### Andy's Morning Stroll

##### July 2022 : Solution

Andy normally strolls across the hexagons of a soccer ball. To find the average length of this stroll, which turns out to be exactly 20, we could set up a series of equations, but the following intuitive argument may be more satisfying.

Imagine Andy took an extremely long stroll on these hexagons without stopping and we chopped that stroll into segments starting and ending at his home square. The symmetry of the ball suggests, in the limit of the length of the stroll going to infinity, exactly 1/20 of the hexagons Andy visits will be his home hexagon. This means the sum of the lengths of all the segments, divided by the number of segments, will approach 20. But this limiting ratio is exactly the average length of a stroll starting and ending at his home hexagon!

So now we need to compute the probability that his stroll on the kitchen floor will strictly exceed twenty steps. This was a bit of a computational exercise, we can construct a transition matrix showing that this probability, to seven significant digits, is 0.4480326…

Congrats to this month’s solvers!