Even though many prototypes of 3D self-reconfiguring modular robots (SRMRs) have been developed in recent years, a demonstration involving 1'000 modules remains a challenge. This is largely due to complex mechanics needed to achieve connection, disconnection and especially actuation in such a system. This work introduces "Kubits," which is, to the best of our knowledge, the first SRMR that achieves these functionalities without moving parts, i.e. in solid-state. Each module contains a kind of programmable magnet whose magnetization can be controlled. The simultaneous control of touching magnet pairs of two modules is used to create attraction (connection), neutrality (disconnection) and actuation (repulsion), which results in self-reconfiguration by a cube pivoting around an edge. We detail the design of the system and demonstrate a series of successful flips, including a jumping mode. The energy-efficient, lightweight and robust (both in terms of mechanics and control) method in Kubits is a promising path for scalable self-reconfiguration.