Remove 4 matches to leave 4 equal triangles with every matchstick as a part of a triangle.
Time to solve 15 minutes.
Have a try. It's a fun puzzle.
Solution to Remove 4 matches to leave 4 equal triangles
Problem analysis and selecting the strategy
The puzzle seems to be awkward, at least it looked awkward to me at first.
By habit, I try to imagine the final solution figure of 4 triangles. You may call it strategy, but I know it to be my very first target objective.
I found it to be not easy to decide what will be the final solution figure with 100% certainty.
The question that I asked myself first,
Will the final figure be 4 equal triangles each with 2-match sides or 1-match sides?
Intuitively, I decided from the final number of matches as 16 - 4 = 12, that with so few matches, 4 equal triangles each with 2-match sides won’t be possible at all.
So it seemed very reasonable to assume that the final figure will be 4 triangles with each triangle having 1-match side only.
I wanted to be clear about the final solution figure.
Let me tell you why I want to know the final figure. My intention is to compare the initial puzzle figure and the final solution figure to identify the 12 common matchsticks that I must keep fixed. I will simply remove the rest 4 matchsticks to get the solution.
Solution to Remove 4 matches to leave 4 equal triangles puzzle
Getting a more clear idea on the final solution figure
I count the number of matches that will be left at the end to imagine what kind of 4 triangles be formed with these matches.
This is an analysis of the number of matchsticks and common matchsticks together, a very effective technique often used in solving matchstick puzzles.
After I remove 4 matches from the 16 match puzzle figure, 12 matches are left. Now I ask the crucial question,
What kind of 4 triangles can be created out of 12 matches?
You know what the answer would be.
All four equal triangles must be independent of each other sharing no single matchstick between two triangles. Also, as all triangles were connected in the initial figure, the 4 independent triangles will also be connected at their corners.
Wasting no time, I create such a candidate final solution figure.
Now I compare this candidate solution figure with the initial puzzle figure.
Look at the two figures I put side by side.
Can you find any similarity between the two figures? Can you identify any matchstick structure that is present in both the figures. It would be great if the whole candidate solution figure is identified as a part of the initial puzzle figure.
But alas, that’s not to be. The largest structure identified as common to both the structures is the 3 triangle 9 match structure colored red in the figure shown.
You may place the fourth triangle connected to any of the six corners of the three interconnected triangles, but you cannot get this fourth triangle as a part of the original puzzle figure.
You may remove matches, but may not move matches, isn’t it?
You may think, what a failure! But I think the other way. I try to extract as much benefit from this failure as possible and immediately conclude,
Conclusion 1: The three centrally connected independent matchstick structure CANNOT BE A PART OF THE FINAL SOLUTION.
And a second more precise conclusion followed,
Conclusion 2: To create the solution figure then, you must destroy at least one of the three CENTRALLY placed interconnected independent matchstick triangles.
But wait, I have one more weapon left in my problem solving armory.
Solution to Remove 4 matches to leave 4 equal triangles
Common matchstick and number of triangle analysis
Really, such a simple truth I overlooked till now,
The number of triangles are 8. So I have to destroy 4 triangles to leave 4.
And what about common matchsticks?
The puzzle figure has 8 common matchsticks. I have to make it zero in destroying 4 triangles by removing 4 matches.
So I conclude,
Conclusion 3: I have to destroy 4 triangles by removing 4 matches leaving no stick hanging.
Next conclusion is a bit more abstract, but it is not difficult to appreciate it,
Conclusion 4: The upper portion of the puzzle figure is a mirror-image of the lower one with a perfect symmetry. So it must be true that removing 4 matches also can be divided into two absolutely similarly placed pairs of matches in the two portions of the mirror images - upper and lower.
And as we have concluded earlier, removals will be concentrated on the three centrally interconnected three triangles.
You can also be sure of another idea. You cannot make any removal from the base of symmetry shared by the mirror images of upper and lower portions of the puzzle figure.
But I will not use this idea. All the ideas on removal that we have gained till now should be enough for me to see the solution when I concentrate on these removals.
Final solution to Remove 4 matches to leave 4 equal triangles puzzle
The following is the solution figure that I could easily create by removing the crossed and faded 4 matches.
Observe that exactly as I have concluded, one pair of matches removed on the left destroyed two triangles and eliminate 4 common sticks. In the same way, the second pair of matches removed on the right also destroyed two triangles and eliminate 4 common matchsticks.
You may be thinking, why take so much trouble of logic and reasoning! I have already seen the 4 matches to be removed in just about a minute. Yes, of course. That’s certainly possible. That’s what we call an intuitive solution.
Won’t you appreciate both the solutions!
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