Matchstick puzzle: Move 4 matches to turn the tower upside down
Move 4 matches in the tower to make it turn upside down.
How many ways can you do it?
Recommended time limit is 20 minutes.
All 10 matches are isolated and freestanding. Yet these are related to each other to form the upright tower structure.
This matchstick puzzle doesn't have any single closed shape of squares and triangles. That's why majority of the matchstick puzzle solving techniques won't be applicable here.
Structural analysis should play the most vital role in solving this puzzle systematically and quickly.
Enjoy solving the puzzle.
Solution to the matchstick puzzle - Move 4 matches to turn the tower upside down
On analysis of the structure of the tower, the specific characteristics of the placement of the matches identified are,
- In each row, the matches are at equal distance from each other.
- A match in a row above is placed at the midpoint between the two sticks below it.
- Vertical separation between the rows is same, and,
- Number of matches in each row reduces by one moving in the direction towards the fourth row at the top with the single stick at the top forming the pinnacle.
These are the placement details of the matches.
For analyzing the problems, the figure of upright tower with matchsticks labeled is shown below.
A very important fifth characteristic of the whole structure is,
The structure is symmetric on both sides of a virtual vertical line passing through it center. This is the vertical axis of symmetry of the tower structure.
Realizing the importance of this property, we quickly form an inverted tower structure by moving a few matches. It is easy. So we do it as an experiment.
Creating an inverted tower Structure by moving a few matches
Before moving any matchstick, we would easily visualize the inverted tower structure with the symmetry of the original maintained.
The inverted tower structure is placed below the original upright one for ease of transformation. To make the inverted tower realistic each of the matchsticks is shown as inverted. This doesn't count towards forming the solution. In the solution, all the sticks will remain upright.
Identify the main CORE of the structure as the sticks 1, 2, 3, 4, 5, and 6.
The top stick 1 is separated by one row and is in same vertical alignment with stick 5 that is the central stick of the third row.
Below stick 2 on its both sides are placed the two sticks 3 and 4.
If we can create this CORE structure in inverted form, we can create the whole inverted structure easily.
To do this, naturally you would move stick 1 below sticks 8 and 9 and between them.
Observed that, the six sticks in the inverted CORE structure are already placed in right positions by the new single stick movement. These are the sticks, 1, 8, 9, 4, 5 and 6.
The sticks 2 and 3 are also in their right places in the inverted structure and so need not be moved.
Move only two sticks 7 and 10 on the bottom row of the original on two sides of sticks 2 and 3 to form the inverted tower structure in quick time.
But you have moved only 3 matches, not four, the sticks 1, 7 and 10.
At this point we realize that,
The tower Structure can be inverted symmetrically about the central axis of symmetry only by moving 3 sticks, not 4, as required in the puzzle.
In other words,
To invert the tower structure by moving 4 sticks, the structure must have its vertical axis of symmetry moved off-axis to the right or left of the original axis of symmetry.
Final solution to the matchstick puzzle - Move 4 matches to turn the tower upside down
Let's start fresh with this clear knowledge on the action required, not in details, but the basics learned will be enough for the solution from this point on.
Original number structure is reproduced below for easy understanding of the final analysis and actions taken.
Objective is to create the part of the central inverted fulcrum by moving match 1.
It is obvious that the top stick must be moved first.
So in the first step, we create the part structure equivalent to the structure formed by sticks 1, 2, 3, and 5, by placing matchstick 1 below and in between matchsticks 9 and 10. The axis of symmetry has to be shifted right or left for the solution, isn't it? We have chosen to shift the axis towards right, and this is the only promising position to place the pinnacle stick 1.
Matches 1, 9, 10 and 6 is the structure equivalent to the original part structure formed by matches 1, 2, 3 and 5.
As we have shifted the axis of symmetry to the right, the matches to be moved now would be from the left flank, the match number 7 first.
Place it on the right of the match 6 to place the third match in the third row from the bottom. The match 4 and 8 would also need to be moved to finally form the inverted tower structure.
The final solution is shown below. The matches in the original position are still shown but faded out to give you an idea about how the new inverted structure has been created from the original one.
The four matches moved to form the inverted tower are 1, 7, 4 and 8.
How many solutions to the puzzle is possible?
Knowing that for creating the inverted tower, the axis of symmetry has to be shifted to the right or left, the answer the question is straightforward.
There are two possible solutions to the puzzle. We have shown one by shifting the axis of symmetry to the right.
The second solution would have the pinnacle single stick placed below and between sticks 7 and 8 to form the inverted tower. Axis of symmetry in this case would be shifted left of the original axis of symmetry.
There is an advantage in studying the structural specialties so that the special characteristics can be replicated in each step when creating the new structure.
Analysis and clear understanding about the positional interrelation of the matches also clearly indicate the way forward to the solution for creating the new structure.
This approach of holding on to the positional interrelations of the matches with other matches with respect to the whole structure is absolutely essential for creating the new structure without taking any wrong step.
The first brief experiment in creating the inverted structure with the same axis of symmetry as the original is also an important part of the process of the solution.
It helped to consolidate the idea about how the final solution needed to have its Central matchstick shifted off-Axis towards right or left.
With this clarity, now we could easily answer the question about number of possible solutions in the end.
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Move 4 to turn the tower upside down Matchstick puzzle