## Move 3 sticks in the figure and form 2 squares

### The Puzzle

**Part I:** *Move 3 matchsticks in the figure shown and 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 limit:** 5 minutes.

Try to solve the puzzle before you go through the solution. It will be fun, we assure.

### Comment

The puzzle is different from the stick puzzles we have solved till now. The matchstick puzzle figure is not made up of complete regular geometric shapes of squares or triangles. The figure is incomplete and unbounded.

How should we proceed?

You can go ahead in 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. That's the first part of the puzzle.

### Solution: Structural analysis and reasoning—Identifying which 3 sticks to move and where to move

#### Analysis of number of sticks used and conclusions drawn

First step of solving any matchstick puzzle is to count number of sticks—it is 10.

Obviously, to make 2 squares out of these 10 sticks,

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

This is the **first conclusion.**

**Think over:** why can't you use the larger square of 3-stick sides.

Number of sticks required to make **such two independent squares** would be, $8+4=12$.

As we have 10 sticks,

There must be 2 sticks common between the two squares reducing the number of sticks needed from 12 to 10.

This is the **second conclusion.**

It follows immediately that,

The small square cannot be left as it is outside the larger square as it would have only 1 stick common with the larger square and you cannot in any way complete the larger square.

That means—you have to move all three sticks of the small fully formed square in the puzzle figure

excluding the vertical stickthat is the part of the incomplete larger square.

This is the **most important third conclusion.**

This settles first issue—**which sticks to move.** You now know for sure that you have to move the three sticks of the small complete square leaving the vertical stick on the right as it is.

The following figure shows the results of reasoning achieved till now. You are now sure that the check-marked sticks are the sticks you have to move.

We have used many words to reach this point, but when actually solving the puzzle, you can reach this conclusion very quickly—by reasoning or by instinct.

A byproduct of this knowledge is—you cannot move any stick other that the three check-marked ones.

#### Final stage of solution: Where to place the 3 sticks?

Two places are to be filled up surely to complete the larger 2-stick side square, isn't it? That's easy to see.

But there are two possible places to move the 3rd stick for completing the second square.

Any of these two would be a solution.

Following figure shows the solutions.

You can easily find the two possibilities of placing the three sticks moved. First solution has the small square on the top right corner, and in the second solution it is on the bottom right corner. There can't be any other possibility because of the initial independently hanging stick at the right middle.

But the question is—are these two solutions rotationally unique?

#### Second part: How many rotationally unique solutions?

Just rotate the first solution figure by 90 degrees clockwise and you will get the second solution. These two are equivalent. **Unique solution is only one.**

#### 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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