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Challenge 399: Ant-ics!

Where will Anton be?

Thanks to Ryan (Y13) for providing this problem.

Anton the ant is situated at a vertex V of a cube. Every second, Anton selects a random edge adjacent to V, and crawls along that edge. After exactly 400* seconds, what is the probability that Anton is again at V?

*Anton thought about stopping after exactly 399 seconds, but found that this made the problem less interesting to do.... 

Note that Anton has lots of energy and has no problem crawling along 400 edges at a rate of one per second!

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