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King's Maths School

Challenge 448: The ant-verage

Anton is not your ant-verage ant!

(a) Anton is at vertex A of a square ABCD. Each second he moves to an adjacent vertex, picking at random. (For example, if he is at vertex B, then he moves to vertex A or C with equal probability.)

He stops when he reaches vertex C. What is the average (ie mean) time he takes to stop?

(b) Now Anton is at one vertex of a cube; he moves each second to a randomly chosen adjacent vertex. He stops when he reaches the opposite vertex (ie the one furthest away from his starting vertex). What is the average time he takes to stop?

Submit your solution

Please do send in your solution to this problem to weeklymaths@kcl.ac.uk You can scan or photograph your written work, or type your solutions. If this is your first weekly maths challenge solution, please include your year group and the name of the school you attend. We'll be happy to provide feedback on your solution, assuming that you are in year 11 or below. If you are older than this, we hope you enjoy trying the problems and reviewing your solutions against those we publish on the website.