Move 3 sticks in the T-shaped 5 square figure and form 4 equal squares
The stick puzzle
Part I: Move 3 matchsticks in the T-shaped figure made up of 5 squares and reduce the number of squares to 4 with no stick left hanging.
Part II: How many unique solutions can you find?
Recommended time is 10 minutes.
This puzzle is interesting but not difficult.
Enjoy solving flawlessly!
Solution to the stick puzzle: Move 3 sticks from the T-shaped 5 squares and form 4 equal squares
Analysis of the structure and knowing precisely what you have to do
At the first step, you need to count total number of sticks as 16. These sticks form 5 squares.
How many sticks are required to form a square? It is 4. So to form 5 independent squares 20 sticks would have been used—4 more than 16 sticks we have.
How could 5 squares have been formed even with number of sticks 4 less than required 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 the shapes inependent of each other by ONE.
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.
Each common stick reduces the requirement of maximum number of sticks by 1.
If you are aware of the common stick concept, just skip the next section.
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 first concept to know—the concept of common stick between two unit shapes.
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 common sticks and form 4 independent squares by moving just 3 sticks.
At this stage, you have precisely and clearly understood what you have to do for solving the puzzle.
Third step: Strategic solution by End state analysis approach
We have explained the use of this approach while solving an important 5 square to 4 square puzzle earlier. For the current puzzle, let us briefly go through how to use this strategic approach and solve the puzzle easily and quickly.
In this strategic approach of problem solving in general,
You have to compare the possible 5 square figures that can be considered as final solutions or what we call end states with the given starting problem figure.
The goal of comparison would be to,
Assess or judge how similar are the possible final figure and the problem figure being compared.
And the outcome or result of comparisons between promising final solution figures and the problem figure would be to,
Identify the one most promising final figure having maximum similarity with the puzzle figure. You would actually try to move 3 sticks to reduce number of squares to 4 on this most promising final solution figure.
If you fail with this first chosen promising final figure, you would select the second such solution and try again.
We are showing comparison of one promising final solution with our problem figure below. Remember, in a promising final figure of 4 squares made up of 16 sticks, all 4 squares will be independent from each other with no stick common between any two.
How many squares are untouched in the possible and promising final figure on the right? Three squares, A, C and D would remain untouched if we move sticks in the problem figure to form this possible final figure.
This is the maximum similarity between the puzzle figure and possible solution figure you can achieve, and so you can be more or less sure that the figure on the right indeed is a solution.
Fourth step: Identifying the sticks to move from the puzzle figure to form the possible final figure
This is the stage of actually identifying and moving the sticks, and will be the final stage if the number of sticks to be moved turns out to be 3. If not, you have to repeat the previous step, and this final step again.
Here the puzzle is simple enough so that while analyzing the possible solutions mentally, we have simultaneously gone through the third and fourth steps identifying not only the solution but also which sticks to move.
The 3 sticks to move are identified below by red, blue and green colored arrows.
You could have moved the three sticks in any sequence. This is your first solution.
Okay, the second part of the question now. Is there any more solution?
Can you identify?
A second solution to the puzzle
Now you are very clear about the puzzle figure. It is easy to see that instead of forming the new square on the right, you could have formed it on the left. This second unique solution is shown below.
Any more solution? How can you be sure that there is no more solution to the puzzle?
Proving is always a bit complex.
Going deeper: Reasoning based on structural analysis—process of transformation from problem to solution
As we have to form 5 squares from 4 squares in 3 stick moves, we conclude by analytical reasoning that,
- All in all in 3 stick moves, we must reduce 2 squares and create 1 new square. Net result will then be 4 squares in the solution.
- In one stick move we must free up a single stick and destroy a square without any stick hanging to be taken care of later.
- In second, what we call compound move, we must free up two sticks and reduce or destroy one square. In addition there must be exactly one stick hanging.
- The three sticks freed up will then be used to form three sides of a new square. The base of the new square will be the free hanging stick created at the previous step.
The first conclusion is obviously true.
What about the second conclusion? At this point only you think of the fact that we have also to eliminate 4 common sticks. If you take away the stick on the upper side of square B which is the blue stick in the first solution, you eliminate 3 common sticks in a single move. This move achieves so much that in every solution this must be a part.
Now you have to reduce one square and and eliminate one common stick. The only square that is feasible to be destroyed is square E. Not any other square because it would create more complications rather than moving towards the solution.
This is why the third conclusion is also true in abstraction.
The fourth conclusion is the action that follows from the earlier steps automatically and completes the solution.
Only two solutions can then be there by creating the new square on either side of square D.
Not satisfied? Okay, let us return to our original approach of End state analysis.
Finding out if there is any more solution by End state analysis
What is the first condition of forming a possible solution of 4 squares made up of 16 sticks in our problem?
We know that the squares must be independent having no common stick between any two. But in addition, the squares must also be corner-connected.
If you think a bit, this becomes obvious. The untouched squares are corner-connected in the problem figure and will remain corner-connected in the solution.
Think again. How many such possible configurations you can imagine in addition to the two already in the bag?
There can only be two more such rotationally unique configurations, and no more. The two are shown below.
Compare each with problem figure. How many common squares do you find? For each only two are common, not three, isn't it? That's why the first two solutions earlier are the only ones that are there.
Each has 3 squares common with the puzzle figure—the maximum.
No end state or possible final figure having degree of similarity with problem figure less than the maximum can be a solution.
Unique solution in 3-dimensional rotation including horizontal or vertical flipping
The two solution figures that you have found are unique if you consider roration on the PLANE OF THE PAPER. But if you consider 3-dimensional rotation including horizontal and vertical flipping, one solution becomes same as the other.
You will have only one unique solution in 3-dimensional rotation.
Imagine a vertical axis through the center of the middle square. If you rotate the second solution about this vertical axis by $180^0$, you will get the first solution. This rotation is equivalent to horizontal flipping.
This is shown in the following figure.
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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