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A and B start at opposite corners of a 4x4 grid and step towards each other's corner at random each second. What is the probability they ever share a position?

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Two people start at opposite corners of a 4x4 grid: A on the bottom-left node, B on the top-right node. Every second A moves one edge right or up at random while B moves one edge left or down at random; each must move every second (if a direction is blocked they take the other). What is the probability that A and B occupy the same position at some moment?

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