## Move 3 matches to make 3 squares matchstick puzzle

Move 3 matches to make 3 squares from 4 matchstick squares in the puzzle figure.

How many unique solutions can you find?

**Total recommended time** is 10 minutes.

It’s not a difficult puzzle and you should enjoy solving it.

Two solutions, both systematic, are presented after the initial analysis. The solutions are,

**Systematic solution by common stick elimination**in details.**Quick solution by End state analysis**, an especially powerful general problem solving concept.

*You should give it a try before going through the detailed solutions.*

### Systematic Solution to the matchstick puzzle: Move 3 matches to make 3 squares

We’ll first take up an analytical approach to arrive at conclusions based on matchstick puzzle concepts and deductive reasoning.

#### First stage: Structural analysis to know what exactly you have to do

In the

first step,you need tocount the total number of sticksand then in the second step, analyze the difference between given puzzle figure and the goal solution figure. Outcome is, clear idea on steps to be taken to reach the solution.

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 the 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 have equilateral triangles. There can be many variations. But the fact remains that,

Each common stick between two unit shapes

reduces the number of sticksrequired 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.*

Now look at our 4 square figure again—this time we have labeled 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 to solve the puzzle.*

*This is the outcome of the 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** for solving this puzzle.

### First solution by common stick elimination technique: Move 3 matches to make 3 squares matchstick puzzle

#### Second phase of puzzle structure analysis

*Your 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 the maximum value of 1. One of the three numbers has to be 2.

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 concept 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 the 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 the 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 to use in forming 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.

If you move any of these sticks, say stick 6, it 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,**

Use of the freed-up stick 4 in square C as the base and forming 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.

The **stick puzzle solution figure** shown.

#### Summary of 3 stick moves: Move 3 matches to make 3 squares matchstick puzzle

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.

#### Special note

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 solve the puzzle quickly.

The detailed analytical reasoning 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 skill-set for identifying and using patterns in general.

### Can we form any other Unique 3 square figure from the figure of 4 squares by moving 3 sticks?

In other words, do we have any more **UNIQUE solution configuration?**

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 equivalen**t as is the third move.

Consider for a moment the logic.

#### Rotational equivalence of four solution figures

The four possible solution configurations are shown.

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 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. Last, 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: Move 3 matches to make 3 squares matchstick puzzle

*The idea about the solution figure that it would have three independent squares is the starting point in this approach.*

Now imagine possible solution figures and compare each with the problem figure,

judging the 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, andthen only identify which sticks to move.

This is kind of **Other way round approach**, starting from the end to select a solution figure first.

**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? 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.

Imagine the second possible promising final solution and compare it with the problem figure shown.

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.**

It is easy to find the sticks to move next—*it will be the 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 will like it more.

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