Matchstick puzzle: Move 3 matches to make 5 equal squares
In the matchstick figure of seven squares, move 3 matches to make 5 equal squares. No match should be overlapping or left hanging.
How many solutions can you find?
Total time to solve: 20 minutes.
Do give it a try. It'll sure be interesting.
Solution: Move 3 matches to make 5 equal squares: Common match analysis to visualize final figure
Two things of the puzzle figure get my attention first - the 7 square figure is made up of as many as 20 matchsticks WITH as many as 8 COMMON MATCHSTICKS. Because of so many common matchsticks, the figure is tight and compact.
With such a large number of squares and matches, I stopped any thought of trying for an intuitive solution first. It would invariably end up into a tangled fiasco.
So I set down to the well-used path of common matchstick analysis for the desired solution and the initial puzzle figure.
The 8 common matchsticks in the 20 matchstick puzzle figure have reduced the need of number of matchsticks for 7 independent squares from 28 to 20. That's clear, but this information doesn't help me towards finding the solution.
What about the common match situation for the desired solution?
I have to form a 5 equal squares figure with the 20 matchsticks.
What does it tell you?
It gives a very important information,
Conclusion 1: The final 5 squares figure won't have any common matchsticks between any two squares as number of matchsticks required for making 5 independent matchsticks is 20 only.
In addition it implies that,
Conclusion 2: The 5 squares will have to be edge-connected because in only 3 match moves no single square can be separated completely from the other squares.
With these insights into the final solution figure I move ahead with,
Strategic method of Analyzing the puzzle figure itself for discovery of maximum number of edge-connected independent squares.
The reason I decide so is,
Reasoning: If I have to form the fairly large figure of 5 edge-connected squares from the 7 squares figure in only 3 match moves, bulk of the 5 edge-connected squares must already be a part of the 7 squares figure.
Check out the reasoning for your own satisfaction.
Solution: Move 3 matches to make 5 equal squares: Identifying edge-connected independent squares in the puzzle figure
It takes just a minute to identify as many as 4 edge-connected independent squares resting comfortably as a part of the puzzle figure.
These are identified by the red-colored sticks in the figure below.
The puzzle figure of 7 squares already has as a part of itself as many as 4 edge-connected independent squares.
Solution: Move 3 matches to make 5 equal squares: Forming a 5 independent square figure as a candidate solution
To form a candidate solution, my job is now very simple,
I just have to add 1 independent square to this 4 independent squares to create the first candidate for the final solution.
It takes practically no time to add such a 5th independent but edge-connected square to the existing 4 squares. Following figure shows it.
On top of the green colored 'Base' matchstick, three more sticks are added to create the 5th independent square edge-connected to the other 4 existing edge-connected squares.
In fact, recognizing the 5 squares figure as the actual solution is as easy as identifying how the solution figure would be formed from the original puzzle figure.
The solution with three matches moved is shown below.
I have colored the base matchstick now with red to differentiate it from the other three matches that are moved to create the new 5th square. Rest of the 4 squares and the base red-colored matchstick were part of the existing puzzle itself.
I had to move just the three matches 2, 3 and 4 that are faded out.
Okay, we have the solution. But what about the second part of the puzzle? Are there any more solution?
Solution to second part of Move 3 matches to make 5 equal squares puzzle: Any more solution?
You already know quite a lot about the puzzle structure by now. To answer this second part, you just have to organize the knowledge gained.
What is the certainty? It is a certainty that whatever be another solution, the four edge-connected independent squares that are part of the puzzle figure must also be a part of the solution figure.
In only 3 match moves, it would be impossible to disturb the 4 square existing structure and still have a solution.
You are now in a position to conclude on the basis of this certain knowledge that,
Conclusion 3: Three of matchsticks 1, 2, 3 and 4 will have to be moved with the fourth used as the base to form the 5th edge-connected independent square.
You are now analyzing the functions of individual matchsticks in a set of matches narrowed down to 4.
Continuing analyzing the function of these 4 matches, you make the next conclusion,
Conclusion 4: Base analysis: Matches 3 and 4 cannot act as the base of the new 5th square because if we create a 5th square on either of these, it will become a common stick between two squares and total number of squares will become 6.
Without wanting it, you would create 6 squares if you use match 3 and 4 as the base of the new square.
This firmly indicates that,
Conclusion 5: Only match 1 or 2 can be used as the base of a new 5th square.
We have the first solution with Base as matchstick 1. Now we will see the second solution below with matchstick 2 as the base of the new 5th square.
And there can be only these two solutions.
You can make this final conclusion with full confidence as you have proceeded step by step systematically, taking care of the possibilities exhaustively as well.
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