Water Jug Puzzle

Given a set of 3 jugs of water that have capacities of a, b, and c liters, find the minimum number of operations performed before each jug has x, y, and z liters. Only jug C will start completely filled.

An operation is any of the following: A jug is emptied, a jug is filled, or water is poured from one jug to another until one of the jugs is either empty or full.

For example, jugs A, B, and C with capacities of 3, 5, and 8, where jugs A and B start empty and C has the full 8, require 2 operations to reach the state of 0, 3, and 5 liters in the jugs.

Create a function that, given an array of jug capacities [A, B, C] and an goal state array [x, y, z], returns the minimum number of operations needed to reach the goal state. If the inputs are invalid or there is no solution, return "No solution."

Examples

waterjug([3, 5, 8], [0, 3, 5]) ➞ 2

waterjug([1, 3, 4],  [0, 2, 2]) ➞ 3

waterjug([8, 17, 20], [0, 10, 10]) ➞ 9

waterjug([4, 17, 22], [2, 5, 15]) ➞ "No solution."

waterjug([3, 5, 8], [0, 0, 9]) ➞ "No solution."

Notes

• The amount of water in a jug can never exceed the capacity of that jug.
• The total liters in the goal state must be equal to the capacity of jug C.