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Two particles start at opposite corners of an octagon. Each second each one moves one edge clockwise or anticlockwise on a coin flip. What is the expected number of coin flips until they meet?

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A particle sits on a vertex of a hexagon. Each second a fair coin moves it one edge clockwise (Heads) or anticlockwise (Tails). What is the expected time to return to its start?A particle on a decagon vertex moves each second on a die roll: 1 or 2 clockwise, 3 or 4 anticlockwise, otherwise it stays. What is the expected time to reach the opposite vertex?A particle starts at the centre of a 10 x 10 grid and moves one step up, down, left or right at random each move. What is the expected number of moves until it leaves the grid?

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