## 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 give an honest 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.

Before going ahead, you should try to solve the problem.

Recommended time is 10 minutes. But we would say, if you are not able to solve withing 20 minutes then you may be going in random way that would take indefinite time.

Okay, let's get on with the solution. First we will state the reason why matchsticks are popular for making puzzles.

If you want you may skip the following section and **jump straight to the solution.**

#### Why matchsticks are popular in puzzles

You may ask, why of all things matchsticks are used in puzzle formations?

Important advantages of using matchsticks in puzzles are,

- Matchsticks are all of equal length, and so it is very easy to physically make regular geometric shapes like triangles or squares by using matchsticks,
- In a figure, the matchsticks can be easily rearranged in any way to change the original shape, and
- Matchsticks are very cheap and easily available.

You can make any complex figure made up of matchstick triangles or squares or even polygons with larger number of sides.

Only your imagination is the limit of how many different types of matchstick figure making or puzzling you can do. Cost will be just your time.

Gain is though great, *because matchstick puzzle solving has the assured effect of improvement of basic pattern recognition, method creation and analytical skills which combine to improve your problem solving skills.*

#### 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 is applicable in this problem also,

Each stick common to two triangles reduces required number of sticks (or sides) by 1 compared to the sticks required for making 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 all 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 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 nature and movement of the first stick

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 of these stringent 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 nature and movement of the second stick

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 different solution to the problem are there.

### Food for more thought

A solution configuration formed a bit differently is shown below,

Any one 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 one 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.*

In the * next matchstick puzzle* we will highlight an altogether different but no less powerful approach of

*strategic pattern based problem solving.*

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