The puzzle: Move 3 sticks in the 4 square figure and form 3 equal squares
This is the eleventh matchstick puzzle with solution.
You have to move 3 matchsticks in the figure made up of 4 squares and reduce the number of squares to 3 with no stick left hanging.
How many unique solutions can you find?
In how many ways can you solve the puzzle?
To solve three parts of the puzzle problem, recommended time is 10 minutes.
One request before we dive into the solution—please try to solve the puzzle before you look through the solution.
Systematic Solution to the puzzle: Move 3 sticks from the 4 squares and form 3 equal squares
We'll first take up an analytical approach to arrive at conclusions based on matchstick puzzle concepts and deductive reasoning.
First Phase Analysis of the Structure of the puzzle to Know Precisely What you have To Do
At the first step, you need to always count total number of sticks. It is 12 for this configuration. These sticks form 4 squares.
How many sticks are required to form a SINGLE square? It is 4. So to form 4 squares 16 sticks would have been used—4 more than the 12 sticks we have.
How could then 4 squares have been formed even with number of sticks 4 less than the number needed to form 4 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 16 sticks to form 4 independent squares to $16-4=12$ sticks.
Mark the word "independent". No square in a 16 stick figure of 4 independent squares would have any stick common with any other square.
If you know this concept just skip the next section where it is explained in a bit more details.
In our problem, 16 sticks certainly would have been required to form 4 squares, but then, the 4 squares would have had no stick common between any two squares—all 4 would have been free-standing independent squares.
4 squares formed by 16 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 15 now—one less than 16.
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—with awareness of 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 4 square figure again—this time we have labelled the squares by A, B, C, and D, so that we can refer to a specific square.
You have exactly 12 sticks, the maximum number required to form 3 independent squares. This is the key pattern identification.
4 common sticks reduced the maximum requirement for 4 squares from 16 to 12. Our job is then clearly to,
- Eliminate all 4 common sticks and,
- Form 3 independent squares,
- By moving just 3 sticks.
What is not mentioned explicitly in these three tasks is—in the process you also reduce number of squares by 1.
In this second step you have precisely and clearly understood what you have to do for solving the puzzle.
This is the outcome of first phase of analysis of the structure using matchstick puzzle domain knowledge of common sticks.
Note that if you manage to eliminate all four common sticks in three stick moves, the four square figure would automatically be converted to a three independent square figure and vice versa.
This gives rise to TWO POSSIBLE APPROACHES to solve this puzzle.
First Solution by Common Stick Elimination Technique
Second phase of puzzle structure analysis
You target is to eliminate all four common sticks in 3 stick moves, and you'll identify the suitable sticks to move by second phase of Structure analysis.
Without looking at the figure, apply your REASONING to realize that,
You can eliminate 4 common sticks in 3 moves ONLY IF you eliminate at least TWO common sticks in a single move.
This is based on mathematical limitation that you cannot split number 4 into three positive integers with maximum value of 1.
Now look at the 4 square problem figure below with 4 common sticks check-marked and eight other sticks numbered as 1 to 8 for convenience of explaining the solution.
We are now nearly ready to identify the 1 stick to move that eliminates two common sticks at one go. And then identify two more sticks to eliminate two more common sticks.
The important side-effect of these steps is,
Two squares will be destroyed and 1 new square will be created by the sticks moved.
This knowledge also plays an important part in solving matchstick puzzles.
Concept of promising stick
This four square figure is interesting in the sense that, if you move any of the 12 sticks first, at least 2 common sticks will lose their common property.
But can you move any of the four check-marked common sticks? Think over.
If you move any of the four check-marked sticks it will create immediately 4 free-standing sticks to settle later. In remaining two moves this would be impossible to do. So a certain conclusion is,
The first stick move eliminating two common sticks must NOT be a common stick itself.
Which of the remaining eight sticks you would move first?
Again this figure is special because,
Any of the remaining eight sticks numbered 1 to 8 can be moved first as all 8 sticks are equivalent in position and role in the structure. These are the corner sticks. These are the PROMISING sticks.
Move stick numbered 1 first.
This eliminates 2 common sticks and destroys 1 square. In addition, stick numbered 2 is also freed for movement and use without any more side-effect. We may classify stick 2 at this stage as a NEUTRAL resource stick. It is a free stick for use in formation of the new square AND it won't have any effect on the remaining structure.
So Move stick numbered 2 second.
Result is: two free sticks gained, two common sticks eliminated and 1 square reduced.
We have only one stick move left.
With this 3rd stick you have to eliminate 2 more common sticks and destroy 1 more square.
Next conclusion you would make is,
You cannot select any of the four sticks numbered 5, 6, 7, 8 belonging to the two squares adjacent to the square destroyed just now.
This is because, moving any of these sticks, say stick 6, would eliminate 1 more common stick, destroy 1 more square, but would create two free hanging sticks and the situation would be impossible to manage.
Even if you use the stick 5 as the base of the new square and form the other three sides by the 3 sticks available in 3 moves, 1 more common stick will still remain between squares C and D as well as 1 stick common between earlier squares A and B will remain free-hanging.
The only feasible action is,
You have to select for 3rd move any of the sticks 3 or 4 belonging to the square C opposite to the square B destroyed by first stick move.
Select stick 3 in for 3rd move.
Finally then the new square will be formed by,
Using the freed-up stick 4 in square C as the base, form the other 3 sides of the new square E by the 3 sticks 1, 2 and 3 moved.
This new figure will satisfy all conditions of the puzzle and is the solution.
Following is the solution figure,
Summary of 3 stick moves
Select any pair of corner sticks for first and second moves, destroying 1 square, eliminating 2 common sticks and freeing up 2 sticks.
Select for the third move, any of the corner sticks of the square opposite to the square just destroyed.
Using the remaining corner stick of this second square as base, form the three sides of a new square by the three sticks freed up in three moves.
Any experienced matchstick puzzle solver can solve this puzzle in 30 seconds. It is apparently an easy puzzle.
An inexperienced puzzle solver may also select two corner sticks instinctively and then after a bit of thinking go on to solve the puzzle quickly.
The detailed analytical reasoning nevertheless forms the basis or reasons behind each action, even if the action is taken instinctively.
Consciously knowing the reason behind pattern based instinctive actions strengthen the skillset for identifying and using patterns in general.
Can we form any other Unique 3 square figure from the given figure of 4 squares by moving 3 sticks?
In other words, do we have any more UNIQUE solution configuration?
What do you think? Just think for a moment.
Answer is: No, there cannot be any more Rotationally UNIQUE solution because the starting pair of stick moves for all four corners are rotationally equivalent as is the third move.
Consider for a moment the logic.
Rotational equivalence of four solution figures
The four possible solution configurations are shown below.
These are the only four possible solution configurations of 3 independent squares that you can create by moving 3 sticks from the problem figure.
Question is—are these unique?
When two solution configurations are considered as unique, you cannot create one from the other by rotation—the solutions must be rotationally unique.
If you rotate the leftmost clockwise by 90 degrees you would get the second from left. Further 90 degrees clockwise rotation will make the third figure from left. Lastly another 90 degrees clockwise rotation will get you the fourth figure from left.
All these four solution configurations are Rotationally equivalent. You have only one unique solution to the puzzle.
We'll conclude by showing you the second way to solve the puzzle by End State Analysis approach that is our favorite and is the quickest methodological solution.
We have shown the analytical approach because, End State Analysis Approach cannot easily be applied for a number of puzzle configurations, but you can always adapt the analytical approach to any puzzle situation.
Solution by End State analysis Approach
The idea about the solution figure that it would have three independent squares is the starting point in this approach.
Second action would be to imagine possible solution figures and compare each with the problem figure, judging degree of similarity between the two.
Finally, you would identify the possible solution figure that has maximum degree of similarity with the problem figure as the solution, and then only identify which sticks to move. This is kind of other way round approach.
First possible solution figure is shown on the right with the problem figure on the left. Judge the degree of similarity between the two.
How many squares and sides are common between the two? It is clear that only square A and square D are common between the two.
Move two pairs of corner sticks of square B and square C to form the new square E, but it will take 4 stick moves, not 3. This is not your solution.
Proceed to imagine the second possible promising final solution and compare it with the problem figure as below.
This time again squares A and D are common. Any more similarity?
Yes, there is one more similarity—the bottom-most side of square C in puzzle figure on the left remains unmoved in the possible solution figure on the right.
Because of this tiny bit of similarity of an additional element, you can form three squares from 4, by moving 3 sticks, not 4. This possible solution is your actual solution.
Finding the sticks to move next is an easy task—it will be rest of the sticks that are not common or similarly placed between the two figures.
This powerful method is clean, elegant and quick. Possibly you would like it more.
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