## Six matchstick triangle puzzle

In the matchstick puzzle figure move 2 matches to form 5 triangles from 6 triangles. Each matchstick in the solution must be a part of a triangle.

How many solutions can you find?

**Recommended time to solve** is 15 minutes.

Solve the puzzle before going through the solution.

### Solution to the move 2 matches to form 5 triangles puzzle: Common stick analysis

For easier explanation, we will show the puzzle figure again with triangles labeled and common sticks marked.

We will first do the **common stick analysis.**

The puzzle figure has 14 sticks and **4 common sticks** that form 6 triangles. Target is to reduce number of triangles to 5.

As the maximum number of sticks required to form 5 triangles is 15, **we will have 1 common stick in the solution**, reducing the need to 14 sticks.

Conclusion 1:In 2 stick moves we have toreduce the number of triangles by 1 and the number of common sticks by 3.

Conclusion 2:As triangles A, B, C and D have the three common sticks,one of the two stick moves must be from these four triangles.The second stick to be moved may be from these four triangles, or the triangles E and F.

### Solution to the move 2 matches to form 5 triangles puzzle: Which sticks to move

Concentrate on the top row of four triangles and ask the question: **which stick move eliminates the maximum number of common sticks?** For easy reference we will use the following figure of the puzzle.

When we examine the top row four triangles, the two marked sticks are identified as the most favorable ones. **If we move either of the two, at one stroke, two common sticks will be eliminated creating no hanging stick.** That is a big advantage.

Conclusion 3:One of these two marked sticks must be moved for a solution.

It is also easy to conclude,

Conclusion 4:None of the four corner sticks of the two triangles can be moved, as moving each pair eliminates only one common stick.Three common sticks remain in the figure.

Conclusion 5:We cannot also move any of the common sticks, asmoving a common stick creates four hanging sticksto take care of, which is impossible.

### Solution to the move 2 matches to form 5 triangles puzzle: First solution

At this point, it is easy to form the **first solution, **as shown.

Between the two most favorable sticks to move, stick 1 and stick 3, stick 3 had the advantage of being used as the base of a new triangle. We kept this stick fixed and with sticks 1 and 2 created a new triangle. The moves destroyed two triangles and eliminated two common sticks as well. Five triangles are formed with no hanging isolated stick. This is the first solution.

### Solution to the move 2 matches to form 5 triangles puzzle: Any more solutions?

By the knowledge gained till now, it is clear that solely from the top four triangles, the first solution is the only possible solution.

Conclusion 6:For a new solution, the sticks of the two triangles below the top four must be considered for movement.

**Question: **Is it possible to have a new solution solely by moving two sticks from the bottom two triangles?

No, it is not possible, because **we must reduce the number of common sticks to 1 in the top four triangles as well.**

Conclusion 7:For a new solution, either of the sticks 1 or 3 must be moved along with one stick from the two triangles below the top four.

The figure with sticks for the final analysis numbered is shown.

We already know either stick 1 or stick 3 must be moved for a new solution. **Question is:** which among the four sticks 4, 5, 6 or 7 to be moved?

#### Requirement for moving the second stick

The second stick will destroy 1 triangle and eliminate 1 common matchstick. But the stringent requirement is,

Conclusion 8:The second stick movemust also leave a hanging stickon which the new triangle will be created using it as the base.

Conclusion 9:Either of the sticks 4 or 5 may then be moved for the new solution.

Moving stick 6 or 7 cannot produce any solution.

With two equivalent move possibilities of stick 1 or 3, combined with another pair of equivalent possibilities of stick 4 or 5 will create **four different solutions.**

One of the solutions **among the four of this class of solutions by moving a stick from one of the bottom two triangles** is shown above.

**Total number of solutions is then 5.** From the top four triangle row, only one solution is possible.

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