## A stick puzzle that provides rich learning in problem solving methods if you give it a serious try

### Preamble

This is the second matchstick puzzle with solution. You have to convert the 5 square figure above into a 4 square figure by moving just 2 matchsticks.

First *we will describe the puzzle precisely* and then explain **three ways to solve the puzzle**. Why three ways? Because,

Finding out many ways to solve a problem improves your ability to think up new promising avenues of solution when you face a new problem. This is the use of

many ways technique.

The **first two solutions take up** similar * pattern based analytical approach*, but with different focus of attention.

In contrast, the third solution approaches the problem from an altogether unusual angle

to reach the solution very quickly.

We have earlier highlighted this approach as * End state analysis approach* which is a general problem solving approach and can be applied in a large variety of problem situations.

### The puzzle: Convert the given 5 square matchstick figure to 4 squares in 2 moves

The following is the five-square figure made of equal-length matchsticks that is to be converted to a 4-square figure made of same matchsticks by moving 2 matchsticks with no overlap of two matchsticks or no matchstick kept hanging independently without being a part of any square. Assume all squares to be of equal size.

Before going ahead, you should try to solve the problem yourself.

**Recommended time is 20 minutes**. But we would say, if you are not able to solve within 30 minutes then you may be going randomly that would take indefinite length of time.

Okay, let's get on with the solutions. First we will state briefly the reason why matchsticks are popular for making and solving puzzles.

If you want, **you may skip this part.**

#### 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.*

### Solution to 5 square to 4 square matchstick puzzle: Core domain concept in formation of geometric shapes made up of matchsticks

The most basic concept that is inherent in any matchstick puzzle made up of geometric shapes is applicable in this puzzle also,

Each stick common to two squares reduces required number of sticks (or sides) by 1 compared to the sticks required for making same two squares independent of each other.

The following figure should make the concept more clear.

This **matchstick puzzle truth**, as we call it, holds for figures made up of matchstick triangles, squares or even regular polygons all of equal sides.

From this commonly applicable concept in solving any matchstick puzzle we will now focus on our 5 square puzzle and present the first solution approach.

### First Solution to 5 square to 4 square matchstick puzzle: Pattern analysis and Requirement specifying approach

#### Problem analysis: Stage 1: forming the broad solution requirement by key pattern identification

The crucial pattern we identify based on the core concept explained above is,

We have five squares with 16 sides which would just be enough for 4 squares with no common sides.

So in the desired configuration of four squares, **there can't be any common side between two squares.**

This is the broad solution requirement specification (or property of the solution configuration).

#### Problem analysis: Stage 2: First conclusion towards solution requirement specification by deductive reasoning

Our **first conclusion** that follows from the broad solution requirement specification is,

By moving two matchsticks,

we must remove all the four common sides in the initial configurationto fulfill the broad solution requirement specification.

This is simple deductive reasoning.

#### Problem analysis: Stage 3: Second conclusion by configuration pattern analysis and deductive reasoning

Now only we look deeper into the structure of the initial puzzle configuration and analyze the relations between the four common sides to determine how we can remove all the four in two moves.

The outcome of this analysis is the **second conclusion**,

As in one stick move more than two common sides can't be nullified, each of the two stick moves must nullify two and only two common sides.

**Reason:** The **crucial first part of the second conclusion is true** because, a matchstick has two ends, and so taking it out can change the status of at most two sticks connected to its two ends from **common side stick** to **just another side stick**. There could have been another pair of sides from its two ends on the other side, but then that would have made the matchstick a common side stick, which is prohibited from moving. **Make your own stick structures** to understand or disprove this logic.

#### Problem analysis: Stage 4: Understanding how the four common sides can be nullified

Mark that we are now using the phrase "nullifiy a common side" rather than "remove a common side". This is because, **if we remove a common side** two squares get affected with number of squares reducing by 2 and creating at least 4 sides not being a part of any square. This is a far too destructive action to be rectified in any way towards the solution with only one more stick move remaining.

Thus we form the **third conclusion** towards the solution,

In any of the two moves,

none of the four common sides can be moved.

That's why we use the phrase "nullify a common side" modifying the initial target phrase, "remove a common side".

How can we nullify a common side by moving a matchstick? The obvious answer is—by destroying a suitable square adjacent to another square, the two having a common side. An example is shown below,

With this knowledge, now we examine the initial puzzle configuration and form the **fourth conclusion**,

We must destroy one suitable square and nullify two common sides in each of the two stick moves.

And then the **fifth conclusion**,

As the two moves together must nullify a total of 4 common sides, the two squares chosen for destroying can't be adjacent to each other.

This is because if the two chosen squares are adjacent, the number of common sides would be reduced by 1 and total common sides nullified would have been 3, not 4.

#### First Solution to 5 square to 4 square puzzle: Stage 5: Identification of two suitable squares to be destroyed

Brief examination of the puzzle configuration yields the only two possible squares to be destroyed nullifying four common sides in the process.

Destroying each of these two squares 1 and 2 nullifies two common sides with two adjacent squares in each case. We are now very near to the solution. The only task left is to identify one suitable stick from each of the chosen squares and form a third square by the two sticks thus freed up. This additional square will make the number of squares exactly 4 (5-2+1=4) as required.

#### First Solution to 5 square to 4 square puzzle: Final stage: Identification of two suitable sticks and moving the two to form a new square

In each of the two chosen squares two possible sides qualify for moving. Which one to choose?

At this last stage we apply deductive reasoning and **identify the prime condition for last choice**, the **sixth conclusion**,

To form a new square with two free sticks there must be two already existing sides

that are not a part of any existing square(otherwise a new common side would be created).

The final **seventh conclusion** is a corollary of the sixth,

The two existing freely hanging sides of an incomplete new square must be the result of moving the two sticks earlier.

These two conditions help to choose the two sticks and form the new square in no time without any confusion at all. The final solution is shown below.

Observe that the **focal point of the solution** was, "**how to make number of common sides zero**". The steps followed systematically by using deductive reasoning and pattern identification.

In the second solution we will use a different approach that is somewhat similar to the first solution approach, but with a different focal point of attention.

Instead of focusing on the key property of the final solution configuration, we will analyze the **key requirements of each of the actions of stick movement** in the second solution.

### Second Solution to 5 square to 4 square puzzle: Action requirement specification based on pattern analysis and deductive reasoning

As mentioned, in this approach we will analyze and form the **specification of actions for movement of the sticks**. In other words, we will gradually narrow down the type of sticks that can be moved to fulfill the puzzle conditions.

Let us show the puzzle configuration with squares labelled for ease of explanation.

#### Problem analysis: Stage 1: Broad action requirement specification

Recognizing that moving one stick from any part of the given configuration will destroy at least one square (moving a common stick destroys two squares), and moving two sticks from same square won't finally reduce the number of squares by 1, we make our **first conclusion**,

The two sticks that can be moved must belong to two different squares and none of the two sticks can be a stick common to two squares.

Moving out a common stick frees up at least four sticks and the large change can no way be restored to the desired solution by one single remaining stick movement.

The **second conclusion** follows as,

With each stick movement we need to destroy (or reduce) 1 and only 1 square, so that with these two sticks freed in two moves, we can form a new square, making the total number of squares as 4.

Together, these two form the **essential but broad action requirement** for solving the puzzle.

*In next stage of analysis we would find this requirement to be not enough. But it is easier to proceed with a lightweight simple requirement than a more complex one.*

#### Problem analysis: Stage 2: Defining the nature of the two sticks that can be moved: More specific requirements

Looking deeper into the effect of the two stick movements, keeping in mind the need of forming a new square with two freed up sticks, we make the **third conclusion**,

After moving and freeing two sticks, a new square can only be formed if

each stick movement leaves exactly zero or one stick hanging unattached as a side to any square. Also, after moving, there should be two sides of an incomplete new square available for creating the new square with the two freed up sticks.

This is a very specific requirement drawn from the earlier requirements, pattern analysis and deductive reasoning.

**Reason:** If no sticks are left hanging because of moving out two sticks and destroying two squares, we cannot create a new square with two freed up sticks unless at least one pair of sides of an incomplete new square and that are part of other squares is available. In this case when the new square is formed with the two freed up sticks, common sides are invariably produced which is prohibited in our puzzle. This special case can arise **in a new puzzle** but not in our given puzzle. **Make such a new puzzle using matchsticks.**

If two sticks are left hanging unattached as sides of any square, but forming two sides of an incomplete new square, the two freed up sticks can conveniently complete the new square. **This must be the case for our puzzle.**

If more than one stick is left hanging by any of the stick movement, total number of sticks left hanging would be more than two and a new square can't be formed in that case.

*Make your own matchstick configuration to understand this reasoning chain.*

As in the given puzzle, movement of two sticks cannot leave zero number of hanging sticks, we draw the fourth conclusion as,

The two squares from which two sticks are to be moved must be connected at a corner.

#### Second solution to 5 square to 4 square puzzle: Identifying the specified pair of squares and sticks

Such a pair of squares can only be squares B, D or C, E. The C, E pair does not satisfy the stringent condition in the third conclusion (moving any eligible side in square E creates not 1, but two freely hanging sides). And it is short work to the solution to identify the vertical left side in square D and vertical right side in square B to be moved to form a new square as shown in the last stage of the first solution.

This approach though a bit heavy analytically, solution can be reached faster.

We will now end the session with the third approach that is completely different from the first two.

### Third solution to 5 square to 4 square puzzle: by End state analysis approach

In this approach we won't go into the structural details of the puzzle configuration. Instead, we would identify the key pattern requirement for the final solution and comparing the initial configuration with possible final configurations derived from the key pattern, we would choose the final configuration that has maximum similarity with the initial configuration.

This approach is justified as, converting the initial configuration to the possible final configuration with maximum similarity should involve minimum number of stick moves.

#### Third solution: Stage 1: Pattern identification and Formulation of the key requirement for the final configuration

We have already gone through this key requirement but will repeat it for ease of understanding and completeness. The key requirement 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 between 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.

#### Third solution to 5 square to 4 square puzzle: Stage 2: 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. This conclusion is based on visual assessment.

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

With three squares similar between the two it is easy to identify the two sticks that are to be moved to create a new square.

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

**Question:** Which of the three approaches do you like and why?

#### Number of solutions

We have shown three approaches to reach one same solution. These are the solution approaches, or ways to the solution.

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

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

### End note

In all the solution approaches we have proceeded systematically, and not in any random way. This is what we call **systematic approach to problem solving**.

Generally, systematic approach to problem solving depends heavily on **identification of key patterns**, **creation of effective methods** and **deductive reasoning** to move towards the solution without any confusion as well as in minimum number of steps, if possible. That's why we call this as **efficient problem solving**.

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.

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