## Two intersecting large triangles matchstick puzzle

The puzzle figure comprises two large triangles intersecting each other. Move 4 matches to make 6 triangles. No matchstick should be left hanging.

#### The Puzzle

In the following matchstick figure, move 4 matches to create 6 triangles. No matchstick must be kept hanging.

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

Try to solve this interesting puzzle.

### Solution to move 4 matches to make 6 triangles puzzle: Matchstick count analysis

Total matchsticks are 12 and 6 triangles of 1-stick side need 18 matchsticks. To make 6 such small triangles of the same size, then 6 matchstick must be common between two triangles. The large number of common sticks in the final solution implies,

Conclusion 1:The small triangles must be enclosed by a larger triangle. Otherwise, common sticks can't occur.

Also,

Conclusion 2:The larger triangle must be of 2-stick side length or 3-stick side length.

Let us first **visualize a larger triangle of 3-stick side length** comprising 9 sticks.

Can we form six triangles from this larger frame of 3-stick side triangle and three free matchsticks?

Conclusion 3:If six triangles are to be made by suitably placing the three sticks, because of the symmetry of the figure we may conclude that, by placing each of the three sticks, we must create two triangles.

Thinking of this simpler problem we ask ourselves,

**Question:** Can we make two triangles from the 3-stick side large triangle and one more stick?

The answer is YES, we can, if we place the single stick parallel to the base to form a small triangle at the apex of the large triangle.

*This is a small 1-stick side triangle inside a large 3-stick side triangle.*

It should be easy to extend the idea to *create two more pairs of triangles by placing the two more free sticks suitably.* Result shown.

At first we thought we have the solution. But when we examined the resulting figure objectively, the truth dawned upon us. Number of triangles is only 4 not 6. The large triangle is the same for all three apex smaller triangles.

Conclusion 4:Solution must have the larger enclosing triangle as a 2-stick side triangle.

A 2-stick side triangle comprises 6 sticks. We have then 6 free sticks. **It is easy to form four small triangles** by placing 3 marches inside the larger triangle.

We have formed four small triangles with 9 matchsticks. But that is not all. We have to count the enclosing larger triangle as well. *The figure above then comprise 5 triangles*.

We still have 3 matchsticks free and **the sixth triangle formed by these three sticks must be separate and independent from the five triangle figure.** The solution figure will then be like the figure below.

### Final solution to move 4 matches to make 6 triangles matchstick puzzle: Stick moves identified

We have an idea of the solution, but unless we identify which sticks to move and where, our solution is not complete.

Let us show you the puzzle figure and the idea figure for the solution side by side to aid stick move identification.

It is now easy to decide which sticks to move and which to keep fixed for the solution.

**Solution:** *Keep marked sticks 5 and 6 fixed and move one of the sticks of 1, 2, 3 and 4, say stick 4, to complete the third side of the isolated triangle. Place the rest three sticks 1, 2 and 3 to form the main 5-triangle component of the solution.*

**Final solution** shown.

The solution uses the upside down 2-stick side triangle of puzzle figure. *You may get the second solution using the upright 2-stick side triangle of the puzzle figure.*

But if you rotate this second solution by 180 degrees it will become same as the first solution. *Rotationally unique solution is only one.*

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