FISH is a fast and simple ideal magnetohydrodynamics code that scales to ~10,000 processes for a Cartesian computational domain of ~10003 cells. The simplicity of FISH has been achieved by the rigorous application of the operator splitting technique, while second-order accuracy is maintained by the symmetric ordering of the operators. Between directional sweeps, the three-dimensional data are rotated in memory so that the sweep is always performed in a cache-efficient way along the direction of contiguous memory. Hence, the code only requires a one-dimensional description of the conservation equations to be solved. This approach also enables an elegant novel parallelization of the code that is based on persistent communications with MPI for cubic domain decomposition on machines with distributed memory. This scheme is then combined with an additional OpenMP parallelization of different sweeps that can take advantage of clusters of shared memory. We document the detailed implementation of a second-order total variation diminishing advection scheme based on flux reconstruction. The magnetic fields are evolved by a constrained transport scheme. We show that the subtraction of a simple estimate of the hydrostatic gradient from the total gradients can significantly reduce the dissipation of the advection scheme in simulations of gravitationally bound hydrostatic objects. Through its simplicity and efficiency, FISH is as well suited for hydrodynamics classes as for large-scale astrophysical simulations on high-performance computer clusters. In preparation for the release of a public version, we demonstrate the performance of FISH in a suite of astrophysically orientated test cases.