## The puzzle: Move 8 sticks in the 5 square figure and form 2 squares

This is the 12th matchstick puzzle with solution. You have to move 8 matchsticks in a figure made up 5 squares and reduce the number of squares to 2.

**In how many ways can you solve the puzzle?** In other words, *what are the different methods or approaches by which you can solve the puzzle?*

The third part of the question is: **how many unique solution can you find? **A unique solution means, if you rotate the solution figure in any way, it will still remain unique. And you have to find ALL such unique solutions.

This third part is not easy.

*To solve three parts of the puzzle problem, recommended time is 30 minutes.*

One request before we proceed to solve the puzzle—please try to solve the puzzle before you look through the solution.

### The Intuitive Solution to the Puzzle by unfocused overview look: Move 8 sticks from the 5 squares and form 2 squares

Look at the puzzle figure below left, absorbing all parts of the graphic simultaneously without focusing on a particular stick. And simultaneously try to mentally move pairs of sticks to the periphery of a three stick long outer square.

This is an exploratory approach and if you can imagine the four stick pair movements from inside the periphery of the three stick long outer square, you will get in 8 moves 2 squares—one large square of 3 stick side length, and the second, small square of one stick side-length.

The Transformation of the puzzle figure on the left to solution figure on the right may be automatic and in a few tens of seconds.

Follow the four pairs of arrows along which the four pairs of sticks are TRANSLATED horizontally and vertically to form the two squares in 8 stick moves. If you can see the solution, you will be able to see it very quickly.

With an overview look on the puzzle figure without focusing on a particular stick, the solution may come to you easily.

This is a powerful way to solve any problem—the intuitive way.

But can you answer the question on number of unique solutions?

#### The Systematic Analytical Solutions to the Puzzle: First Phase: Move 8 sticks from the 5 squares and form 2 squares

Analytical solution of any matchstick puzzle consisting of regular geometric shapes invariably starts with COUNTING of number of sticks and then trying to imagine what kind of figure the solution would be.

For the moment forget that you already know one solution.

The five squares in the puzzle are made up of 16 matchsticks, and you have to make 2 squares out of 16 matchsticks.

But you know that standalone independent two squares require at most $4+4=8$ matchsticks.

*Then how could the puzzle be solved? Is it wrongly described?*

In a moment you discover **the first key pattern**—it seems that there is no way you can form two EQUAL squares from 16 sticks, but yes, it should be possible to form one large square and the second a smaller square.

As the number of sticks to be moved are quite large we won't resort to common stick analysis which is very useful for puzzles with 2 or 3 stick moves.

Instead, we'll do a bit of **number analysis to understand what kind of two squares CAN be formed.**

### Second phase Analysis of the structure of the puzzle to discover nature of shape that will be formed—Number analysis

We'll split 16 into two numbers, one for the number of sticks required for the first square and the second for the second square. At this point we'll assume that the 2 solution squares would have NO COMMON STICKS.

When you start this analysis, it becomes clear to you that,

Number of sticks required to form a square

must be an even integer divisible by 4because all 4 sides will be of same length.

So 16 will be a sum of two numbers, **each a multiple of 4**. We are just analyzing the possibilities by simple and inviolable mathematics.

The possibilities of 16 as a sum of two multiples of 4 are only two,

4 + 12, and

8 + 8.

4 sticks form a square of side length of 1 stick, and 12 sticks form a square of side length 3 sticks. Right?

Continuing this thread of reasoning, you can quickly visualize that the larger square must be the square that will be formed by **NOT MOVING the 4 outermost sticks** **and the four 4 inner sticks forming an existing small square**.

* Move rest 8 sticks *and you will get the solution you got intuitively just now. This is the

**first unique solution.**

#### Exploring the second possibility of 16 = 8 + 8 sticks to form two squares

We'll now explore how the second possibility of 16 sticks = 8 sticks + 8 sticks, can be converted to a solution figure in 8 stick moves.

**Analysis** reveals immediately that a square made up 8 sticks means, each side is made up of 2 sticks, $2\times{4}=8$, isn't it? That is obvious.

Now comes the **second phase of analysis**—what form would the two squares take?

Answer to this question is also simple,

As we have assumed no common stick between the two squares, the two middle-sized squares must be corner-connected and standalone.

It will be no problem for you to imagine such a configuration. And when you form the final solution figure as shown below, with a little bit of trying you should also be able to form this new figure from the given puzzle figure by moving just 8 sticks.

The 8 sticks that are moved are identified by red check marks and these 8 sticks in the old figure are faded out. The sticks not check-marked or faded out are 8 in number and those are the sticks that remained unmoved.

This is the **second rorationally unique soluion.** On four corners of the old figure you can form a new square of side length of 2 sticks, but these four configurations would be equivalent to each other on rotation by multiples of $90^0$. So instead of four unique solutions, you get one unique solution of this configuration.

Without systematic analysis, it is not easy to discover this possibility let alone form this solution.

You may feel the solution is complete. But no, we are yet to answer the troublesome part of the question—**Can we have any more solutions?**

Only when we explore all possibilities and find ALL Solutions with certainty, our answer is exhaustively complete.

### Exploring ALL POSSIBLE solutions to the matchstick puzzle of 2 squares from 5 squares in 8 moves

We'll use number analysis, trial and deductive reasoning to answer this awkward question.

Take the first combination of two squares from 12 + 4 = 16 sticks.

There *could have been a second rotationally unique solution of this combination* with the smaller square located outside the larger square and connected at one of the four corners. The following is such a figure.

Offhand you may think it requires 12 stick moves. But if you try, you should be able to form this figure from the puzzle figure in 10 stick moves, not 8.

This is NOT a valid solution.

What about the combination of 16 = 8 + 8?

We have already found out the only possible rotationally unique solution for this combination. This path of exploration is exhausted.

**But yes, we have not considered the possibility of a solution with COMMON sticks.**

#### Exploring the possibility of solution to the matchstick puzzle 5 square to 2 square in 8 moves with common sticks

Again we would start with number analysis.

If you think a bit you would realize that there can be **only one probable configuration with common sticks** that can be explored for solution to the puzzle—a figure consisting of,

$16 \text{ sticks} = 12 \text{ sticks} + 8 \text{ sticks} - 4 \text{ common sticks}$.

**Number of common sticks must be a multiple of 4**, and for each common stick number of maximum stick requirement will reduce by 1.

*You cannot consider a larger square of side length 4 sticks because that itself would consume all 16 sticks leaving nothing for the second square even after taking account of common sticks.*

This means, the probable solution would consist of a larger square of 3 stick side length and a middle-sized square of 2 stick side length inside the larger square. The two squares would have 4 sticks common between them.

From number analysis you can be certain that there cannot be any other possibility of two squares formed by 16 sticks with a few common sticks between the two.

The following is the **third probable solution figure** made up of a 3 stick side long square and a 2 stick side long square with **4 sticks common** between them.

Can you form this figure from the puzzle figure in 8 stick moves? Go ahead. Give it a try.

You'll find that this also is a solution. We leave this small task forming this figure from puzzle figure in 8 stick moves to you as a job.

**Hint:** Concentrate on keeping UNMOVED maximum number of sticks that would belong to the 3 stick side length and 2 stick side length new squares.

Total number of unique solutions to the puzzle is 3.

We have followed first, intuitive approach using defocused overview look to SEE the solution pattern, and then by a combination of Number analysis, Matchstick concepts, Deductive reasoning and concept based trial, all three possible unique solutions are created.

This is a beautiful puzzle rich in learning potential.

#### End Note

If you form your own matchstick puzzle and solve it **exhaustively** using all methods you know and can create, it will be a richly rewarding experience.

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