The paper combines two topics belonging to the general theme of the spontaneous symmetry breaking (SSB) in systems including two basic competing ingredients: the self-focusing cubic nonlinearity and a double-well-potential (DWP) structure. Such systems find diverse physical realizations, chiefly in optical waveguides, made of a nonlinear material and featuring a transverse DWP structure, and in models of atomic BEC with attractive inter-atomic interactions, loaded into a pair of symmetric potential wells coupled by tunneling across the separating barrier. With the increase of the nonlinearity strength, the SSB occurs at a critical value of the strength. The first part of the paper offers a brief overview of the topic. The second part presents a model which is designed as the simplest one capable to produce the SSB phenomenology in the one-dimensional geometry. The model is based on the DWP built as an infinitely deep potential box, which is split into two wells by a delta-functional barrier at the central point. Approximate analytical predictions for the SSB are produced for two limit cases: strong (deep) or weak (shallow) splitting of the potential box by the central barrier. Critical values of the strength of the nonlinearity at the SSB point, represented by the norm of the stationary wave field, are found in both cases (the critical strength is small in the former case, and large in the latter one). For the intermediate case, a less accurate variational approximation (VA) is developed.