Matchstick puzzles: Move 2 matchsticks to make 6 squares and move 8 matchsticks to make 6 squares
Part I: In the figure shown, move 2 matches to make 6 squares with no matchstick kept hanging.
Recommended time limit: 10 minutes.
Part II: In the same figure shown, move 8 matchsticks to form 6 squares with no matchstick kept hanging.
Recommended time limit: 10 minutes.
Just remember, however much improbable the solution may seem, there must be a solution.
Solution to the first part of the matchstick puzzle: Move 2 matches to make 6 squares
How to make 6 squares with 12 matches?
It is bewildering at first. Especially when you know that with 12 matchsticks you can make a maximum number of 4 squares with 4 common sticks between them.
Rather than remaining clueless and confused or trying randomly, you can spot the solution intuitively and immediately.
Or you can apply your reasoning and a very powerful method that produces assured solution in such cases.
Proceeding logically, we will focus first on discovering a 6 square figure made up of 12 sticks and the way that is possible.
Question 1: How can 6 squares be formed with 12 sticks?
When you don't have any clue to form the 6 squares from 12 matches in the usual way, you take up the Method of Dimension exploration and discovery, or Dimension discovery technique in short.
Dimension discovery technique: When you face an impossible situation and you know that a solution exists, search for any DIMENSION or characteristic in the problem that is TAKEN FOR GRANTED or ASSUMED, and when you find one such, change it suitably for reaching the solution.
With this focus on discovering what you have ASSUMED about the solution, it takes little time for you to discover that,
Though not specified in the puzzle description, you have IMPLICITLY ASSUMED the squares to be of equal size.
With this realization, you become sure that the SIZE OF THE SQUARES WOULD HAVE TO BE UNEQUAL for the solution to be possible, and the key conclusion is made,
Conclusion 1: To have 6 squares with 12 matchsticks, size of all squares cannot be equal.
This is a deviation from conventional way of forming squares with matches.
This departure from usual approach is also a necessity as otherwise forming 6 squares of equal size with 12 matches would have been IMPOSSIBLE.
Now being on the right path to the solution, the second question you ask is,
Question 2: As 1 square would be destroyed by moving 2 matches, how can these 2 matches be used for creating at least 3 additional squares?
Conclusion 2: The easiest and the only way to do this is to go for $2 + 4 = 6$ squares, equivalent to the existing two of three squares and the single remaining square divided into 4 squares by the two sticks moved.
The action is absolutely focused and solution very clear.
Move any two corner sticks and place across the middle of two perpendicular sides of one square thus dividing it into four squares.
Obviously you can't move any common stick, you would only move two corner sticks of a square.
The solution figure for the first part of the puzzle is shown below,
How many solutions are possible?
Whichever two corner sticks you move to form the additional 4 squares, all solutions will be same when you rotate them on the plane around an axis perpendicular to the plane passing through the center of the solution figure.
Rotationally unique solution will only be one.
Ok you have solved the first puzzle.
Now we will take up the second puzzle.
Solution of the matchstick puzzle: move 8 matches to make 6 squares
By now you know how to form 6 squares with 12 matches on your flat table. The puzzle and the first solution are shown in the figure below side by side.
Identified the DIMENSION taken for granted as equal size of squares and changing the dimension to different square sizes, the puzzle of forming 6 squares using 12 matches moving 2 matches is solved.
But now you face the second impossibility, and a harder one,
To form 6 squares using 12 matches moving 8 matches.
With experience of solving the seemingly impossible first puzzle by identifying one dimension that you had assumed to be same sized squares, you are sure that you have to,
Identify a second DIMENSION also taken for granted and unknowingly assumed to be true.
If you can identify such a new changeable dimension, you would just manipulate the dimension for the second solution. This is what we call Dimension discovery technique.
Different sizes already exploited it doesn't take you long to realize that, you have again assumed without being conscious about it, that,
The 6 squares are to be formed on the same plane, say on the flat surface of your table.
Conclusion: So in this case, the solution must a three dimensional figure with six squares.
A change of 2 dimension to 3 dimension.
The solution figure would simply be a cube with 12 edges.
Holding up the structure would be another problem but using proper glue it can be made. The solution is shown below,
You have visualized the solution, but the actual job of forming the cube by moving 8 matches remains. That is also easy.
You would keep one of the four squares, say the upper left corner square of the puzzle figure fixed. It will form the front face of the cube. Rest 8 sticks of the puzzle figure you would move to create the cube.
The following figure shows the situation. Matches 1, 2, 3 and 4 are kept untouched. These four matches of the upper left corner square of the puzzle square and becomes the front face of the cube.
Rest 8 matches of the puzzle figure are moved to proper position forming the rest 8 edges of the cube on the right.
For easy visualization, the faded out front face of the solution cube is overlapped with its mother square in the puzzle square. The following figure explains,
This is what we LATERAL THINKING, an inventive way of thinking.
You have followed a systematic way to discover the new dimension (not physical dimension) or characteristic you have implicitly assumed or taken for granted. Changing the newly discovered dimension to your advantage, you have reached the solution that is inventive.
Effectively, you have reached an inventive solution following a methodological approach of dimension discovery technique that is generally known to be infeasible.
In solving real life problems also, when you face an apparently impossible situation, without feeling dejected, push ahead to identify any dimension or characteristic of the problem that you have assumed to be true and unchangeable, but actually you can change. Use the newly discovered changeable dimension and the inventive solution of the otherwise impossible to solve problem will be yours.
Happy problem solving.
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Move 2 matches to make 6 squares and move 8 matches to make 6 squares - a pair of matchstick puzzles