A Note on Geomancy

The Wikipedia article on Geomancy discusses the mathematical structure of the process. A passing mention to the fact that [d]ue to the mathematics of the chart, only figures that have an even number of points total can become Judges piqued my curiosity. I verified it by computer in an instant, but my curiosity was not satisfied: how can this fact be mathematically proven? It is rather simple, actually: the Judge has the parity of both Witnesses combined. The Witnesses in turn have the overall parity of the four Nieces. The Nieces have themselves the parity of the Mothers and the Daughters combined. But the Daughters have the overall parity of the Mothers, as the former are a rearrangement of the latter. Hence the parity of the Judge is twice the parity of the four Mothers combined, and thus even.