## Move 3 matchsticks to make 4 squares from the 5 squares puzzle figure

In the 5 squares matchstick puzzle figure you have to move 3 matchsticks to make 4 squares. You have 10 mins to solve this 3rd 5 squares matchstick puzzle.

#### The 5 squares matchstick puzzle

**Part I:** Move 3 matchsticks to make 4 squares from the T-shaped 5 squares matchstick puzzle figure with no stick left hanging.

**Part II:** How many unique solutions can you find?

**Recommended time **is 10 minutes.

This puzzle is interesting, but not difficult.

Enjoy solving flawlessly!

### Solution to the third 5 squares matchstick puzzle: Move 3 matchsticks to make 4 squares

#### Analysis of the structure and knowing precisely what you have to do

**At the first step,** you need to count the total number of sticks as 16. These sticks form 5 squares.

How many sticks are required to form a square? It is 4. So to form 5 independent squares 20 sticks would have been used—4 more than 16 sticks we have.

How could 5 squares have been formed even with a number of sticks 4 less than required to form 5 squares?

This is where the key concept in matchstick puzzle solving comes in—the **concept of sticks common between two adjacent unit shapes.**

In our problem, the unit shape is a square. In another puzzle, the figure could comprise equilateral triangles. There can be many variations. But the fact remains,

Each common stick between two unit shapes reduces the number of sticks required to form the shapes independent of each other by ONE.

In our problem, 20 sticks certainly would have been required to form 5 squares, but then, the 5 squares would have had no stick common between any two squares—all 5 would have been **free-standing independent squares.**

Each common stick reduces the requirement of maximum number of sticks by 1.

Now look at our 5 square figure again—this time we have labeled the squares by A, B, C, D and E, so that we can refer to a specific square.

You have exactly 16 sticks, the maximum number required to form 4 independent squares. 4 common sticks reduced the maximum requirement for 5 squares from 20 to 16. **Our job is then clearly to,**

Eliminate all common sticks and form 4 independent squares by moving just 3 sticks.

**At this stage,** *you have precisely and clearly understood what you have to do to solve the puzzle.*

#### Third step: Move 3 matchsticks to make 4 squares 3rd puzzle: Strategic solution by End state analysis approach

Let us briefly go through how to use this strategic approach and solve the puzzle easily and quickly.

In this **strategic approach of problem solving, in general,**

You have to compare the possible 5 square figures that can be final solutions or what we call

end stateswith thestarting problemfigure.

The **goal of comparison would be to,**

Assess or judge how similar are the possible final figure and the problem figure being compared.

And the **outcome or result of comparisons** between *promising final solution figures and the problem figure would be to,*

Identify the one most promising final figure having the maximum similarity with the puzzle figure. You would actually try to move 3 sticks to reduce the number of squares to 4 on this most promising final solution figure.

If you fail with this first chosen promising final figure, you would select the second such solution and try again.

We are showing a comparison of one promising final solution with our problem figure below. *Remember, in a promising final figure of 4 squares made up of 16 sticks, all 4 squares will be independent from each other with no stick common between any two.*

How many squares are untouched in the possible and promising final figure on the right? Three squares, A, C and D, would remain untouched if we move sticks in the problem figure to form **this possible final figure.**

This is the maximum similarity between the puzzle figure and possible solution figure you can achieve, and so you can be more or less sure that the figure on the right indeed is a solution.

#### Fourth step: Move 3 matchsticks to make 4 squares 3rd puzzle: Identifying the sticks to move from the puzzle figure to form the possible final figure

This is the stage of actually identifying and moving the sticks, and will be the final stage if the number of sticks to be moved turns out to be 3. If not, you have to repeat the previous step, and this final step again.

Here, the puzzle is simple enough so that while analyzing the possible solutions mentally, we have simultaneously gone through the third and fourth steps identifying not only the solution but also which sticks to move.

The 3 sticks to move are identified below by red, blue and green colored arrows.

You could have moved the three sticks in any sequence. This is your first solution.

Okay, the second part of the question now.

**Can you identify any more solution?**

#### A second solution to the puzzle

Now you are very clear about the puzzle figure. It is easy to see that instead of forming the new square on the right, you could have formed it on the left. This second unique solution is shown.

Any more solution? How can you be sure that there is no more solution to the puzzle?

Proving is always complex.

### Going deeper: Reasoning by on analysis of structure

As we have to form 5 squares from 4 squares in 3 stick moves, we conclude by analytical reasoning that,

- In 3 stick moves, we must reduce 2 squares and create 1 new square. Net result will then be 4 squares in the solution.
- In one stick move, we must free up a single stick and destroy a square without any stick hanging to be taken care of later.
- In second, what we call compound move, we must free up two sticks and reduce or destroy one square. In addition, there must be exactly one stick hanging.
- The three sticks freed up will then form three sides of a new square. The base of the new square will be the free hanging stick created at the previous step.

The first conclusion is obviously true.

What about the second conclusion? At this point only, you think of the fact that we have also to eliminate 4 common sticks. If you take away the stick on the **upper side of square B, which is the blue stick in the first solution**, you **eliminate 3 common sticks in a single move.** This move achieves so much that in every solution this must be a part.

Now you have to reduce one square and eliminate one common stick. The only square that is feasible to be destroyed is square E. Not any other square because it would create more complications rather than moving towards the solution.

This is why the third conclusion is also true in abstraction.

The fourth conclusion is the action that follows from the earlier steps automatically and completes the solution.

Only two solutions can then be there by creating the new square on either side of square D.

Not satisfied? Okay, let us return to our original approach of End state analysis.

### Finding out if there is any more solution by End state analysis

What is the *first condition of forming a possible solution of 4 squares made up of 16 sticks in our problem?*

We know that *the squares must be independent, having no common stick between any two.* But in addition, **the squares must also be corner-connected.**

If you think a bit, this becomes obvious. The untouched squares are corner-connected in the problem figure and will remain corner-connected in the solution.

Think again. How many such possible configurations you can imagine besides the two already in the bag?

There can only be two more such *rotationally unique configurations,* and no more. The two are shown.

Compare each with the problem figure. How many common squares do you find?** For each only two are common**, not three, isn’t it? That’s why the first two solutions earlier are the only ones that are there.

Each has 3 squares common with the puzzle figure—the maximum. No

end stateorpossible final figurehaving thedegree of similaritywith problem figureless than the maximumcan be a solution.

### Unique solution in 3-dimensional rotation, including horizontal or vertical flipping

The two solution figures that you have found are unique if you consider rotation on the PLANE OF THE PAPER. But if you consider 3-dimensional rotation including horizontal and vertical flipping, one solution becomes same as the other.

You will have *only one unique solution in 3-dimensional rotation.*

Imagine a **vertical axis** through the center of the middle square. If you rotate the second solution about this vertical axis by 180^{0}, you will get the first solution. **This rotation is equivalent to horizontal flipping.**

This is shown in the following figure.

#### End note

Last, **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.

*The way to the solution, the approach, the thinking are more important than the solution itself. The concepts and methods stay with you and are enriched as you solve more and more problems.*

And you can take even a short break of fifteen minutes to create a new puzzle of your own and spend the time solving it. If you do it regularly it will sharpen your pattern based problem solving skill, which is an extremely valuable skill.

### Know how to solve difficult problems easily without wasting time on random attempts

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