document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197385>Question 1197385</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An ant moves on the following lattice, beginning at the dot labeled $A$. Each minute he moves to one of the dots neighboring the dot he was at, choosing from among its neighbors at random. What is the probability that after $5$ minutes he is at the dot labeled $B$? Image is attached below. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #830631 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13198)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=830631\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The analysis by tutor @ikleyn to find the number of ways to end at B is good.  But her answer to the question is flawed, because there are not always 4 choices of a direction to go from a given vertex; the total number of possible paths of length 5 <b>starting from point A on the lattice</b> is far less than 4^5.<br><BR>\n" );
document.write( "In the end, I believe the problem has a very simple solution and answer.<br><BR>\n" );
document.write( "In 5 moves on the given lattice, the only points where the ant can end up are the four points 1 unit away from A.  Point B is one of those four points; and the symmetry of the lattice means there is equal probability of ending up at any one of those four points.<br><BR>\n" );
document.write( "So the probability of ending at B after 5 moves is 1/4.<br><BR>\n" );
document.write( "ANSWER: 1/4<br><BR>\n" );
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document.write( "Response to tutor @ikleyn....<br><BR>\n" );
document.write( "What an absurd note from you regarding my response to the question!<br><BR>\n" );
document.write( "Did you take any time to read my response? or were you just reacting to the idea that anyone would dare to disagree with you?<br><BR>\n" );
document.write( "I will not agree with you when you are WRONG!<br><BR>\n" );
document.write( "The path EEEEE, <b>starting from point A on the lattice</b>, is not possible -- it goes off the lattice.  In fact, it is obvious (to anyone who is paying attention to the problem) that a path EEEEE is not possible from ANY point on the lattice.<br><BR>\n" );
document.write( "In your effort to show your superiority (?) in all things mathematical, you might try paying attention to the problem that you are working on.<br><BR>\n" );
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