## 2nd Move 3 matches to make 2 squares puzzle

### The Matchstick Puzzle

**Part I:** *In the puzzle figure, move 3 matchsticks to make 2 squares.*

**Part II:** **How many unique solutions 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:** 5 minutes.

Enjoy!

### Comments

The matchstick puzzle figure is not made up of usually encountered complete regular geometric shapes of squares or triangles. It does not have any such single closed shape. And it consists of a fairly large number of 15 sticks.

How should you proceed?

You can go ahead by using trial and error, but usually that takes more time. Instead, if you do a bit of analysis, you can reach at least one solution quickly. And that'll be the solution to the first part of the puzzle.

### Solution to 2nd Move 3 matches to make 2 squares puzzle: Structural analysis and reasoning—Identifying which sticks to move

**First step** of solving any matchstick puzzle is to count number of sticks—it is 15.

Obviously, to make 2 squares out of these 15 sticks,

The

2 squares must be of unequal size—one large square with 3 stick side length and one small square of 1 stick side length.

**This is the first conclusion.**

#### Why not 4-stick side larger square?

Four-stick long sides of a square would require 16 matchsticks. Even if you create the 1-stick side smaller square attached in a corner of the larger square, number of common sides will be 2, and total number of sticks required would be $16+4-2=18$, more than what you have.

And if you attach the smaller square anywhere else inside the larger square, number of common sides will be only 1, and total number of sticks used would further increase by 1 to 19.

**Think over:** Why you cannot use 3-stick side larger square and 2-stick side smaller square. Use similar logic.

#### How can you use 15 sticks to form two squares?

Number of sticks required to make **two such independent squares**, one 3 stick side and the other 1 stick side, would be, $12+4=16$.

As you have 15 sticks,

There must be

1 stick common between the two squaresreducing the number of sticks needed from 16 to 15.

**This is the second conclusion.**

The **third conclusion** is easy to make,

You have to

close the gap of the larger square, AND form the1-stick side smaller squareattached to the middle stick of any side of the large square.If you form the small square at any corner of the large square, there would be 2 common sticks, not 1.

Sticks used would then be 14not 15.

#### Which sticks to move?

*Candidates obviously are the four sticks inside the larger square.*

**It seems,** you may keep stick 1 undisturbed, but *sticks 2, 3 and and 4 you should move* as these are well inside the larger square. These three will be isolated and left hanging if stick 1 closes the gap to complete the larger square.

The following figure shows the four candidate sticks numbered.

Which 3 of the four sticks are to be moved?

#### Final stage of solution: Convenient position of stick 1 helps quick solution

You have already concluded that the **new smaller square must be formed attached to the middle stick of a side of the larger square (so that number of common side is 1, not 2). **

This is what we call as a BINDING CONDITION that cannot be violated.

You have *stick 1 already waiting for the other two sides of the smaller square to be formed.* **So it is convenient not to be moved**.

Solution is now clear,

Move any of the 3 sticks numbered 2, 3 and 4 to close the gap of the larger square, and form the smaller square with the rest two free sticks and already existing stick number 1.

Following figure shows the solution.

**How many unique solutions?**

*For a moment forget the convenience of keeping stick 1 unmoved. Open your mind to all possibilities without any bias.*

If instead of stick 1, you keep stick 2 fixed can you not make another similar solution shape?

Why, yes. You can.

Close the gap of the large square by stick 1 and complete the two other sides of small square by moving stick 3 and stick 4 with stick 2 already waiting as an existing side.

The small square is formed in this case on the left vertical side of the large square.

This will be the second solution.

Well, well, you can also form **two other solution shapes** by **keeping fixed** stick 3 and stick 4 respectively.

With stick 3 fixed, the small square is formed on the bottom horizontal side and with stick 4 fixed, the small square is formed on the right vertical side of the larger square.

There can then be **four solutions that look different.**

**If you don't rotate, you will have 4 unique solutions, but if you rotate any of the solution shapes it'll be same as the other three.**

Rotationally unique solution is only 1.

You have taken care of all possibilities! Exhaustively.

### 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 as well as **interesting pastime.**

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