## 6 triangles matchstick puzzle

In the 6 triangles matchstick puzzle figure move 2 matches to make 5 triangles. How many solutions can you find? Total time to solve 20 minutes.

### Preamble

Matchstick puzzles fascinated young and old for ages. Sticks are easily available, shapes can be made without any effort, and if you are curious, you can make a structure with many matchsticks and see what happens to the structure if you remove, move or add one and then more than one stick.

The first matchstick puzzle with its solution we'll present now with the **dual objectives** of;

- Offering you the satisfaction of solving the puzzle once more, and,
- If you like to go through the solution, showing you how a problem puzzle can be solved in
**systematically using analysis and reasoning.**

### The 6 triangles matchstick puzzle: Move 2 matches to make 5 triangles

#### First part of the puzzle

There are six triangles as in the following figure made up of matchsticks. You have to MOVE 2 (two) sticks to new positions and transform the structure to a five triangle structure.

You cannot throw away any stick, add any new stick and in the new structure five complete triangles will only be there with no stick loose. All sticks must be a part of one or more than one triangle.

**Recommended time:** Ten minutes, though you may take more time, but measure the time and remember the way you have solved it.

#### Second part of the puzzle

*How many such possibilities are there? If you think there is more than one possibility, find the others.*

**Time to solve** 10 minutes.

The solution to the first matchstick puzzle involves clarity in analytical reasoning on geometric shapes.

Do try to solve before going through the solution.

### Solution to 6 triangles matchstick puzzle: Move 2 matches to make 5 triangles

The following is the six-triangle figure that is to be converted to a 5-triangle figure made of same matchsticks by moving 2 matchsticks with no overlap of two matchsticks or no matchstick kept hanging independently without being a part of any triangle.

First, let us refresh our knowledge on the core concept of matchstick puzzles.

#### Core concept in formation of geometric shapes made up of matchsticks

The most basic concept that is inherent in any matchstick puzzle of geometric shapes applies in this problem also,

Each stick common to two triangles reduces the required number of sticks (or sides) by 1 compared to the sticks required for making the same two triangles independent of each other.

The following figure should make the concept more clear.

This matchstick puzzle truth, as we call it, holds for figures made up of matchstick triangles, squares or even regular polygons of equal sides.

#### Problem analysis: Stage 1: forming the broad requirement: solution requirement specification by pattern and requirement analysis

In our problem, we have six triangles and 14 sides where 18 sides would have been required to form 6 independent triangles. So the six triangles in total have 4 sides common to or shared by two triangles.

In the target formation, we have to form 5 similar triangles with these same 14 sticks by moving just two sticks. Only 1 side will then be common in this 5 triangle figure.

Thinking from a broader perspective now we decide,

Essentially, with each stick movement, we need to destroy (or reduce) 1 and only 1 triangle, so that with these two sticks freed in two moves, we can form a new triangle, making the total number of triangles 5.

This conclusion originates from **pattern and requirement analysis** for solving the problem. We would classify this conclusion as a part of **solution requirement specification**. Without going into details, we form this essential action requirement for solving the problem. Think over and **try to defeat the logic**.

*In the next stage of analysis we would find this requirement to be not enough. But it is easier to proceed with a lightweight, simple requirement than a more complex one.*

#### Problem analysis: Stage 2: defining the characteristics of the first stick to be moved

When we move out the first stick,

Only one triangle must be destroyed. Also, none of the other two sides of the triangle destroyed should be kept hanging; the two must be securely fastened as sides of other triangles.

This in fact, is the second part of requirement specification that we discover when trying to choose a suitable first stick. The requirement specification is still not complete.

Think over. **Try to defeat the conclusion.**

We find only two sticks in the whole figure satisfy both conditions.

We pick up the one marked 1 and move it to a new position as below. **Try to find the other one.**

#### Problem analysis: Stage 3: defining the characteristics of the of the second stick to be moved

Choice of second stick is easier.

It should destroy a single triangle and keep one side hanging and the third side secure as a side of another triangle. This hanging side will form the base of the new triangle.

This conclusion is the last missing part of the requirement specification, that again we discover while choosing the second stick.

**Try again to defeat the logic.**

We select the stick marked 2 and move it to form the new triangle.

Problem solved.

The first stick could have been moved to a different position so that the second stick choice would have been different. Try to find such possible solutions.

We have not directly answered the question of how many solutions to the problem are there.

#### Food for more thought

A solution configuration formed a bit differently is shown.

Any of the two sticks could have been moved first, and this awareness helped us to fine-tune the requirement specifications for first and second stick. We just remove the distinction of first and second stick from the specifications stated earlier.

Three part complete **solution requirement specification** is then,

- Only one triangle must be destroyed in each of the two stick moves.
- In one stick move, none of the other two sides of the triangle destroyed should be kept hanging; the two must be securely fastened as sides of other triangles.
- In the other stick move, keep one side hanging and the third side secure as a side of another triangle. This hanging side will form the base of the new triangle.

Any of the moves may be the first move.

Finding more than one solution improves your problem solving skills in line with ** many ways technique** we defined some time back.

This is *pattern identification and solution requirement specification approach*. Problem analysis, pattern identification and method formation play key roles.

This approach gives you deep insight into the problem in specific and such types of problems in general. Final gain will be *improvement in your problem solving ability.*

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