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

### Puzzle

**Part I:** Move 8 matchsticks in the figure made up 5 squares and reduce the number of squares to 2.

**Part II:** **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.*

**Total recommended time:** 20 minutes.

Try to solve the puzzle before you go forward. It'll be interesting for sure.

### Part I: Immediate intuitive Solution to the Stick Puzzle by bird's eye view look

Look at the puzzle figure, 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, a 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.

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

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

### Part II: Number of unique solutions: Systematic Analytical Solutions to the Puzzle

You'll now stop for a moment and reorient your mind to analytical mode.

#### First Phase: Number analysis: 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.

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

First conclusion is easy to make—*it seems* there is no way an equal sized two-square solution from 16 sticks can be reached, but yes,

It should be possible to form one large square and the second a smaller square from given figure in 8 stick moves.

As the number of sticks to be moved are quite large, avoiding common stick analysis, you'll resort to **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

First truth in forming squares,

Number of sticks required to form a square

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

Now,

Split 16 into two numbers, number of sticks required for the first square and the second square. Assume that the 2 solution squares would have NO COMMON STICKS.

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

$16=4 + 12$, and,

$16=8 + 8$.

**First combination of 16=4 + 12:** One 1-stick side square plus a second 3-stick side square.

**Second combination of 16 = 8 + 8:** Two numbers of 2-stick side squares. Right?

At the moment keeping the second possibility of equal sized two squares stored away in memory for later use, continue the thread of reasoning for unequal size of squares and visualize that,

The larger square must be the square that will be formed by

NOT MOVING the 4 outermost sticksand the four 4 innermost sticksthat form an existing small square.

* Move rest 8 sticks *and you will get the solution you have reached intuitively in the beginning. This has been your

**first unique solution.**

On this occasion, you have reached the same solution but through another, a bit slower but more assured, path of analytical reasoning.

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

Now explore how the second possibility of 16 sticks = 8 sticks + 8 sticks, can be utilized to form a solution figure in 8 stick moves.

Truth is, 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.

But **what relative position would the two squares take?**

Answer to this question is also simple,

As there must be

no common stick between the two squares(all 16 sticks having been used up in making the sides of the two squares), the two mid-sized squaresmust 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 below,** with a little bit of trial you should also be able to form this new figure from the given puzzle figure by moving exactly 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 solution.**

*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, it still remains to answer the troublesome question—**Can we have any more solutions?**

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

### Exploring ALL POSSIBLE solutions: Trial with possible solution configurations

Now you have to use number analysis, trial and deductive reasoning to answer this relatively difficult question.

**Strategy adopted,**

Form possible solution configurations by number analysis and matchstick puzzle concepts and explore whether the possible solution figure can be reached from the given puzzle figure by moving just 8 sticks.

First take the combination of two squares with 12 + 4 = 16 sticks.

There *could have been a second rotationally unique solution for 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 a **minimum of 10 stick moves**, not 8.

**This is NOT a valid solution.**

What about the combination of 16 = 8 + 8?

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

But yes, you have not considered the possibility of a solution with

COMMON sticks.

#### Exploring the possibility of solution to the matchstick puzzle 5 squares to 2 squares 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 square and another 2-stick side 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 of forming the solution figure in 8 stick moves to you.

**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.

So, **total number of unique solutions to the puzzle is 3.**

### Summing up

1. First,you have followed intuitive approach using defocused bird's eye view to SEE the solution pattern,

2. Second,by a combination of Number analysis, Matchstick concepts, Deductive reasoning and concept basedtrial on Possible solutions, all three unique valid solutions are reached.

This is a beautiful stick 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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