I haven’t posted a puzzle in quite a while, so…
Every morning, I drive from my home at 1st and A to my work at 9th and J (8 blocks east and 9 blocks north), always heading either east or north so as not to backtrack… but I choose a random route each time, with all routes being equally likely. How many turns should I expect to make?
And a follow-up: on busy city streets, left turns are typically much more time-consuming than right turns. On average, how many left turns should I expect to make?
The first problem is presented in Zucker’s paper referenced below, and solved using induction on a couple of recurrence relations. But it seems to me there is a much simpler solution, that more directly addresses the follow-up as well.
- Zucker, M., Lattice Paths and Harmonic Means, The College Mathematics Journal, 47(2) March 2016, p. 121-124 [JSTOR]