  We explore the minimum energy configurations of repelling particles
distributed over n possible locations forming a toric grid. We conjecture that
the most energy-efficient way to distribute n/2 particles over this space is to
place them in a checkerboard pattern. Numerical experiments validate this
conjecture for reasonable choices of the repelling force. In the present paper,
we prove this conjecture in a large number of special cases---most notably,
when the sizes of the torus are either two or multiples of four in all
dimensions and the repelling force is a completely monotonic function of the
Lee distance between the particles.
