## The puzzle: Move 2 sticks in the 6 square figure and form 5 equal squares

This is the 9th matchstick puzzle with solution. You have to move 2 matchsticks in a figure made up 6 squares and form a 5 square figure, clean with no stick overlap and no stick unattached to a shape.

In how many ways can you do it?

*Recommended time is 10 minutes.*

Enjoy solving the problem.

By chance if you can't find the solution or feel curious about how we have solved the puzzle step by step, the solution for you follows.

### Solution to the puzzle: Move 2 sticks from the 6 square figure and form 5 equal squares

#### Analysis of the structure and you get assured of less difficulty in solving

At first the problem seems to be complex because of relatively large number of squares and sticks. Do you think the problem would be difficult to solve?

Instead of just looking at the puzzle figure you focus on what figure finally you have to form—*it is just a 5 square figure*. There is a *small difference in number of squares* between the starting figure and the target solution figure. On top of this, now you notice, *you have to move only 2 matchsticks.*

These two facts assure you that the problem should not be difficult to solve after all. This is what we call *confidence boosting assurance of not so difficult solution.*

It helps you to go ahead and solve the problem quickly and easily by boosting your confidence.

#### Analysis of the structure to evaluate possible approaches and deciding on the most promising approach

You know of two main approaches of solving matchstick puzzles—*End State Analysis Approach* and *Common stick analysis approach coupled with deductive reasoning*.

The **first** is, *End State Analysis Approach* in which the *promising possible end configurations are compared one by one with the starting puzzle figure for maximum similarity*. This approach gives you quick and clean solution especially when the

**possible end states are few**.

More frequently this condition is satisfied when the number of sticks are just enough to form independent squares or triangles without any sharing or common stick between two closed shapes, square or triangle.

Two of the notable puzzles of this type we have solved earlier are the * 5 squares to 4 squares in 2 stick moves* and

*.*

**5 squares to 4 squares in 3 stick moves**In this puzzle of ours, the total number of sticks is 18 which is 2 short of maximum number of sticks required to form 5 independent squares. There will then be 2 shared or common sticks in the final solution figure. Imagining possible solution figures may not be easy at all in this case.

So we decide to follow the second approach of common stick analysis coupled with deductive reasoning.

#### Initial conclusions from common stick analysis and deductive reasoning

In the **second approach** the **number of common sticks** in the starting puzzle figure is compared with the number in the solution figure.

The number of common sticks is 6, quite a large number, in the starting figure. And the number of common sticks is 2 in the solution figure.

The figure below identifies the common sticks in the starting 6 squares.

As you know—each common stick reduces the number of sticks required to form 5 squares by 1. As the number of sticks required to form 5 independent squares with no common stick is, $5\times{4}=20$, when 5 squares are to be formed by the existing 18 sticks, $20-18=2$ number of sticks must be common between two pairs of squares.

So it is a bit surprising that *in only 2 stick moves you have to reduce the number of common sticks by 4*.

What does it signify? It implies that at least * in 1 stick move you should target to nullify maximum number of common sticks*—more the merrier.

With this initial knowledge, when you try to find out how you can reduce number of common sticks to the maximum extent with 1 single single stick move, it takes you just a few moments to identify* the square marked "A" contributing as many as 4 common sticks* to the puzzle figure of 6 squares. *If you remove any of the four sides destroying this square, 4 common sticks would be nullified at a single move.*

So you take up* stick move analysis* on this square A that you have identified as your

*attention target*from

**common stick analysis.**You decide not to disturb the stick number 3 and 4 as both would create * 3 orphan sides of a square*—impossible to set right in 1 remaining move.

What about the sticks 1 and 2?

Removing any of these two would create * 2 numbers of orphan sticks*, and you notice in addition,

*in each of these two options, as many as 5 common sticks are nullified.*

*Each of these two choices should result in one solution.* We call these two possible actions as * most promising actions for solution*. We haven't thought through the final solution yet, but feel confident that

*there is no other alternative path to the solution.*

*You decide to select stick 1 first for moving as trial.*

#### Second stage: selecting stick 1 as first stick to move and selecting the second stick to move: First solution

The result of selecting the stick 1 as first move is shown below. The sticks around have been numbered for easy reference.

Now with stick 1 free, you have no other option than to select the stick 5 for second move. Notice that now the figure has only 1 common stick.

The solution is now clear. * Just complete the incomplete square E with the two sticks 1 and 5 moved*. It already had two existing sides, 6 and 7. This is the

**first solution**.

Notice that stick 7 has taken up a *new role of the second common stick so that condition for creating 5 squares with 18 sticks are fully satisfied*.

#### How many solutions?

You already know that if you had moved stick 2 first instead of stick 1, there could have been a second solution. And indeed it is so. The second and last solution by moving stick 2 first and then stick 8 next is shown below.

To us, these are **the only two possible solutions and **we have exercised **many ways technique** in solving the same problem in more than one way.

First and **most critical reason that these two are the only possible solutions** is,

No solution would be possible without destroying the square A we identified in the beginning.

*You can try by destroying any square other than square A first.*

**Try out.** If you can suggest a third or fourth solution, we would be too happy to know about it.

After all, these puzzle solving is not maths!

### Think other way round: Create a new puzzle for your friends

Now that you know clearly how 5 squares can be formed from 6 squares by moving two sticks, it should also be possible to form 6 squares from any of the two of our final solution figures made up of 5 squares, isn't it?

And that would easily be a new puzzle that you may entertain your friends with.

### End note

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.

*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 proceed to 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*,

*an extremely valuable skill*.

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