Matchstick puzzle, 5 squares to 4 squares in 3 moves

Submitted by Atanu Chaudhuri on Sat, 05/01/2019 - 01:45

move 3 matches to make 4 squares

Matchstick Square Puzzle - Move 3 matches to make 4 squares

To solve this matchstick square puzzle made up of 5 squares, move 3 matches and make 4 squares.

The process of identifying patterns and drawing conclusions, gradually moving on to the solution in sure steps, is presented after the puzzle description.

The puzzle: Move 3 matches to make 4 squares

In the figure below squares are formed by matchsticks. Move just 3 matchsticks to make 4 squares with no matchstick unattached to a square.

Recommended time: 15 minutes.

5 squares to 4 squares in 3 moves matchstick puzzle

Comment

Give it a try—it will sure be interesting to solve. In fact, most of the matchstick puzzles made up of squares are fun to solve.

If you wish, you may go through our solution that won't just be the steps.

We will explain the why's and how's of the systematic steps that finally reach the solution.

We firmly believe that,

The process of solving a problem is more important than the solution itself.

If you are aware of the fundamental general concepts of matchstick puzzle solving, you may skip the following section and move straight on to the solution. To skip, just click here.

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.

core concept in matchstick puzzles

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 matchstick square puzzle - Move 3 matches to make 4 squares: Pattern analysis and deductive reasoning approach

Solution stage 1: Initial Problem analysis: Observations

For convenience of understanding the puzzle figure is shown again.

5 squares to 4 squares in 3 moves matchstick puzzle

First, as usual, we count the total number of sticks. The 16 sticks are just enough to make 4 squares of equal size with no stick common to two squares. So,

The 4 squares in the new configuration must be independent of each other with no common stick between two squares.

This is the first important conclusion made out of number of stick analysis.

Second observation is,

Movement of any single stick from the fully formed configuration will destroy at least 1 square.

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

Solution stage 2: Chained reasoning towards solution—conclusions

Third conclusion based on the number of initial and final number of squares is,

To get 4 squares from 5 squares in 3 stick moves, we must destroy in total 2 squares and create 1.

Fourth conclusion follows,

As every stick movement from a fully formed square formation destroys at least 1 square, for moving 3 sticks and 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 and last move again at least one hanging stick will be created, so that 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 with 4 sticks, 1 stick would be left hanging. That's why in moving 2 sticks together we cannot leave any stick hanging.

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

Fifth conclusion follows from the fourth conclusion,

The single square from which the two sticks would be moved out must have the other two sides part of two existing squares.

This will ensure no hanging stick.

Sixth conclusion specifies the only way the solution can be achieved,

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.

We have reached stage where we can easily identify the 2 sticks to move that would destroy one and only one square without leaving any unattached hanging stick.

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

Seventh Conclusion is,

The 2 sticks belonging to same square must be two corner sticks with two remaining sides part of two independent squares.

There are ONLY 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 rotationally unique solution.

The solution is shown below.

5 squares to 4 squares in 3 moves matchstick puzzle first solution

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 the matchstick square puzzle - Move 3 matches to make 4 squares: 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 independent 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.

5 to 4 squares 3 in moves comparing with possible final solution first trial

The squares in the this possible final solution (on the right) are separated horizontally wide and so the possible final 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.

5 to 4 squares in 3 moves comparing with possible final solution second trial

With three squares 'A', 'C' and 'E' common between the two configurations, the similarity between the two is maximum (because if we had 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? That is your decision to make.

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.

5 square to 4 square in 3 moves matchstick puzzle second solution

End note

In both the solution approaches we have proceeded systematically, and not in any random way. This is what we call systematic 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 systematic and 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 make it a habit to solve problems using different approaches and also look for other possible solutions, it would surely 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.