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Add 9 Matches to Make 8 Nodes Matchstick Puzzle

Add 9 Matches to Make 8 Nodes Matchstick Puzzle

Add 9 matches to make 8 nodes in the puzzle figure of a triangle of 3 nodes made up of 3 matchsticks. Time to solve: 5 minutes. Try applying reason.

Matchstick puzzle Add 9 matches to make 8 nodes

In the figure below, a triangle comprises 3 matchsticks. Add 9 more matchsticks to these 3 to make the number of nodes exactly 8. A node is the point of joining of two sticks. There must not be any open node. In other words, at least two matchsticks must join at each node.

Add 9 to make 8 nodes matchstick puzzle 1 triangle

Furthermore, one must be able to reach any node starting from any other node.

Time to solve: 10 minutes.

Solution to the Add 9 matches to make 8 nodes matchstick puzzle

As a painless, simple trial, two matches are added to the existing three nodes, as shown.

Add 9 to make 8 nodes matchstick puzzle 2 triangles 4 nodes

What is really achieved by the experiment?


Conclusion 1: by adding two matches to an existing side, a new triangle with one extra node is created.

Effectively then, by adding two matches and creating a new triangle, ONE ADDITIONAL NODE IS CREATED.

It follows by simple arithmetic,

Conclusion 2: By adding 2 x 4 = 8 additional matches each pair of matches forming a new triangle, 4 nodes besides the existing 3 are created. Total number of nodes this way becomes 7.

One node then remains to be created by adding only one more matchstick. AND THIS IS IMPOSSIBLE.


Conclusion 3: It is impossible to create 8 nodes by adding 9 matches and creating ONLY NEW TRIANGLES.

This is the discovery of the key constraint.

To overcome the key constraint then,

Conclusion 4: By adding 9 more matches 1 quadrilateral (that must be a rhombus because of equal length oblique sides) must be created along with 3 triangles with 6 matches.

The single additional match unused in the previous experiment of creating 4 more nodes with 8 more matches has now been effectively used for TRANSFORMING AN EXISTING TRIANGLE TO A RHOMBUS THUS ADDING ONE MORE NODE TO THE TOTAL OF 7.

The solution shown.

Add 9 matches to make 8 nodes solution

Additional 9 matches are colored, 5 as sides of equilateral triangles as green and the rest 4 colored brick red forming a rhombus.

This rhombus with 3 additional matches provided two additional nodes and played the role of the breakthrough component.

End note

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