Two Jugs of Water Riddle: Systematic Problem Solving Approach

Submitted by Atanu Chaudhuri on Fri, 09/10/2020 - 00:55

The Two Jugs of Water Riddle Solved by Goal Directed Systematic Problem Solving Techniques

Measure 4 liters of water with two empty jugs of 5 liters and 3 liters capacity and a tank full of water. Fill and pour as many times as you want.

How many ways can you measure 4 liters of water by repeating no fill or pour move?

Recoommend time to solve the riddle is 15 minutes.

The two jugs of water riddle is an intriguing puzzle that sparks inventive thinking. Here we'll explore multiple approaches to solve this puzzle following not random, but systematic problem solving techniques.

Solution of two jugs of water riddle

Understanding the Puzzle: To begin, imagine the objects—the jugs—and the actions—filling and emptying them. This initial abstraction lays the foundation for solving the puzzle.

Stage 1: Visualizing Outcomes of Moves

This way you won’t have any difficulty in deciding the outcome of a move—all in your mind. Do a trial of 2 moves.

Fill the 3-liter jug from the tank.
Empty the 3-liter jug into the 5-liter jug.

The OUTCOME of these two moves at first glance,

3-liter jug empty and,
3 liter water in the 5-liter jug.

Did we consider all possible outcomes? No, we just missed the 2 liters of empty volume in the 5-liter jug. Let us include it.

Stage 2: Precise Outcome Specification

With precision, all outcomes of first two trial moves:

3 liters of empty volume in the 3-liter jug.
3 liters of water in the 5-liter jug.
2 liters of empty volume in the 5-liter jug.

Stage 3: Creating Equations as second level abstraction

Translate outcomes of moves into simple arithmetic equations of single digit numbers that are the volumes. Take an example: If you fill the 3-liter jug from the fully filled 5-liter jug, the outcomes of volume status of two jugs will be,

5-liter jug: 5 - 3 = 2 (2 liters of water and 3 liters empty)
3-liter jug: 0 + 3 = 3 (fully filled to 3 liters)

Stage 4: Final goal and Last Move

Now we will use the powerful problem solving technique End State Analysis to define the final goal of the solution and the move that will give us this solution.

Define the final goal: 5-liter jug with 4 liters of water. Achieve this by:

Pouring 3 liters from the 3-liter jug into the 5-liter jug containing 1 liter.

Stage 5: Work backwards to discover how you get 1 liter water in the 5-liter jug: Intermediate goal

Take help of arithmetic equations of volumes to get 1 liter water in 5-liter jug. Result of 1 can be achieved in the equation:

(3 + 3) - 5 = 1.

This breaks down to two volume equations,

3 + 3 = 6, and,
6 - 5 = 1.

Stage 6: Intermediate Solution: Getting 1 liter water in 5-liter jug

Convert the last two volume equations to moves.

3 + 3 = 6 can only mean,

Fill 3-liter jug.
Empty 3-liter jug into the 5-liter jug.
Fill 3-liter jug again.

The 6 liters of water can only be distributed in the two jugs.

Now, 6 - 5 = 1 translates to,

Fill the 5-liter jug (containing 3 liter water) by pouring water from the 3-liter jug. Exactly 2 liters of water will be needed to fully fill the 5-liter jug and 1 liter of water will remain in the 3-liter jug. Now you have to transfer this 1 liter into the 5-liter jug.
Empty 5-liter jug into the tank.
Pour 1 liter from 3-liter jug into the empty 5-liter jug. Intermediate goal reached.

Stage 7: Final solution: Fill the empty 3-liter jug again and pour 3 liters of water to the 5-liter jug that has now 1 liter water: Result: 4 liters of water in 5-liter jug: Easy

Fill the empty 3-liter jug.
Empty it into the 5-liter jug, resulting in 4 liters of water in 5-liter jug.

Solution reached in 8 moves.

Key Techniques Used:

Abstraction to convert outcomes of pour and fill moves into equations of water volumes in the jugs.
Awareness of empty volumes after fill moves.
Final Goal definition and working backward to define intermediate goal. Use of End state analysis and Working backwards problem solving techniques.
Abstraction in reverse to translate volume equations into moves.

Part 2: Finding All Solutions to Two Jugs of Water Riddle

The key idea in the first solution was to form the final goal move as, 1 + 3 = 4. Extend this approach to find all solutions:

Consider all possible volume equations resulting in 4 liters in 5-liter jug: The first number in the LHS must be water in 5-liter jug and the second, water poured from either of the jugs.

1 + 3 = 4 (your first solution)
5 - 1 = 4
3 + 1 = 4 (Not feasible because, after you get 1 liter water in 3-liter jug you need 3 liter water in 5-liter jug for which you need one more empty 3-liter jug. You don't have it.)
2 + 2 = 4 (Infeasible. You may get 2 liters easily in 5-liter jug, but to get another 2 liters in 3-liter jug you need one more empty 5-liter jug.)

Only other solution can be given by 5 - 1 = 4.

Second Solution: Final goal 5 - 1 = 4: Find first intermediate goal

To pour 1 liter from fully filled 5-liter jug the intermediate goal is,

2 liter water in 3-liter jug. Next you will fill this 3-liter jug fully from filled up 5-liter jug leaving 4 liters of water in 5-liter jug.

The moves to reach the intermediate goal first and then the final goal:

Fill 5-liter jug.
Fill 3-liter jug from 5-liter jug. 2 liter water remains in 5-liter jug.
Empty 3-liter jug in the tank.
Pour 2 liter water from 5-liter jug to empty 3-liter jug. 3-liter jug has 2 liter water and 1 liter empty volume.
Fill 5-liter jug.
Fill 3-liter jug pouring 1 liter from filled up 5-liter jug. 4 liter water remains in 5-liter jug.

Solutions to the puzzle are only two. Mathematically proved :).

End Note

Goal directed systematic problem solving techniques comprising of End state analysis technique, Abstraction technique and Working backwards technique offer assurance and efficiency (producing all solutions without any random trial and error), ensuring a thorough exploration of all possibilities.

Solution also aided by Goal definition and precise outcome specification. Precision in defining is important in any problem solving.