Question

Design a minimum depth decision tree to solve each of the following problems: (a) Take the water in an eight-gallon jug, and divide it into two four-gallon portions using a three-gallon jug and a five-gallon jug that initially are empty. (b) Take the water in a six-gallon jug. and end with two gallons in a four- gallon jug using a four-gallon jug and a three-gallon jug that initially are empty.

Short answer

(a) and (b) require sequential pouring to achieve the desired measurements.

Step-by-step solution

Understand the Problem (a)

We have three jugs: an 8-gallon jug filled with water, and a 3-gallon and a 5-gallon jug, both empty. Our goal is to divide the water so that there are two portions of exactly 4 gallons.

Initial State (a)

Start with the jugs as follows: 8-gallon jug filled (8 gallons), 3-gallon jug empty (0 gallons), and 5-gallon jug empty (0 gallons). Represent the initial state as (8, 0, 0).

Transfer and Fill (a)

Pour water from the 8-gallon jug into the 5-gallon jug until it is full, resulting in (3, 0, 5).

Adjust with Transfers (a)

Pour water from the 5-gallon jug to the 3-gallon jug until the 3-gallon jug is full, resulting in (3, 3, 2).

Redistribute for Equal Portions (a)

Pour the water from the 3-gallon jug back into the 8-gallon jug, resulting in (6, 0, 2). Then pour from the 5-gallon jug to the 3-gallon jug, resulting in (6, 2, 0).

Final State (a)

Pour from the 8-gallon jug to the 5-gallon jug until it's full, resulting in (1, 2, 5). Then, pour from the 5-gallon jug to the 3-gallon jug to get (4, 0, 4). Now, we have two portions of 4 gallons each.

Understand the Problem (b)

We have a 6-gallon jug filled with water, and a 4-gallon and a 3-gallon jug, both initially empty. Our goal is to end with exactly 2 gallons in the 4-gallon jug.

Initial State (b)

Start with the jugs as follows: 6-gallon jug filled (6 gallons), 4-gallon jug empty (0 gallons), and 3-gallon jug empty (0 gallons). Represent the initial state as (6, 0, 0).

First Transfer (b)

Pour from the 6-gallon jug into the 4-gallon jug until the 4-gallon jug is full, resulting in (2, 4, 0).

Adjust with Emptying (b)

Empty the 4-gallon jug back into the 6-gallon jug, resulting in (6, 0, 0). Then pour from the 6-gallon jug into the 4-gallon jug again, resulting in (2, 4, 0).

Final Adjustment for 2 Gallons (b)

Pour from the 4-gallon jug to the 3-gallon jug until the 3-gallon jug is full, resulting in (2, 1, 3). Then, pour from the 3-gallon jug back into the 6-gallon jug and then pour from the 6-gallon jug to the 4-gallon jug, resulting in (0, 2, 3). Thus, 2 gallons remain in the 4-gallon jug.

Key concepts

Jug Problem

To understand the jug problem, you first need to grasp the basic concept: how can you transfer liquid between containers of different sizes to reach a desired measurement? This problem is a classic example of a decision-making puzzle in discrete mathematics that involves logical reasoning and strategic planning. The key is to use the jugs in a clever way to split the initial amount of water into the required portions.
The beauty of the jug problem lies in its simplicity and the challenge it poses. Though it seems straightforward, it requires a step-by-step approach to achieve the desired results. With given amounts of water and jug capacities, the task is to think about every possible move thoroughly before taking action.
This not only allows you to see the range of options available but also to identify the most effective strategy to reach the endpoint quickly.

Problem Solving Strategy

The problem solving strategy for the jug problem focuses on breaking down the task into smaller, manageable steps. First, start by understanding the initial conditions: which jugs you have and the total amount of water. Then, work out several possibilities by transferring water between jugs.
One effective approach is to record each step with notations like \(x, y, z\) where \(x\), \(y\), and \(z\) represent the amounts of water in each jug. This helps keep track of your actions and their outcomes.

• Step 1: Define the problem and its constraints clearly. Identify the initial and desired conditions.
• Step 2: Experiment with different sequences of actions, such as transferring water from one jug to another.
• Step 3: Consider reversing actions if needed, like pouring water back or emptying a jug to make more room.
• Step 4: Ensure every step taken moves closer to the goal. Each move should either fill a jug to its potential or leave space necessary for future moves.

This methodical problem-solving approach not only aids in solving jug problems efficiently but can also be applicable to various other logical puzzles.

Water Jug Puzzle

The water jug puzzle is an enthralling test of logical thinking and resourceful problem-solving. This puzzle has been around for ages and remains a captivating exercise for students and problem solvers all over the world.
In the specific puzzles we looked at, different capacities of jugs were used, requiring a strategic transfer of water to achieve equal portions or end with a specific amount. This puzzle is not just about the end solution; it is about navigating through constraints and systematically overcoming each obstacle.
Through these scenarios, we see how repetition, strategic transfers, and reversals are critical. For instance, filling a jug completely, then transferring some of it back to achieve a half-full state and so forth. It is this art of aligning available resources (jugs and their capacities) with the desired outcome (measured water amounts) that builds strong analytical skills.
Indeed, the water jug puzzle is a wonderful exercise that challenges one’s logical acuity while sharpening problem-solving abilities.