We examine a generalization of the one-dimensional Ising model involving interactions among neighbourhoods of "k" adjacent spins. The model is solved by exploiting a connection to an interesting computational problem that we call ""k"-SAT on a ring", and is shown to be equivalent to the nearest-neighbour Ising model in the absence of an external magnetic field. The aim of this paper is to explore an interplay of ideas in the context of a toy problem, while introducing a way of thinking about exactly solvable models in terms of "computational analogues". (Contains 1 table.)