## 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 techniquescomprising ofEnd state analysis technique,Abstraction techniqueandWorking backwards techniqueoffer 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.*

### More puzzles to enjoy

From our large collection of interesting puzzles enjoy: * Maze puzzles*,

**Riddles**,

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**Mathematical puzzles***,*

**Logic puzzles***,*

**Number lock puzzles***,*

**Missing number puzzles***,*

**River crossing puzzles***and*

**Ball weighing puzzles***.*

**Matchstick puzzles**You may also look at the full collection of puzzles at one place in the **Challenging brain teasers with solutions: Long list.**

*Enjoy puzzle solving while learning problem solving techniques.*