In the puzzle figure, move 2 matches to form 4 equal squares. No matchstick should be left hanging. Recommended time to solve 15 minutes.
Do give it a try before you go through the solution.
Solution to the move 2 matches to form 4 equal squares puzzle: Which sticks to move
From a brief examination of the puzzle shape we can conclude,
Conclusion 1: No stick of the three complete squares can be moved.
Reason: Moving any stick from a complete square will destroy a square. To form 4 squares by moving 2 sticks then, 2 new squares are to be formed. This is an impossible task.
Conclusion 2: The ideal candidates for movement are the two sticks hanging below the bottom left complete square.
Moving these two sticks won’t disturb the other three complete squares. We are only to form one new complete square by moving these two sticks.
But is it possible at all?
Solution to the move 2 matches to form 4 equal squares puzzle: Stick count and common matchstick analysis
For convenience of explanation, the puzzle figure shown again.
We’ll analyze the three complete squares.
The three squares comprise 10 matches and two common matchsticks. The base match of the top square is a common match shared by the top square and the two squares below the top square.
Three independent squares need 12 matchsticks, and the two common matches reduces the need by 2 so that with 10 matches the three squares could be formed.
Challenge is now to form a fourth square with the two additional matches moved.
In whatever way we place the two moved matches OUTSIDE the area of the three complete squares, a new fourth square cannot be formed. We need THREE free matches to form a new square.
Again, four independent squares need 16 matches. To form 4 squares with 12 matches then, four common sticks must be formed. And that is the most compact four square figure, as shown.
It would have been easy to form this figure only if we could move the three sticks of the top square instead of two.
This also is a dead end.
Solution to the move 2 matches to form 4 equal squares puzzle: Problem transformed to move 2 sticks to form three squares
In the puzzle, as the top square is asymmetrically placed on the two squares below it, no way can it contribute to forming the extra square by moving the two free sticks. Our job is then to form three squares from two squares by moving two hanging free sticks, as shown.
This simplified problem is the CORE of the puzzle and is an apparent impossibility. When we face this type of situation, often we get the breakthrough idea by Property analysis and change technique.
In this method we analyze all the properties of the problem that we know and that we implicitly assume as true without being aware that it can be changed.
Here, in adding two matches to the two square figure, we assume
The two sticks can be added only on the OUTSIDE area of the two complete squares.
By the property change analysis method, we are urged to question this assumption and ask,
Is it possible to add the two sticks in any other way? For example, inside the two squares itself?
The emphatic answer is YES.
We can place the two sticks one inside the left square and the second inside the right square both vertical and separated exactly by one stick distance. This will create a third square by adding just two free sticks.
This additional square will be created inside the two squares. We’ll thus call this new result achieved as the Square in Square technique and add it to our arsenal of matchstick puzzle techniques.
The following figure shows the formation of three squares by 9 sticks using the Square in Square technique.
The third square is highlighted by coloring its four sticks.
Three squares are formed by 9 sticks. This implies presence of 3 common sticks (maximum 12 minus 9 is 3). Can you find the two additional common sticks?
The first is obviously the common vertical side shared by the original two squares.
The two additional common sticks are the two horizontal sides of the square in the two squares. Sharing is with the two existing squares.
The following figure highlights the shared common horizontal sides.
Final solution to the move 2 matches to form 4 equal squares puzzle
By using the breakthrough concept of square in square, the solution to the original puzzle formed, as shown.
The two free sticks are placed just below the two vertical sticks of the top square. But this is not a necessary condition.
You can form the fourth square by
placing the first moved stick inside the left square vertically anywhere and the second inside the right square vertically, the two separated by one stick distance.
There could thus be infinite many solutions.
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