document.write( "Question 1197385: 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. \n" ); document.write( "

Algebra.Com's Answer #830621 by lotusjayden(18) $\"\"$ $\"About$

You can put this solution on YOUR website!
Image is here: https://latex.artofproblemsolving.com/d/0/4/d04c14a38efb837151549f4b864fde841fa50f95.png
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document.write( "Answer(since I originally answered the question in Latex, and since It's almost time to sleep, I will edit it tomorrow.)(The original post is here: https://artofproblemsolving.com/texer/yglroond)\r\n" );
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document.write( "Color the dots red and blue shown below.. Notice that whenever the ant moves, it moves from a red dot to a blue dot or a blue dot to a red dot. So since  is a red dot, it must move to a blue dot, then a red dot, then a blue dot, then a red dot, and end up on a blue dot. There are only four blue dots, and the ant is equally likely to end up on any one of these four, since the diagram is symmetric to a  rotation. The probability that the ant ends on  after five minutes is therefore .\r\n" );
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\n" ); document.write( "Note: @Ikleyn is not correct unlike the other times I saw her answer questions. The correct ones are me and @greenstamps. \n" ); document.write( "

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