## This time again 5 to 4 square puzzle, but it's a different story now

This is the third matchstick puzzle with solution. You need to transform a 5 square configuration of matchsticks to 4 square configuration again, but this time you move 3 sticks without any stick overlap or hanging.

In the first section we will describe the puzzle. This will be followed by analytical solution.

Major part of the solution will be observation and analysis that will quickly move onto the solution.

### The puzzle: Convert the given 5 square matchstick figure to 4 squares in 3 moves

The following is the five-square figure made up of equal-length matchsticks that is to be converted to a 4-square figure made up of same matchsticks by moving 3 matchsticks with no overlap of two matchsticks and no matchstick kept hanging independently without being a part of any square. Assume all squares to be of same size.

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

**Recommended time is 15 minutes**. But we would say, if you are not able to solve within 20 minutes then you may be going randomly that would take indefinite length of time.

Okay, let's get on with the solution. First we will first go through the most fundamental concept in a figure of geometric shapes made up of matchsticks.

If you are aware, you may skip the section and move on to the solution directly.

### Fundamental concept in a figure 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.

### Solution to 5 square to 4 square matchstick puzzle in 3 moves: Pattern analysis and deductive reasoning approach

#### Solution stage 1: Initial Problem analysis: Observations

**First,** as usual, we count the total number of sticks 16 just enough to make 4 squares of equal size with no stick common to two squares. The 4 squares in the new configuration must be independent of each other. Any stick common to 2 squares reduces the stick requirement by 1. For example, to form 5 independent squares we would have needed 20 sticks. But as 4 sticks are common between two squares here, 16 sticks became sufficient to form 5 squares.

**Second,** movement of any single stick from the fully formed configuration will destroy at least 1 square.

**Third,** if a common stick is moved, 2 squares will be destroyed with at least 5 sticks not forming any side of a square. This move is thus impracticable and we won’t consider it for solution.

#### Solution stage 2: Chained reasoning towards solution—conclusions

**First:** to get 4 squares from 5 squares in 3 stick moves, we must destroy in total 2 squares and create 1.

**Second:** As every stick movement from a fully formed square formation destroys at least 1 square, for 3 stick movements to destroy 2 squares, it follows that 2 sticks are to be moved to destroy one and only one square without leaving any hanging stick.

*If this move creates a hanging stick*, in total 3 sticks would become free for placement in this move. In the third move again at least one hanging stick will be created, and a total of 2 sticks would become free in this move. After two moves then we would have 5 sticks free for placement. Even after creating the new square 1 stick would be left hanging. That's why in the move of 2 sticks together we cannot leave any stick hanging.

This conclusion we identify as the **focal point of the solution** or **key pattern identification**.

**Third:** In the third move, 1 stick will be moved destroying one square, but it will also leave a third side hanging, that will fit into the new independent square. Three sticks moved will be three sides and the fourth side will the stick hanging in the second move.

**Fourth:** The single square to which two sticks selected for movement belong, must have the other two sides part of two existing squares. This will ensure no hanging stick.

#### Solution Stage 3: Observation: Identification of 2 sticks to be moved from same square

**Conclusion:** So, the 2 sticks belonging to same square must be two corner sticks with two remaining sides part of two independent squares. There are 2 such possibilities.

#### Solution Stage 4: Final solution

Either move two corner sticks from upper left square or from the lower left square. Let us select the two corner sticks from upper left square. Select the third side to be moved as the lower horizontal stick of the lower left square. With three free sticks form three remaining sides of a new square with left vertical stick of the erstwhile lower left square as the fourth side.

As the figure is horizontally symmetric, movement of first 2 sticks in two ways will contribute to only 1 unique solution.

The solution is shown below.

As decided we have selected and moved the two sticks from square 'B', and lower horizontal stick from square 'D' thus destroying both the squares. These three free sticks with the left vertical stick of erstwhile square 'D' form the new square 'F'. In three stick moves we have transformed the 5 square puzzle configuration to 4 square configuration.

**Question:** Do you think this is the only solution configuration?

We will answer the question later. For now we will adopt a second powerful approach altogether different from the first analytical approach and solve the puzzle again.

Let us see how.

### Second approach to solve 5 square to 4 square by 3 moves puzzle: by End state analysis approach

In this approach we won't go into analyzing the 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**.

#### Second solution approach: 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.

This is what we call **solution requirement specification**, a set of binding conditions that the solution must satisfy.

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*.

#### Second solution approach: 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 this possible final solution (on the right) are separated wide horizontally and so the configuration is very dissimilar with the initial puzzle configuration. Only two squares 'A' and 'C' can be considered common between the two configurations. No way can we convert the puzzle configuration to this possible final configuration by moving three sticks. 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 'A', 'C' and 'E' common between the two configurations, the similarity between the two is maximum (because if we have 4 squares common, the puzzle becomes unsolvable). So we examine how to move three sticks from initial configuration to reach this final configuration and arrive at the same solution as before.

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

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

#### Number of solutions and solution approaches

We have shown two approaches to reach the same solution configuration. These are the solution approaches, or ways to the solution.

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

The answer is yes, of course. If you examine further possibilities, especially the second possibility of moving 1 stick from square 'D', you will have the second solution configuration immediately.

Instead of selecting the lower horizontal stick of square 'D' we would select left vertical stick.

The second solution is shown below.

### End note

In both 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**.

We have *solved the same problem adopting two approaches as well as discovered two solution configurations to the puzzle*. This is use of **Many ways technique** in two dimensions—in approaches and in solutions. If you can make it a habit to solve problems using different approaches and also look for other possible solutions, it improve your ability to discover new possibilities in any problem situation. This practice gives you great gains in terms of improved innovative skills.

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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