For every rational value of the reduced magnetization of the array, m = µ /µ < 1, (ignoring sign) there exist very numerous possible arrangements of up and down dots, one (or more) of which must have minimum dipolar interaction energy W m . We have calculated that energy for a range of excellent candidates for these ground states at various values of m, in particular those of the chessboard AFM state of m = 0 and the uniform FM state with m = 1. We suggest and argue that, for the sequence of optimal configurations, W m and dW m /dm increase monotonically with m, and this is supported by our **data** in Fig. 4. (dW m /dm is likely to be locally discontinuous). An analytic formula for W m (m) is derived on the assumption that for m → 1, the minority down dots are located near points on an equilateral triangular super-lattice. A similar formula is derived for m → 0 relating to the excess up dots. Remarkably, these formulae agree at m = 1/2 (though their gradients differ) and fit the **data** for specific states very well, especially near m = 0 and 1. A similar formula involving a square super-lattice instead of the triangular super-lattice gives results in precise agreement with the **data** for specific structures with m = 0, 0.6 and 0.8, these being those structures in which the minority dots do indeed lie on square super-lattices. It differs from the first asymptotic formula by little more than 1.11% over the appropriate range 0.5 ≤ m ≤ 1.