Since we are only interested in the modular arithmetic , we may overlook the need for secondary moves on the western side of the parallelogram and keep applying only the basic moves. First, together with the relative primality of A and B, it insures that all three capacities share no common factor, safe 1. Were they not, the quantities that could be measured with three vessels of the specified capacities would share their common factor. For the problem to be generally solvable the mutual primality of all three capacities is a necessary condition. However, it is not sufficient.
Anomalies also arise when the three jugs are rather big .
This completes the analysis of existence of a solution to the 3 Jugs problem. That of the 2 and 4 Jugs problems is left as an enticement for the future Poissons. The puzzle in barycentric coordinates Three Jugs Problem. Two Pails Puzzle. Alexander Bogomolny 2789908 .