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 is made up of 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.
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 below.
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 in addition to 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.
Following is the 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.
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