## Matchstick star puzzle

In the matchstick star puzzle figure, move 2 matches to form 6 and only 6 triangles. Each matchstick in the solution must be a part of a triangle.

How many solutions can you find?

**Recommended time to solve** the puzzle: 25 minutes.

This is an interesting puzzle and you should try to solve the puzzle before going through the solution.

### Solution to the matchstick star puzzle - move 2 matches to create 6 triangles: Getting an idea of the solution figure

Before selecting matches to move we will first form an idea of the solution figure.

**How many triangles comprise the puzzle figure?** You may say 6. But that will be wrong. You have missed the two large triangles of three-stick length sides—one upright and the other inverted. The six small triangles are formed by the intersection of these two large triangles only.

The total number of triangles in the puzzle figure is then 8—six small 1-stick side triangles and two large 3-stick side triangles.

We have to move 2 matches to reduce the number of triangles by 2. There must be exactly six triangles of any size in the solution figure.

We will make a small experiment to move two corner sticks of a small triangle first.

### Solution to the matchstick star puzzle - move 2 matches to create 6 triangles: Making a small trial and conclusions

If one stick of an external triangle is moved, the other stick must be moved as well.

What is the net result of the movement?

**Count the number of triangles.** That is the most important parameter here. From 8 triangles earlier, you are left with 6 now. Two triangles, one large and one small, are destroyed.

Wherever you move these two sticks, a new triangle will be created and the total number of triangles will rise to 7.

From the trial we can conclude,

Conclusion 1:No matchstick from an external small triangle can be moved, especially because, if one stick from an external triangle is moved, one stick becomes hanging and you have to move it too.

Conclusion 2:Two of the 6 internal sticks numbered 1 to 6 can only be moved.

We may also conclude,

Conclusion 3:By moving two internal sticks MORE THAN TWO TRIANGLES MUST BE DESTROYED.

Why? Simply because, with two sticks moved you will create a new triangle. Unless you destroy at least 3 triangles from 2 stick moves, the final solution figure can’t have 6 triangles.

**Realize that this last conclusion is the most important one.**

### Solution to the matchstick star puzzle - move 2 matches to create 6 triangles: How to destroy more than 2 triangles by 2 stick moves: second trial

If you move sticks 1 and 2 how many triangles are destroyed? The two small triangles will surely be destroyed and along with the two, the two large triangles will also be destroyed. Total reduction in number of triangles will be 4.

Now the challenge is changed to creating two new triangles by the two moved sticks.

Can you place these two sticks moved to create two new triangles?

How about the following figure?

Instead of forming one new 1-stick side triangle, we have adopted a new approach of forming a two-stick side triangle. But still number of new triangles created is one and the total number of triangles 7.

We may conclude,

Conclusion 4:No two adjacent inner sticks can be moved.

Observe that we are making trials, but **following a systematic approach.** By the same approach, **next we will move sticks 2 and 6, the two sticks separated by one stick.**

### Solution to the matchstick star puzzle - move 2 matches to create 6 triangles: Third trial produces the first solution

For ease of explanation, the star puzzle figure with the inner sticks numbered is shown again.

Here we have moved stick 2 and 6. As these two are parts of the same inverted large triangle, the **number of triangles destroyed is 3 not 4.**

Solution is easy. *Create a 1-stick side new triangle inside or outside.*

In both these solutions, two stick moves destroyed two small triangles and one large triangle, whereas in the first solution on the left, the new 1-stick side triangle is created outside and in the second solution, the new triangle is created inside. Total number of triangles in both cases is 6, five small and one large.

You may place the new small triangle at different locations, but the main classes of solution are two—the new triangle outside and the new triangle inside.

### Move 2 matches to create six triangles puzzle: Can there be any more solution?

With the same systematic approach, **we will now move stick 2 and stick 5, with a separation of 2 sticks.** This move also creates a solution, as shown.

The **two moves destroyed four triangles, two small and both the large ones.** When we placed the two moved sticks horizontally as shown,** in a single stroke two new triangles of 2-stick side length are created.** Total number of triangles becomes six as desired.

This solution represents the third type where 4 small triangles and two 2-stick side medium-sized triangles make up the solution.

Because of the systematic approach, **you get the exhaustive view of the possible solutions.**

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