Combinatorics and snakes.

Although this might feel very simple and playful, this is an unsolved problem that has been considered. In particular, this is equivalent to the following problem:

Start at the origin in the plane. How many self-avoiding 'walks' are there on the integer lattice of prescribed length? Some results include this paper, which relates the number of walks in a strip with fibonacci numbers. Equivalently, the number of snakes that are 'very tall' (so that they're in a strip) is related to the fibonacci numbers.

Madras and Slade have a book, The Self-Avoiding Walk dedicated largely to this problem.