Water Jug Problem with Python

The Water Jug Problem is one of the oldest puzzles in computer science and Mathematics. It is worked out using two jugs of different volumes, where you have to measure out a certain target volume of water through a series of steps. This document also offers an algorithm in python which will identify whether the target amount can or cannot be measured as well as the states that will get to the solution.

Question

Given two jugs with capacities jug1_capacity and jug2_capacity, and a target amount target, how can you determine if it is possible to measure exactly target liters of water using the two jugs? Furthermore, if it is possible, what is the sequence of states (steps) from the initial state (0, 0) to a state where one of the jugs contains exactly target liters of water?

Installation

To run the provided Python code, make sure Python 3.x is installed on your system. If it is not installed, download and install it from the official Python website.

Python Code to Solve Water Jug Problem

from collections import deque

def water_jug_problem_trace_path(jug1_capacity, jug2_capacity, target):
   # Queue to keep track of states and the path to reach them
   queue = deque([((0, 0), [(0, 0)])])
   # Set to keep track of visited states
   visited = set()
   visited.add((0, 0))

   while queue:
      (jug1, jug2), path = queue.popleft()

      # Check if we have reached the target amount of water in either jug
      if jug1 == target or jug2 == target:
         return path

      # List of possible actions
      actions = [
         (jug1_capacity, jug2), # Fill jug1
         (jug1, jug2_capacity), # Fill jug2
         (0, jug2), # Empty jug1
         (jug1, 0), # Empty jug2
         (min(jug1_capacity, jug1 + jug2), jug2 - (min(jug1_capacity, jug1 + jug2) - jug1)), # Pour jug2 into jug1
         (jug1 - (min(jug2_capacity, jug1 + jug2) - jug2), min(jug2_capacity, jug1 + jug2)) # Pour jug1 into jug2
      ]

      for action in actions:
         if action not in visited:
            visited.add(action)
            queue.append((action, path + [action]))

    return None

# Example usage
jug1_capacity = 4
jug2_capacity = 3
target = 2
path = water_jug_problem_trace_path(jug1_capacity, jug2_capacity, target)

if path:
   print("Path of states followed:", path)
else:
   print("No solution found.")

Output

Instead, the code will display the transition of states from the initial state to a state in which one of the juges has exactly the desired amount. In the considered example where there are two jug with the capacities 4 liters and 3 liters respectively and with the goal to get 2 liters

This result shows the sequence of states and the operations performed in order to successfully obtain the value 2 liters of water using the jugs and from an empty state.

Code Explanation

• Initial State − They both are initially empty and so the initial position of the two jugs is (0, 0).
• Operations − These include; an action of putting water into a jug transferring water from one jug to another or taking water out of a jug. In the process of function activation each operation leads to the formation of a new state.
• BFS Approach − It is used in the game or simulation to find out all possible states and all transitions possible between these states such as Breadth-First Search (BFS). The BFS makes certain that the absolute minimum number of steps is taken to get to the target state.
• Path Tracing − The solution keeps track of the paths taken to get to each of the states thus produces the steps to get to the required amount.

Conclusion

The Water Jug Problem can be associated with algorithms and state space as a valuable example of the state space exploration. This matter is granted with help of the provided Python code not only to determine measurability of the target amount, but also to identify the sequence of states, which would allow to define its solution. It eliminates ones vision from the operations and states, and from this perspective, one is in a better position to understand the problem and find a solution to it.