Geologic evidence suggests that the Earth may have been completely covered in ice in the distant past, a state known as Snowball Earth. This is still the subject of controversy, and has been the focus of modeling work from low-dimensional models up to state-of-the-art general circulation models. In our present global climate, the ocean plays a large role in redistributing heat from the equatorial regions to high latitudes, and as an important part of the global heat budget, its role in the initiation a Snowball Earth, and the subsequent climate, is of great interest. To better understand the role of oceanic heat transport in the initiation of Snowball Earth, and the resulting global ice covered climate state, the goal of this inquiry is twofold: we wish to propose the least complex model that can capture the Snowball Earth scenario as well as the present-day climate with partial ice cover, and we want to determine the relative importance of oceanic heat transport. To do this, we develop a simple model, incorporating thermohaline dynamics from traditional box ocean models, a radiative balance from energy balance models, and the more contemporary "sea glacier" model to account for viscous flow effects of extremely thick sea ice. The resulting model, consisting of dynamic ocean and ice components, is able to reproduce both Snowball Earth and present-day conditions through reasonable changes in forcing parameters. We find that including or neglecting oceanic heat transport may lead to vastly different global climate states, and also that the parameterization of under-ice heat transfer in the ice–ocean coupling plays a key role in the resulting global climate state, demonstrating the regulatory effect of dynamic ocean heat transport.