A method for rapidly predicting the formation and stability of undiscovered single phase high-entropy alloys (SPHEAs) is provided. Our software implementation of the algorithm uses data for 73 metallic elements and rapidly combines them - 4, 5 or 6 elements at a time - using the Miedema semi-empirical methodology to yield estimates of formation enthalpy. Approximately 186,000,000 compositions of 4, 5 and 6 element alloys were screened, and ∼1900 new equimolar SPHEAs predicted. Of the 185 experimentally reported HEA systems currently known, the model correctly predicted the stability of the SPHEA structure in 177. The other sixteen are suggested to actually form a partially ordered solid solution – a finding supported by other recent experimental and theoretical work. The stability of each alloy at a specific temperature can also be predicted, allowing precipitation temperatures (and the likely precipitate) to be forecast. This combinatorial algorithm is described in detail, and its software implementation is freely accessible through a web-service allowing rapid advances in the design, development and discovery of new technologically important alloys.