Hexagonal wheel Matchstick triangle puzzle: Remove 4 matches to get 3 triangles
You have to remove 4 matches from the hexagonal wheel below to leave 3 equilateral triangles.
How many ways can you do it?
Time limit: 5 minutes.
Give it a try. It won't be difficult.
Instead of any random approach we'll solve the puzzle systematically. We'll use common stick analysis on the structure to discover the key information needed to reach the solution. We'll ask ourselves a series of important questions, analyze the puzzle for answer and make conclusions.
Solution to the matchstick triangle puzzle: Remove 4 matches to get 3 equilateral triangles: By Common stick analysis and Question analysis answer technique
The first job when solving any matchstick puzzle is to count the total number of matchsticks. It is 12 in six triangles forming the wheel of a hexagon.
Then we ask two questions,
Question 1: How many matchsticks are needed to form 6 independent triangles? It is simply 18.
Question 2: How could then only 12 matchsticks form the six triangles?
Answer 2: As the 6 triangles have between them also 6 common sides of matchsticks, by the concept of common matchstick, each common matchstick would reduce the number of sticks needed to form independent triangles by 1. Six common matchsticks would reduce number of matchsticks needed from 18 to 12.
The common stick concept is explained in the following section in more details. To skip this section click here.
Concept of common stick between two triangles
The following figure shows two independent triangles formed by 6 sticks on the left joined to form a single structure of two triangles separated by one common stick on the right.
The requirement of number of sticks to form the two triangles is reduced from 6 to 5 because of the single common stick between two triangles. Being a common side, it plays the role of two sides of the two triangles. This reduces the actual requirement of number of sides by 1.
Thus the requirement of number of sticks to form the two triangles is reduced from 6 to 5 because of 1 common stick.
Next question is crucial. It asks about the solution. We have to know about the solution also, isn't it?
Question 3: What would be the total number of sticks and the number of common sticks in the solution figure?
- After removing 4 from 12, there would be 8 matchsticks left. Fine. What about common sticks?
- As three independent triangles would need 9 matchsticks, with 8 matchsticks, the solution figure would have to have 1 common stick between two triangles with the third triangle as independent.
Question 4: How would the solution figure look like? We are trying to actually find a solution figure just by analysis and reasoning.
At this point, let us have a look at a figure of three equilateral triangles, but made up of 9 sticks by the very simple action of removing three alternate sticks on the periphery as shown below.
Well, reducing need of one more stick is dead easy from this point. Just merge the top two triangles, and you will have your solution figure shown below.
Don't stop at this point. we have not solved the puzzle yet, rather we are now quite past the halfway mark. We do not know yet which 4 matchsticks are to be removed to get our solution.
That shouldn't be a problem at all. Just compare the solution on the right and the puzzle figure on the left and you will know.
Fix top two adjacent triangles of the puzzle figure, A and B, as not to be touched along with a third triangle C not having any common side with either of the two fixed. Leaving these three, simply remove the rest of the matchsticks from the puzzle figure. The number of matchsticks removed would certainly be 4 and you would be left with three triangles. It is logic and an account of common matchsticks.
The solution is shown below.
You can solve the puzzle in any way you may feel. Do try. And try to beat this simple path to the solution.
How many solutions would be possible?
That also is easy to see.
The solution figure satisfies the simple but stringent criterion,
Out of the three triangles, two triangles must have one common side but not the third.
As the puzzle figure is rotationally symmetric, whatever way you choose these three triangles, all such choices would be same when rotated.
Rotationally unique solution is only 1.
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Hexagonal wheel to 3 triangles by removing 4 sticks