Matchstick puzzles: Move 2 matches to make 7 squares and move 8 to make 6 squares
Move 2 matches to make 7 squares and move 8 to make 6 squares. Total time to solve the pair of challenging matchstick puzzles is 20 minutes.
The puzzles
Part I: In the figure shown, move 2 matches to make 7 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.
Comment
Just remember, however much improbable the solution may seem, there must be a solution.
Hint: You need lateral thinking, which is thinking in a totally different way compared to conventional thinking.
Solution to the first part of the matchstick puzzles: Move 2 matches to make 7 squares
How to make 7 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 powerful method that produces an assured solution in such cases.
Proceeding logically, we will focus first on discovering a 7 square figure made up of 12 sticks and the way of making it.
Question 1: How can 7 squares be formed with 12 sticks?
When you don't have any clue to form the 7 squares from 12 matches in the usual way, you apply Property change analysis technique.
Property change analysis technique: When you face an impossible situation and you know that a solution exists, search for any PROPERTY 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 7 squares with 12 matchsticks, the size of all squares cannot be equal.
This deviates from the conventional way of forming squares with matches.
This departure from the usual approach is also a necessity as otherwise forming 7 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 4 additional squares?
Conclusion 2: The easiest and the only way to do this is to go for 3+4=7 squares, equivalent to the existing three of the four squares and one of these three squares divided into 4 squares by the two sticks moved.
The action is absolutely focused and the 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.
How many solutions are possible?
Whichever two corner sticks you move to form the additional 4 squares, all solutions will be the 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.
You have solved the first puzzle.
Now we will take up the second puzzle.
Solution to the second part of matchstick puzzles: 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 PROPERTY taken for granted as equal size of squares and changing the property 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 apparently impossible first puzzle by identifying one property that you had assumed to be the same sized squares, you are sure that you have to,
Identify a second PROPERTY also taken for granted and unknowingly assumed to be true.
If you can identify such a new changeable property, you would just manipulate the property for the second solution. This is the Property Analysis and Discovery Technique in action again.
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 be a three dimensional figure with six squares.
Breakthrough idea: 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.
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 the 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 call LATERAL THINKING, an inventive and totally different way of thinking.
End note
You have followed a systematic way to discover the new property you have implicitly taken for granted. You have changed the newly discovered property to your advantage, and reached the inventive solution.
Effectively, you have reached an inventive solution following a methodological approach of discovery of a property 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 property of the problem that you have assumed to be true and unchangeable, but actually you can change. Use the newly discovered changeable property and the inventive solution of the otherwise impossible to solve problem will be yours.
Happy problem solving.
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