## The puzzle: Remove 4 sticks from a hexagonal wheel and form 3 equilateral triangles

This is the sixth matchstick puzzle with solution. You have to remove 4 sticks from a hexagonal wheel and form 3 equilateral triangles. The puzzle is not difficult and so recommended time for solving is 10 minutes.

This puzzle is interesting but relatively easy. So we recommend 10 minutes for solving the puzzle. We assure you that you will enjoy solving the problem.

By chance if you can't find the solution in time or feel curious about how we have solved the puzzle and what are the reasons behind our selecting the four sticks to remove, then you should go ahead and go through the solutions.

### Solution to the puzzle: Remove 4 sticks from the hexagonal wheel and form 3 equilateral triangles

Total number of sticks is 12 and the figure is perfectly symmetric. Without any analysis **quickly we make an experiment to see how 3 equilateral triangles can be formed**.

That is easy. We have just removed the three marked sticks, all in the periphery, and formed 3 equilateral triangles. It is not the solution though—we should have removed one more stick to form three triangles. The experimental figure is shown below.

So the puzzle can't be solved in a few seconds, we have to go for requirement analysis to form 3 triangles. No random method for us.

We have to **analyze the total number of sticks and number of common sticks in the final solution**.

#### Requirement analysis based on number of sticks and number of common sticks in the final figure

After we remove 4 sticks, the final figure will have, $12-4=8$ sticks. As independent 3 equilateral triangles require $3\times{3}=9$ sticks, conclusion is—**one stick must be common between two triangles so that requirement of number of sticks is reduced by 1.**

The concept of 1 common stick reducing the required number of sticks by 1 is explained in the section below. If you already know the concept, just skip the section.

#### Concept of common stick between two triangles

The following figure shows two independent triangles formed by 6 sticks on the left joined to form a single structure of two triangles separated by one common stick on the right.

The requirement of number of sticks to form the two triangles is reduced from 6 to 5 because of the single common stick between two triangles. Being a common side, it plays the role of two sides of the two triangles. This reduces the actual requirement of number of sides by 1.

Thus the requirement of number of sticks to form the two triangles is reduced from 6 to 5 because of 1 common stick.

Coming back to our puzzle, first conclusion is—**we have to form three triangles by 8 sticks, so that one stick will be common between two triangles**. In our first experiment, the three triangles formed were independent with no common stick. That's why it couldn't be the solution.

All the three sticks we have removed in the experiment were from the periphery. This approach won't do, we have to remove at least one stick from inside the figure, that is, remove a spoke of the wheel.

Moreover, **thinking from the standpoint of destruction of triangles by removing sticks**, in the experiment we have destroyed just 1 triangle by each stick removed. To solve the puzzle,

We have to destroy two triangles by three sticks removal and rest 1 triangle destroyed by 1 stick removal.

This **second conclusion** is the **requirement specification for action** and the puzzle is nearly solved.

Just

remove the left hand side two triangles formed by 3 sticks.That will be thecrucial first move. This move destroys 2 triangles.

** The fourth stick removal in second move is straightforward. Remove the bottom right stick on the periphery destroying the 3rd triangle.**

The following is the figure where 4 sticks to remove are identified by arrows.

We have selected the right bottom stick on the periphery as the 4th stick to remove. **Instead, we could have chosen the stick on the upper right on the periphery.** This would have given us the **second solution**. The two solutions are nearly similar though.

The following is the **final solution figure**. The sticks removed have been faded to give an idea how the final figure is formed.

In this puzzle, we have thought in terms of removing **three sticks in 1 move**. This is **group move** concept.

We **could have removed the three sticks in first move in 6 ways**—but as the hexagonal wheel is rotationally symmetric, all these choices won't have created any new solution rotationally.

**Task for you:** Find out how you could have chosen the three sticks in first move in six ways.

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