## Matchstick square puzzle - Turn five into four in two moves

#### The puzzle

Below are squares formed by matchsticks. **Turn five into four in two moves.** All the matchsticks must be used up in the final 4 squares with no matchstick hanging, each attached to a square as a side of the square.

**Recommended time to solve:** 20 minutes.

### What follows

Solutions and the reason for popularity of matchstick puzzles form the sections,

**Why matchstick puzzles are so popular.**based on question, analysis and answer to solve finally by**First quick solution****End state analysis approach.****Solution by direct use of End State analysis approach.**

**Go to any of the sections directly**, skipping the other sections, *by clicking on the link of the desired section above.*

### Why matchsticks are popular in puzzles

You may ask, why of all things matchsticks are used in puzzle formations?

Important advantages of using matchsticks in puzzles are,

- Matchsticks all are of equal length, and so it is very easy to physically make regular geometric shapes like triangles or squares by using matchsticks,
- In a figure, the matchsticks can be easily rearranged in any way to change the original shape, and
- Matchsticks are very cheap and easily available.

You can make any complex figure made up of matchstick triangles or squares or even polygons with larger number of sides.

Your imagination is the only limit to how many different types of matchstick puzzles you can create and solve. Cost will be just your time.

**Gain is though great**, *because matchstick puzzle solving has the assured effect of improvement of basic pattern recognition, method creation and analytical skills which combine to improve your general problem solving skills.*

### First Solution to Matchstick square puzzle - turn 5 to 4 in 2 moves: By Question, analysis and answering conclusion

We will take a totally systematic approach of asking revealing questions, analyzing for answer and making conclusions. Puzzle figure is shown again for ease of explanation.

The first step is to put down what we know about the problem in more details,

- The given figure is made up of 5 squares,
- 16 matchsticks are used up in the figure, and,
- A total of 4 matchsticks are common between two squares.

The **common stick identification** is important for solving matchstick puzzles.

**First question** that is asked,

Question 1:What about the final figure? What are its characteristics? How would it look like?

*Answering intermediate conclusion* is,

Conclusion 1:The solution figure is to be made up of four squares, 1 less than 5 in the puzzle figure, and,The same

16 matchsticksare to be used for making the final 4 square figure.

Based on this knowledge the *second question raised* is,

Question 2:How can the 16 sticks be used in forming the four squares? Will there be any common sticks?

If we get *a clear answer to this questions we would have a good idea about our target figure.*

It is always important not to walk with eyes closed and instead, form a good idea of the destination at the beginning of the journey itself.

*Answering conclusion* is easy to make,

Conclusion 2:As 1 independent square requires 4 matchsticks, 16 matchsticks would just be enough to makefour independent squares. Therecan't be any common stick between any two squaresin this 4 square figure, as a common matchstick connects two squares by definition itself.

This is a **key pattern discovery** easily made.

But if this is the case, *there can't be many different ways the four independent squares be formed.*

So let's explore this path of knowing the nature of the solution figure in more details. This should give us a quick solution.

There are only two ways the four independent squares can be formed,

- With at least one square UNCONNECTED to any of the other squares, or
- All the four squares CONNECTED at their corners.

Have a look at the puzzle figure.

There is *only one way an independent square can be formed by moving two sticks:*

By moving the two sticks tick-marked in the following figure,

But this way no other independent squares can be created at all.

Conclusion 3:thefour independent squares must be corner-connected.

With this additional constraint, the possibilities must only be a few. Let us see what these possibilities are.

As we formed the three possible final figures we have simultaneously drawn a **vertical line of symmetry** through the three possible final figures as well as the puzzle figure.

This is a new method we are using that will surely help us in getting the final solution.

#### Comparing the three final figures with the puzzle figure

**Key pattern** identified,

In both the possible final figure 1 and 3, one square in each side of the vertical line of symmetry is

separated from the line of symmetry by one full square.But

in the puzzle figure all the squares are attached to the line of symmetry.

**Conclusion 4:** These two possible final figures are too much different from the puzzle figure to be reached in 2 stick moves.

The only possible figure that have to be the **solution is the possible final figure 2.**

The last question is simply,

Question 3:which two sticks are to be moved to form thesolution figure 2on the middle from the puzzle figure just above it?

It takes just a brief analysis to identify the two sticks that are to be moved to form the solution figure. Following is the solution.

Now we will use the same End state analysis approach without the reasoning part and with awareness of how the method is to be used.

### 2nd Solution to Matchstick square puzzle - turn 5 to 4 in 2 moves: By direct application of End state analysis approach

The key pattern and requirement in the final solution configuration is,

The given 5 squares are made up of 16 sticks that are just enough for exactly 4 squares. So in the final configuration there can't be any common side (or matchstick) between any two squares.

Keeping in mind that we have to convert the initial configuration to this final configuration, we will form possible final configurations and compare each with the given configuration assessing how much similar the two configurations are.

#### Comparison between possible final configurations with initial configuration

**First trial** is shown below with initial configuration on the left and a possible final configuration on the right.

The squares in the first possible final solution are separated wide horizontally and so the configuration is very dissimilar with the initial puzzle configuration. No way can we convert the puzzle configuration to this possible final configuration in 2 stick moves. This conclusion is based on visual assessment.

If you mentally superimpose one figure on the other, you would find only the two squares B and D common between the two.

We would make the possible final configuration in the **second** trial more compact. It is shown below.

In this case the common squares between the initial and final figure are 3, square A, C and E. With maximum similarity with the puzzle figure then, this possible final solution IS the final solution.

It is easy to identify the two sticks that are to be moved to create a new square as shown in the first solution.

This approach should yield the fastest solution and is one of the most powerful general problem solving approaches.

#### Number of solutions

**Question:** is there any more solution configuration?

You need to convince yourself that the **solution configuration in this case is unique**.

### End note

The two solutions are similar as, both finally uses the same End state analysis approach of comparing possible final solutions with the puzzle figure.

Only difference has been, in the first case, **the method is created practically from scratch** by* question, analysis and answer technique*, and in the second case, *End state analysis approach has been applied directly* with clear intent and knowledge.

The method of problem solving by asking a series of revealing questions, analyzing and getting answers step by step is a

general systematic problem solving approachand its scope of application is wider than theEnd state analysis approach.Both approaches are powerful analytical general methods for solving a large number of different types of problems,

including real life problems.

These solutions though are quick and elegant with **no trial and error**, *you gain little insight into the puzzle structure or the two stick moves.* In fact, bypassing the details of the problem contributed most towards the elegance and speed of solution.

In contrast, **solving this same matchstick puzzle by stick move analysis** takes longer, but imparts a thorough knowledge on various aspects of the puzzle. That's an inevitable trade-off.

**Your challenge:** Solve the puzzle *without using End state analysis approach.*

Lastly, **to solve matchstick puzzles you don't need to know maths or any other subject**—you just have to identify key patterns and use your inherent analytical reasoning skills to home in to the solution with assurance and speed.

### Special note

This matchstick square puzzle with a rich learning potential is used as a learning resource in teaching the subject of Innovation and Problem solving.

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