On optimal solutions of a family of two jugs problem

The two jugs problem is a classic problem in mathematics, computer sciences, problem solving and cognitive psychology, etc. The objective of the problem is to measure a specific amount of water (w) by using the given two jugs with capacities m and n (m < n), where the jugs do not have markings on them to allow measuring the smaller quantities directly. The existing methods of solutions are often heuristic in nature and there is no guarantee that the solution found is optimal, in the sense that the number of pouring steps involved is least possible. In this paper, we are interested in solving a special family of two jugs problem, where w is equal to the average of m and n. Two new formulas for direct computation of the particular solutions of the associated Diophantine equation of such a problem are introduced, together with the proofs of their correctness. We also introduce a decision theorem for finding more optimal solutions for such a special family of two jugs problems. Some illustrative examples are included. Copyright © 2015 Research India Publications.