used in contest solutions:**\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;### **Correct solution (short)**\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Final step: **13 → 26 → 52 → 104 → 208 → 416 → 832 → 1664 → 3328 → 6656 → 665 → 1290 → 129**\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;If you'd like, I can produce a clean, minimal, polished answer for submission.\r<BR>\n" );
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document.write( "BUT &nbsp;NO &nbsp;ONE &nbsp;of his &nbsp;ATTEMTS &nbsp;was successful and &nbsp;NO &nbsp;ONE &nbsp;was a correct answer.\r<BR>\n" );
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document.write( "This his &nbsp;\"Final step: **13 → 26 → 52 → 104 → 208 → 416 → 832 → 1664 → 3328 → 6656 → 665 → 1290 → 129**\r<BR>\n" );
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document.write( "is erroneous, &nbsp;since &nbsp;665*2 &nbsp;is not &nbsp;1290;  &nbsp;it is &nbsp;1330, &nbsp;which is irrelevant.\r<BR>\n" );
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document.write( "So, &nbsp;this part of his solution is nothing else as deceiving a reader, very persistent and verbose.\r<BR>\n" );
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document.write( "After this critic, &nbsp;in my next post, &nbsp;I will give a complete explicit solution to part &nbsp;(a) &nbsp;(under my other nickname &nbsp;\" n2 \").\r<BR>\n" );
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document.write( " </SPAN>\n" );
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