What is the Water Jug Problem in AI?

Nitika Sharma 01 Nov, 2023 • 6 min read


The water jug problem, also known as the ‘water-pouring problem’ or ‘die hard problem,’ is a classic challenge in artificial intelligence and computer science. This puzzle revolves around measuring a specific quantity of water using multiple jugs, each with varying capacities. It’s not merely a brain teaser; it’s a fundamental problem frequently employed to exemplify various problem-solving strategies and algorithms, notably search and optimization techniques.

In the following sections of this article, we’ll delve into the intricacies of the water jug problem. We’ll explore how artificial intelligence approaches and tackles this puzzle, shedding light on applying AI techniques.

Defining the Problem

The Water Jug Problem is a classic puzzle in artificial intelligence involving two jugs, one with a capacity of ‘x’ liters and the other ‘y’ liters, and a water source. The goal is to measure a specific ‘z’ liters of water using these jugs, with no volume markings. It’s a test of problem-solving and state space search, where the initial state is both jugs empty and the goal is to reach a state where one jug holds ‘z’ liters. Various operations like filling, emptying, and pouring between jugs are used to find an efficient sequence of steps to achieve the desired water measurement.

Water jug problem in AI

Solving the Water Jug Problem requires a systematic approach. This is where the concept of state space search comes into play. State space search is a fundamental concept in AI that involves exploring possible states of a problem to reach a desired goal state.

Each state represents a specific configuration of water in the jugs. The initial state is when both jugs are empty, and the goal state is when you have ‘z’ liters of water in one of the jugs. The search algorithm explores different states by applying various operations like filling a jug, emptying it, or pouring water from one jug into the other.

Production Rules for Water Jug Problem

In AI, production rules are often used to represent knowledge and make decisions. In the case of the Water Jug Problem, production rules define the set of operations that can be applied to transition from one state to another. These rules include:

  • Fill Jug A: Fill jug A to its full capacity.
  • Fill Jug B: Fill jug B to its full capacity.
  • Empty Jug A: Empty the jug A.
  • Empty Jug B: Empty the Jug B.
  • Pour from A to B: Pour water from jug A to jug B unless you get an empty jug A or full jug B.
  • Pour from B to A: Pour water from jug B to jug A until either jug B is empty or jug A is full.

Using these production rules, we can construct a solution path to move from the initial state to the goal state.

Algorithm to Solve Water Jug Problem

Now, we will follow the Breadth-First Search (BFS) approach to solve the problem:

  1. Start with the initial state where both jugs are empty.
  2. Create a queue. Next, add the initial state to it.
  3. While the queue is not empty, opt for the following:
    • Pop the front state from the queue.
    • Apply all possible production rules to generate new states.
    • Check if any of these new states match the goal state.
    • If a goal state is found, the problem is solved.
    • If not, add the new states to the queue for further exploration.
  4. BFS ensures that you find the shortest path to the goal state, which is efficient for solving the Water Jug Problem.

Python Program to Solve the Problem

Let’s see a Python program to solve the Water Jug Problem using the BFS algorithm. Here’s a simple implementation:

Python program to solve the Water Jug Problem using BFS

from collections import deque

def water_jug_BFS(x, y, z):
    visited = set()
    queue = deque([(0, 0)])
    while queue:
        jug_a, jug_b = queue.popleft()
        if jug_a == z or jug_b == z or jug_a + jug_b == z:
            return True
        if (jug_a, jug_b) in visited:
        visited.add((jug_a, jug_b))
        # Fill jug A
        if jug_a < x:
            queue.append((x, jug_b))
        # Fill jug B
        if jug_b < y:
            queue.append((jug_a, y))
        # Empty jug A
        if jug_a > 0:
            queue.append((0, jug_b))
        # Empty jug B
        if jug_b > 0:
            queue.append((jug_a, 0))
        # Pour from A to B
        if jug_a + jug_b >= y:
            queue.append((jug_a - (y - jug_b), y))
            queue.append((0, jug_a + jug_b))
        # Pour from B to A
        if jug_a + jug_b >= x:
            queue.append((x, jug_b - (x - jug_a)))
            queue.append((jug_a + jug_b, 0))
    return False

x = 4  # Capacity of jug A
y = 3  # Capacity of jug B
z = 2  # Desired amount of water

if water_jug_BFS(x, y, z):
    print(f'You can measure {z} liters of water using {x}-liter and {y}-liter jugs.')
    print(f'You cannot measure {z} liters of water using {x}-liter and {y}-liter jugs.')

Also Read: 14 Exciting Python Project Ideas & Topics for Beginners

Explanation for Water Jug Problem

This Python program uses BFS to search for a solution to the Water Jug Problem. It starts with empty jugs and explores all possible states by applying the production rules. If it finds a state where one of the jugs contains ‘z’ liters of water, it concludes that a solution exists.

Mirror to Practical Scenarios

The Water Jug Problem is a classic puzzle that, while simple in its setup, mirrors practical scenarios in various real-world situations. Here’s how it reflects practical scenarios:

  • Resource Management: In practical situations, it often relates to resource management. The water in the jugs can represent resources, such as time, money, or materials. The goal is to manage these resources to meet specific needs or objectives efficiently. For example, in project management, you may need to allocate time and budget effectively to achieve project goals.
  • Capacity Constraints: The jug capacities mirror capacity constraints commonly encountered in real life. In manufacturing, for instance, factories have a limited capacity to produce goods. Just like in the water jug problem, you need to optimize the use of these resources to meet production targets.
  • Optimization: The problem requires optimizing actions to achieve a goal. In supply chain management, companies optimize logistics and distribution to minimize costs and maximize efficiency. Similarly, the water jug problem involves finding the optimal sequence of actions to measure a specific amount of water.
  • Decision-Making: Practical scenarios often involve decision-making processes. Decision-makers must choose between various options to achieve objectives. The water jug problem reflects this by requiring decisions on which jug to fill, pour, or empty at each step to reach the target amount.
  • Trial and Error: Just like solving the water jug problem may involve trial and error, practical situations often require experimentation and adaptation. In marketing, for instance, strategies are tested, refined, and adapted based on customer responses. Similarly, solving the water jug problem may involve trying different actions until a solution is found.

AI and Computer Science Applications of Water Jug Problem

The Water Jug Problem is a fundamental case study in AI and computer science for problem-solving techniques. It helps illustrate concepts like search algorithms, state-space exploration, and optimization.

  • Search Algorithms: AI algorithms like Depth-First Search (DFS) and Breadth-First Search (BFS) are used to solve the Water Jug Problem. These algorithms find practical applications in routing, path planning, and network optimization.
  • State-Space Exploration: The problem represents state-space exploration, which is essential in AI planning and robotics. Algorithms used in robotics often explore possible robot states to plan movements and actions.
  • Optimization: The problem demonstrates the core concept of optimization. In AI, optimization techniques are applied in fields such as machine learning for hyperparameter tuning, scheduling, and operations research for resource allocation.

Examples of Similar Problems

The Water Jug Problem serves as a simplified model for various real-world problems. AI and computer programs are applied to address analogous challenges in different domains:

  • Network Routing: In network routing, algorithms determine the best path for data packets to travel from source to destination, considering network constraints and optimizing data flow.
  • Inventory Management: Businesses use AI to optimize inventory management, determining the right quantities to order and maintain while minimizing carrying costs and shortages.
  • Game AI: Game development employs AI algorithms for pathfinding, character behavior, and game strategy optimization. These applications involve state-space exploration similar to the Water Jug Problem.
  • Machine Learning: In machine learning, optimization problems include training deep neural networks with hyperparameter tuning, selecting features, and optimizing model parameters.
  • Transportation and Logistics: AI is utilized for route optimization, resource allocation, and demand forecasting in transportation and logistics, including ride-sharing services and delivery logistics.


The Water Jug Problem is a classic puzzle that has entertained puzzle enthusiasts and challenged AI researchers worldwide. By employing state space search, production rules, and search algorithms like BFS, it is possible to find an efficient solution to this problem.

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Frequently Asked Questions

Q1. What is the objective of the water jug problem?

A. The objective is to find a sequence of actions to measure a specific quantity of water using jugs of different capacities while respecting constraints.

Q2. What is the solution to the water jug problem?

A. The solution involves determining a series of actions like filling, emptying, and pouring to accurately measure the desired volume of water within the constraints of the jug capacities and operations.

Q3. What is the solution to the three water jug problem?

A. The three water jug problem’s solution is akin to the standard version but involves three jugs with varying capacities. The goal remains the same: measuring a specific volume using the three jugs.

Q4. Which search strategy is appropriate for the water jug problem in AI?

A. Appropriate search strategies for solving this problem include depth-first search, breadth-first search, and heuristic search methods like A*. The choice depends on the problem’s complexity and optimization criteria.

Nitika Sharma 01 Nov 2023

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