The puzzle: Move 3 sticks in the 5 square figure and form 4 equal squares
This is the tenth matchstick puzzle with solution.
You have to move 3 matchsticks in a figure made up of 5 squares and reduce the number of squares to 4 with no stick left hanging. In how many ways can you do it?
Recommended time is 10 minutes.
Before this we have solved three more 5 square to 4 square puzzles with different starting configurations.
Even if you have gone through any of those, this puzzle should give you new food for thought. Just give it a try.
One request before we dive into the solution—please try to solve the puzzle before you look through the solution.
Solution to the puzzle: Move 3 sticks from the 5 squares and form 4 equal squares
We'll take up a completely analytical approach to arrive at conclusions based on basic matchstick puzzle concepts and deductive reasoning.
Analysis of the structure and knowing precisely what you have to do
At the first step, you need to always count total number of sticks. It is 16 for this configuration. These sticks form 5 squares.
How many sticks are required to form a SINGLE square? It is 4. So to form 5 squares 20 sticks would have been used—4 more than the 16 sticks we have.
How could 5 squares have been formed even with number of sticks 4 less than the number needed 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 been made up of equilateral triangles. There can be many variations. But the fact remains that,
Each common stick between two unit shapes reduces the number of sticks required to form a figure with the two shapes independent from each other—by ONE.
The 4 common sticks in the puzzle figure reduced the requirement of 20 sticks to form 5 independent squares to $20-4=16$ sticks.
If you know this concept just skip the next section where it is explained in a bit more details.
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.
5 squares formed by 20 sticks is shown on the left of the figure below. In the figure on the right, one stick is common between two squares. The result? Just count. The number of sticks is 19 now—one less than 20.
It is simple common sense, and understanding of this concept clearly is the fundamental requirement for solving any matchstick problem, however complex, easily and quickly—One stick common between two unit shapes reduces maximum number of sticks required by 1.
This is the very first step—understanding the first principle of common stick between two unit shapes, count the total number of sticks and total number of common sticks in the puzzle figure.
Now look at our 5 square figure again—this time we have labelled 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 4 common sticks and form 4 independent squares by moving just 3 sticks.
In this second step you have precisely and clearly understood what you have to do for solving the puzzle.
Solution by analyzing the figure
The 5 square problem figure is now shown below with 4 common sticks check-marked and numbered 1 to 4. Three more sticks are also numbered.
The central square C has each of its 4 sides as a common stick. In the solution, there has to be NO common stick. The first certain conclusion you can reach is then,
In the solution you cannot have this square C—you must destroy it in the process of moving 3 sticks— also destroying 2 squares and creating 1 new square.
This is the key pattern and must-do action identification.
You may wonder—why destroy 2 squares? The answer is simple—you have to destroy square C by moving any one of its side sticks. Say you move the right vertical stick 4 in square C. What happens?
Not only does square C gets destroyed but square D also is destroyed. It is natural.
As each side stick of square C is a stick common to two squares, as soon as you move one of these 4 common sticks two squares are gone.
So by moving the right vertical side stick 4 of square C you have made 3 sticks of square D—sticks numbered 5, 6 and 7 free to move.
Out of these 3 free sticks you can move only two and using these two with the first one moved you have to form the new fourth square.
This is the second certain conclusion.
Which two would you move now and which one would you keep untouched?
You have only one choice.
You have to keep the right vertical stick of square D—stick 6 untouched.
This is the third certain conclusion.
Your task: Check the truth of this third conclusion and find the reasons behind it.
There is your solution below.
The red, blue and green colored sticks of now gone square D are moved to form the NEW fourth square with its fourth side as stick 6 untouched in its previous state.
Not difficult at all. Isn't it?
Point is—the solution is not the most important thing—more important is how you reach the solution—the reasoning, conclusions and the actions.
Okay, this is one solution of the puzzle.
Now you have to find the answer to the second part of the puzzle—is there any more solution(s).
Can we form 4 squares from the given figure of 5 squares in any other UNIQUE way by moving 3 sticks?
As you know various aspects of the puzzle figure quite well now, you would immediately be able to identify three more possible solutions in which earlier numbered sticks 1, 2 or 3 is moved first. The NEW squares would have to be formed on the single untouched stick of square E, B and A respectively.
A possible solution in which the new 4th square is formed on the vertical untouched side of earlier square B is shown below. In this configuration, squares A, D and E are not touched.
Is it a new solution? Give it a thought.
If you flip the first solution figure horizontally by 180 degrees you'll get this figure. This solution is not then a rotationally UNIQUE solution.
To get the other two possible solution figures, you have to rotate the first solution by 90 degrees, clockwise and anti-clockwise.
There in only ONE UNIQUE solution to the puzzle.
You would appreciate that 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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